Advances in Computer Science, Part I - CSEE 2011

Communications in Computer and Information Science

214

Song Lin Xiong Huang (Eds.)

Advances in Computer Science, Environment, Ecoinformatics, and Education International Conference, CSEE 2011 Wuhan, China, August 21-22, 2011 Proceedings, Part I

13

Volume Editors Song Lin International Science & Education Researcher Association Wuhan Branch, No.1, Jiangxia Road, Wuhan, China E-mail: [email protected] Xiong Huang International Science & Education Researcher Association Wuhan Branch, No.1, Jiangxia Road, Wuhan, China E-mail: [email protected]

ISSN 1865-0929 e-ISSN 1865-0937 ISBN 978-3-642-23320-3 e-ISBN 978-3-642-23321-0 DOI 10.1007/978-3-642-23321-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Applied for CR Subject Classification (1998): I.2, C.2, H.4, H.3, D.2, H.5

© Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The International Science & Education Researcher Association (ISER) puts its focus on the study and exchange of academic achievements of international teaching and research staff. It also promotes educational reform in the world. In addition, it serves as an academic discussion and communication platform, which is beneficial for education and scientific research, aiming to stimulate the interest of all researchers. The CSEE-TMEI conference is an integrated event concentrating on the field of computer science, environment, ecoinformatics, and education. The goal of the conference is to provide researchers working in this field with a forum to share new ideas, innovations, and solutions. CSEE 2011-TMEI 2011 was held during August 21–22, in Wuhan, China, and was co-sponsored by the International Science & Education Researcher Association, Beijing Gireida Education Co. Ltd, and Wuhan University of Science and Technology, China. Renowned keynote speakers were invited to deliver talks, giving all participants a chance to discuss their work with the speakers face to face. In these proceeding, you can learn more about the field of computer science, environment, ecoinformatics, and education from the contributions of several researchers from around the world. The main role of the proceeding is to be used as means of exchange of information for those working in this area. The Organizing Committee made a great effort to meet the high standards of Springer’s Communications in Computer and Information Science (CCIS) series. Firstly, poor-quality papers were rejected after being reviewed by anonymous referees. Secondly, meetings were held periodically for reviewers to exchange opinions and suggestions. Finally, the organizing team held several preliminary sessions before the conference. Through the efforts of numerous people and departments, the conference was very successful. During the organization, we received help from different people, departments, and institutions. Here, we would like to extend our sincere thanks to the publishers of CCIS, Springer, for their kind and enthusiastic help and support of our conference. Secondly, the authors should also be thanked for their submissions. Thirdly, the hard work of the Program Committee, the Program Chairs, and the reviewers is greatly appreciated. In conclusion, it was the team effort of all these people that made our conference such a success. We welcome any suggestions that may help improve the conference and look forward to seeing all of you at CSEE 2012-TMEI 2012. June 2011

Song Lin

Organization

Honorary Chairs Chen Bin Hu Chen Chunhua Tan Helen Zhang

Beijing Normal University, China Peking University, China Beijing Normal University, China University of Munich, Germany

Program Committee Chairs Xiong Huang Li Ding Zhihua Xu

International Science & Education Researcher Association, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China

Organizing Chairs ZongMing Tu Jijun Wang Quan Xiang

Beijing Gireida Education Co. Ltd, China Beijing Spon Technology Research Institution, China Beijing Prophet Science and Education Research Center, China

Publication Chairs Song Lin Xiong Huang

International Science & Education Researcher Association, China International Science & Education Researcher Association, China

International Program Committee Sally Wang Li Li Bing Xiao Z.L. Wang Moon Seho Kongel Arearak Zhihua Xu

Beijing Normal University, China Dongguan University of Technology, China Anhui University, China Wuhan University, China Hoseo University, Korea Suranaree University of Technology, Thailand International Science & Education Researcher Association, China

VIII

Organization

Co-sponsored by International Science & Education Researcher Association, China VIP Information Conference Center, China

Reviewers Chunlin Xie Lin Qi Xiong Huang Gang Shen Xiangrong Jiang Li Hu Moon Hyan Guang Wen Jack H. Li Marry. Y. Feng Feng Quan Peng Ding Song Lin XiaoLie Nan Zhi Yu Xue Jin Zhihua Xu Wu Yang Qin Xiao Weifeng Guo Li Hu Zhong Yan Haiquan Huang Xiao Bing Brown Wu

Wuhan University of Science and Technology, China Hubei University of Technology, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China Wuhan University of Technology, China Linguistic and Linguidtic Education Association, China Sungkyunkwan University, Korea South China University of Technology, China George Mason University, USA University of Technology Sydney, Australia Zhongnan University of Finance and Economics, China Hubei University, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China International Science & Education Researcher Association, China Wuhan University of Science and Technology, China, Wuhan University of Science and Technology, China Hubei University of Technology, China Wuhan University, China Sun Yat-Sen University, China

Table of Contents – Part I

Convergence of the Stochastic Age-Structured Population System with Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dongjuan Ma and Qimin Zhang

1

Parallel Computer Processing Systems Are Better Than Serial Computer Processing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zvi Retchkiman Konigsberg

8

Smooth Path Algorithm Based on A* in Games . . . . . . . . . . . . . . . . . . . . . . Xiang Xu and Kun Zou

15

The Features of Biorthogonal Binary Poly-scale Wavelet Packs in Bidimensional Function Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhihao Tang and Honglin Guo

22

The Traits of Dual Multiple Ternary Fuzzy Frames of Translates with Ternary Scaling Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ShuKe Zhou and Qingjiang Chen

29

The Characteristics of Multiple Affine Oblique Binary Frames of Translates with Binary Filter Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . YongGan Li

36

Generation and Characteristics of Vector-Valued Quarternary Wavelets with Poly-scale Dilation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ping Luo and Shiheng Wang

42

A Kind of New Strengthening Buffer Operators and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ran Han and Zheng-peng Wu

49

Infrared Target Detection Based on Spatially Related Fuzzy ART Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BingWen Chen, WenWei Wang, and QianQing Qin

55

A Novel Method for Quantifying the Demethylation Potential of Environmental Chemical Pollutants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yan Jiang and Xianliang Wang

62

Study of Quantitative Evaluation of the Effect of Prestack Noise Attenuation on Angle Gather . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junhua Zhang, Jing Wang, Xiaoteng Liang, Shaomei Zhang, and Shengtao Zang

72

X

Table of Contents – Part I

A User Model for Recommendation Based on Facial Expression Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quan Lu, Dezhao Chen, and Jiayin Huang An Improved Sub-pixel Location Method for Image Measurement . . . . . . Hu Zhou, Zhihui Liu, and Jianguo Yang The Dynamic Honeypot Design and Implementation Based on Honeyd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuewu Liu, Lingyi Peng, and Chaoliang Li

78

83

93

Research of SIP DoS Defense Mechanism Based on Queue Theory . . . . . . Fuxiang Gao, Qiao Liu, and Hongdan Zhan

99

Research on the Use of Mobile Devices in Distance EFL Learning . . . . . . Fangyi Xia

105

Flood Risk Assessment Based on the Information Diffusion Method . . . . Li Qiong

111

Dielectric Characteristics of Chrome Contaminated Soil . . . . . . . . . . . . . . . Yakun Sun, Yuqiang Liu, Changxin Nai, and Lu Dong

118

Influences of Climate on Forest Fire during the Period from 2000 to 2009 in Hunan Province . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ZhiGang Han, DaLun Tian, and Gui Zhang

123

Numerical Simulation for Optimal Harvesting Strategies of Fish Stock in Fluctuating Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lulu Li, Wen Zhao, Lijuan Cao, and Hongyan Ao

131

The Graphic Data Conversion from AutoCAD to GeoDatabase . . . . . . . . Xiaosheng Liu and Feihui Hu

137

Research on Knowledge Transference Management of Knowledge Alliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jibin Ma, Gai Wang, and Xueyan Wang

143

The Study of Print Quality Evaluation System Using the Back Propagation Neural Network with Applications to Sheet-Fed Offset . . . . . Taolin Ma, Yang Li, and Yansong Sun

149

Improved Design of GPRS Wireless Security System Based on AES . . . . TaoLin Ma, XiaoLan Sun, and LiangPei Zhang Design and Realization of FH-CDMA Scheme for Multiple-Access Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abdul Baqi, Sajjad Ahmed Soomro, and Safeeullah Soomro

155

161

Table of Contents – Part I

XI

Design and Implement on Automated Pharmacy System . . . . . . . . . . . . . . HongLei Che, Chao Yun, and JiYuan Zang

167

Research on Digital Library Platform Based on Cloud Computing . . . . . . Lingling Han and Lijie Wang

176

Research on Nantong University of Radio and TV Websites Developing Based on ASP and Its Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shengqi Jing Analysis of Sustainable Development in Guilin by Using the Theory of Ecological Footprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hao Wang, GuanWen Cheng, Shan Xu, ZiHan Xu, XiaoWei Song, WenYuan Wei, HongYuan Fu, and GuoDan Lu Analysis of Emergy and Sustainable Development on the Eco-economic System of Guilin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ZiHan Xu, GuanWen Cheng, Hao Wang, HongYuan Fu, GuoDan Lu, and Ping Qin Multiple Frequency Detection System Design . . . . . . . . . . . . . . . . . . . . . . . . Wen Liu, Jun da Hu, and Ji cui Shi

181

187

194

201

The Law and Economic Perspective of Protecting the Ecological Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chen Xiuping and Liang Xianyan

207

Research on the Management Measure for Livestock Pollution Prevention and Control in China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yukun Ji, Kaijun Wang, and Mingxia Zheng

214

The Design of Supermarket Electronic Shopping Guide System Based on ZigBee Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yujie Zhang, Liang Han, and Yuanyuan Zhang

219

The Research of Flame Combustion Diagnosis System Based on Digital Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . YuJie Zhang, SaLe Hui, and YuanYuan Zhang

225

Design and Research of Virtual Instrument Development Board . . . . . . . . Lin Zhang, Taizhou Li, and Zhuo Chen

231

Substantial Development Strategy of Land Resource in Zhangjiakou . . . . Yuqiang Sun, Shengchen Wang, and Yanna Zhao

239

Computational Classification of Cloud Forests in Thailand Using Statistical Behaviors of Weather Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peerasak Sangarun, Wittaya Pheera, Krisanadej Jaroensutasinee, and Mullica Jaroensutasinee

244

XII

Table of Contents – Part I

Research on Establishing the Early-Warning Index System of Energy Security in China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yanna Zhao, Min Zhang, and Yuqiang Sun Artificial Enzyme Construction with Temperature Sensitivity . . . . . . . . . . Tingting Lin, Jun Lin, Xin Huang, and Junqiu Liu

251 257

An Efficient Message-Attached Password Authentication Protocol and Its Applications in the Internet of Things . . . . . . . . . . . . . . . . . . . . . . . . . . . An Wang, Zheng Li, and Xianwen Yang

263

Research on Simulation and Optimization Method for Tooth Movement in Virtual Orthodontics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhanli Li and Guang Yang

270

An Interval Fuzzy C-means Algorithm Based on Edge Gradient for Underwater Optical Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shilong Wang, Yuru Xu, and Lei Wan

276

A Generic Construction for Proxy Cryptography . . . . . . . . . . . . . . . . . . . . . Guoyan Zhang VCAN-Controller Area Network Based Human Vital Sign Data Transmission Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atiya Azmi, Nadia Ishaque, Ammar Abbas, and Safeeullah Soomro Study on the Some Labelings of Complete Bipartite Graphs . . . . . . . . . . . WuZhuang Li, GuangHai Li, and QianTai Yan

284

290 297

An Effective Adjustment on Improving the Process of Road Detection on Raster Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yang Li, Xiao-dong Zhang, and Yuan-lu Bao

302

Multi-objective Optimized PID Controller for Unstable First-Order Plus Delay Time Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gongquan Tan, Xiaohui Zeng, Shuchuan Gan, and Yonghui Chen

309

Water Quality Evaluation for the Main Inflow Rivers of Nansihu Lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yang Liyuan, Shen Ji, Liu Enfeng, and Zhang Wei

316

Software Piracy: A Hard Nut to Crack—A Problem of Information Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bigang Hong

324

Study of Bedrock Weathering Zone Features in Suntuan Coal Mine . . . . . XiaoLong Li, DuoXi Yao, and JinXiang Yang

330

Mercury Pollution Characteristics in the Soil around Landfill . . . . . . . . . . JinXiang Yang, MingXu Zhang, and XiaoLong Li

336

Table of Contents – Part I

XIII

The Research on Method of Detection for Three-Dimensional Temperature of the Furnace Based on Support Vector Machine . . . . . . . . Yang Yu, Jinxing Chen, Guohua Zhang, and Zhiyong Tao

341

Study on Wave Filtering of Photoacoustic Spectrometry Detecting Signal Based on Mallat Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yang Yu, Shuo Wu, Guohua Zhang, and Peixin Sun

347

Ontology-Based Context-Aware Management for Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Keun-Wang Lee and Si-Ho Cha

353

An Extended Center-Symmetric Local Ternary Patterns for Image Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaosheng Wu and Junding Sun

359

Comparison of Photosynthetic Parameters and Some Physilogical Indices of 11 Fennel Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mingyou Wang, Beilei Xiao, and Lixia Liu

365

Effect of Naturally Low Temperature Stress on Cold Resistance of Fennel Varieties Resource . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beilei Xiao, Mingyou Wang, and Lixia Liu

370

Survey on the Continuing Physical Education in the Cities around the Taihu Lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . JianQiang Guo

375

The Gas Seepage and Migration Law of Mine Fire Zone under the Positive Pressure Ventilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HaiYan Wang, ZhenLong Zhang, DanDan Jiang, and FeiYin Wang

381

Study on Continued Industry’s Development Path of Resource-Based Cities in Heilongjiang Province . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ying Zhu and Jiehua Lv

390

Cable Length Measurement Systems Based on Time Domain Reflectometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianhui Song, Yang Yu, and Hongwei Gao

396

The Cable Crimp Levels Effect on TDR Cable Length Measurement System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianhui Song, Yang Yu, and Hongwei Gao

402

The Clustering Algorithm Based on the Most Similar Relation Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wei Hong Xu, Min Zhu, Ya Ruo Jiang, Yu Shan Bai, and Yan Yu

408

Study of Infrared Image Enhancement Algorithm in Front End . . . . . . . . Rongtian Zheng, Jingxin Hong, and Qingwei Liao

416

XIV

Table of Contents – Part I

Influence of Milling Conditions on the Surface Quality in High-Speed Milling of Titanium Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaolong Shen, Laixi Zhang, and Chenggao Ren Molecular Dynamics Simulation Study on the Microscopic Structure and the Diffusion Behavior of Methanol in Confined Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hua Liu, XiaoFeng Yang, Chunyan Li, and Jianchao Chen

423

430

Spoken Emotion Recognition Using Radial Basis Function Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiqing Zhang, Xiaoming Zhao, and Bicheng Lei

437

Facial Expression Recognition Using Local Fisher Discriminant Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiqing Zhang, Xiaoming Zhao, and Bicheng Lei

443

Improving Tracking Performance of PLL Based on Wavelet Packet De-noising Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . YinYin Li, XiaoSu Xu, and Tao Zhang

449

Improved Algorithm of LED Display Image Based on Composed Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xi-jia Song, Xi-qiang Ma, Wei-ya Liu, and Xi-feng Zheng

457

The Development Process of Multimedia Courseware Using Authoware and Analysis of Common Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LiMei Fu

464

Design of Ship Main Engine Speed Controller Based on Expert Active Disturbance Rejection Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weigang Pan, Guiyong Yang, Changshun Wang, and Yingbing Zhou

470

Design of Ship Course Controller Based on Genetic Algorithm Active Disturbance Rejection Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hairong Xiao, Weigang Pan, and Yaozhen Han

476

A CD-ROM Management Device with Free Storage, Automatic Disk Check Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daohe Chen, Xiaohong Wang, and Wenze Li

482

An Efficient Multiparty Quantum Secret Sharing with Pure Entangled Two Photon States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Run-hua Shi and Hong Zhong

489

Problems and Countermeasures of Educational Informationization Construction in Colleges and Universities . . . . . . . . . . . . . . . . . . . . . . . . . . . Jiaguo Luo and Jie Yu

496

Table of Contents – Part I

Synthesis and Characterization of Eco-friendly Composite: Poly(Ethylene Glycol)-Grafted Expanded Graphite/Polyaniline . . . . . . . . Mincong Zhu, Xin Qing, Kanzhu Li, Wei Qi, Ruijing Su, Jun Xiao, Qianqian Zhang, Dengxin Li, Yingchen Zhang, and Ailian Liu

XV

501

The Features of a Sort of Five-Variant Wavelet Packet Bases in Sobolev Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yujuan Hu, Qingjiang Chen, and Lang Zhao

507

The Features of Multiple Affine Fuzzy Quarternary Frames in Sobolev Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hongwei Gao

513

Characters of Orthogonal Nontensor Product Trivariate Wavelet Wraps in Three-Dimensional Besov Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jiantang Zhao and Qingjiang Chen

519

Research on Computer Education and Education Reform Based on a Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianhong Sun, Qin Xu, Yingjiang Li, and JunSheng Li

525

The Existence and Uniqueness for a Class of Nonlinear Wave Equations With Damping Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bo Lu and Qingshan Zhang

530

Research on the Distributed Satellite Earth Measurement System Based on ICE Middleware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jun Zhou, Wenquan Feng, and Zebin Sun

534

The Analysis and Optimization of KNN Algorithm Space-Time Efficiency for Chinese Text Categorization . . . . . . . . . . . . . . . . . . . . . . . . . . Ying Cai and Xiaofei Wang

542

Research of Digital Character Recognition Technology Based on BP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xianmin Wei

551

Image Segmentation Based on D-S Evidence Theory and C-means Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xianmin Wei

556

Time-Delay Estimation Based on Multilayer Correlation . . . . . . . . . . . . . . Hua Yan, Yang Zhang, and GuanNan Chen Applying HMAC to Enhance Information Security for Mobile Reader RFID System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fu-Tung Wang, Tzong-Dar Wu, and Yu-Chung Lu

562

568

XVI

Table of Contents – Part I

Analysis Based on Generalized Regression Neural Network to Oil Atomic Emission Spectrum Data of a Type Diesel Engine . . . . . . . . . . . . . ChunHui Zhang, HongXiang Tian, and Tao Liu

574

Robust Face Recognition Based on KFDA-LLE and SVM Techniques . . . GuoQiang Wang and ChunLing Gao

581

An Improved Double-Threshold Cooperative Spectrum Sensing . . . . . . . . DengYin Zhang and Hui Zhang

588

Handwritten Digit Recognition Based on Principal Component Analysis and Support Vector Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rui Li and Shiqing Zhang

595

Research on System Stability with Extended Small Gain Theory Based on Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuqiang Jin and Qiang Ma

600

Research on the Chattering Problem with VSC of Supersonic Missiles Based on Intelligent Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junwei Lei, Jianhong Shi, Guorong Zhao, and Guoqiang Liang

606

Research on Backstepping Nussbaum Gain Control of Missile Overload System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianhong Shi, Guorong Zhao, Junwei Lei, and Guoqiang Liang

612

Adaptive Control of Supersonic Missiles with Unknown Input Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jinhua Wu, Junwei Lei, Wenjin Gu, and Jianhong Shi

616

The Fault Diagnostic Model Based on MHMM-SVM and Its Aplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FengBo Zhu, WenQuan Wu, ShanLin Zhu, and RenYang Liu

621

Analysis of a Novel Electromagnetic Bandgap Structure for Simultaneous Switching Noise Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . Hua Yang, ShaoChang Chen, Qiang Zhang, and WenTing Zheng

628

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

635

Convergence of the Stochastic Age-Structured Population System with Diffusion Dongjuan Ma and Qimin Zhang* School of Mathematics and Computer Science, Ning Xia University 750021,Yinchuan, China [email protected], [email protected]

Abstract. In this paper, stochastic age-structure population system with jump are studied. It is proved that the semi-implicit Euler approximation solutions converge to the analytic solution for the stochastic age-structured population system with Poisson jump. The analysis use Itoˆ ′s formula, Burkholder-DavisGundy's inequality, Gronwall's lemma and some inequalities for our purposes. Keywords: Semi-implicit Euler method, Poisson jump, Numerical solution.

1 Introduction The theories of stochastic partial differential equations have been extensively many areas, such as economics, finance and several areas of science and engineering and so on. There have been mach researched in deterministic age-structured population with diffusion system and discussed the existence, uniqueness, stability regularity and localization of the solution of this system [1-3]. In recent years, it is more necessary to consider the random behavior of the birthdeath process and the effects of the stochastic environmental noise for Age-structured population systems. Most of papers are concerned about stochastic population system. The random element is considered, there have been many results from stochastic agestochastic age-structured population system. For instance, Zhang discussed the existence and uniqueness for a stochastic age-structured population system with diffusion [4]. When the diffusion of the population is not considered, Zhang studied the existence, uniqueness and exponential stability of a stochastic age-dependent population system ,and numerical analysis for stochastic age dependent population have been studied in [5-8]. Interest has been growing in the study of stochastic differential equations with jumps, which is extensively used to model many of the phenomena arising in the areas [9-10]. In general, most of stochastic age-structured population system with Poisson jump have not analytic solutions, thus numerical approximation schemes are invaluable tools for exploring their properties. In this paper, a numerical analysis for stochastic age-structured population system which is described by Eqs(1) will be developed. The first contribution is to study the Semi-implicit Euler approximation solutions converge to the analytic solution. The second contribution is to consider diffuse form div( P∇u ). In particular, our results extend those in [6-8]. *

Corresponding author.

S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 1–7, 2011. © Springer-Verlag Berlin Heidelberg 2011

2

D. Ma and Q. Zhang

2 Preliminaries and the Semi-implicit Euler Approximation Let:

O = ( 0, A) × T , and V ≡ {ϕ | ϕ ∈ L2 ( O ) ,

∂ϕ ∂ϕ ∈ L2 ( O ) , where are ∂xi ∂xi

generalized partial derivatives}. Then V ′ the dual space of V .we denote by ⋅ and

⋅

the norms in V and V ′ respectively; by 〈⋅, ⋅〉 the duality product between V , V ′ , and

(⋅, ⋅) the scalar product in H .Let ( Ω, F , P ) be a complete probability space with filtration {Ft }t≥ 0 satisfying the usual conditions (i.e., it is increasing and right

by a

F0 contains all P-null sets). In this paper, we consider the convergence of stochastic systems with diffusion kdiv ( P ∇ u ),

continuous while

⎧ ∂P ∂P ⎪ ∂t + ∂r = kdiv( P∇u ) − μ (r , t , x) P + ⎪ ⎪ f (r , t , x, P ) + g (r , t , x, P ) ∂Wt + h(r , t , x, P ) ∂N t , inQ = (0, A) × Q, A ⎪ ∂t ∂t ⎪⎪ (1) A in(0, T ) × Γ, ⎨ P (0, t , x ) = ∫ β (r , t , x ) P (r , t , x )dr , 0 ⎪ P r x P = ( , 0, ) in(0, A) × Γ, ⎪ 0 ( r , x ), ⎪ P (r , t , x) = 0, onΣ A = (0, A) × (0, T ) × ∂Γ, ⎪ A ⎪u (t , x ) = P (r , t , x)dr , inQ, ∫0 ⎪⎩

div be the divergence operator. Let W(t) be a Wiener process, hr ( ,t, x,P) poisson jump procession. For system (1), the discrete time semi-implicit Euler approximation on by the iterative scheme:

∂Nt is the ∂t

t is defined

∂Qn+1 ∂Q + kdiv(Qn+1∇u) − μ(r, t, x)Qn + f (r, t, x, Qn )]+t +α[− n+1 ∂r ∂r +kdiv(Qn+1∇u) − μ(r, t, x)Qn+1 + f (r, t, x,Qn+1)]+t + g(r, t, x, Qn )ΔWn + h(r, t, x, Qn )ΔNn ,

Qn+1 = Qn + (1−α)[−

Here

α ∈ [0,1], Qn

is the approximation to

P (tn , r , x) , for tn = nΔt , the time

increment is Δt = T  1 , and Brownian motion increment is N Possion process increment is +N n = N (t n +1 ) − N (t n ).

+Wn = W(tn+1) −W(tn ),

Convergence of the Stochastic Age-Structured Population System with Diffusion

3

For (1),we define the continuous-time approximation : t ∂Q Qt = P0 + ∫ (1 − α )[− s + kdiv(Qs∇u ) − μ (r , s, x) Z1 ( s ) + f (r , s, x, Z1 ( s ))]ds 0 ∂r t ∂Qs + ∫ α [− + kdiv(Qs ∇u ) − μ (r , s, x) Z 2 ( s) + f (r , s, x, Z 2 ( s))]ds 0 ∂r t

t

0

0

+ ∫ g (r , s, x, Z1 ( s )d W ( s ) + ∫ h(r , s, x, Z1 ( s )d N ( s ),

Z1 (t ) = Z1 (t, r, x) = ΣkN=−01Qk I[k Δt ,(k +1)Δt ] , Z2 (t ) = Z2 (t, r, x) = ΣkN=−01Qk +1I[kΔt ,(k +1)Δt ] , There I G is the indicator function for the set G and Z1(tk ) = Z2 (tk−1) = Qk = Q(tk , r, x). To establish the convergence theorem we shall use the following assumptions: (i)(Lipschitz condition ) here exists a positive constant K such that P1 , P2 ∈ C

| f (r , t , x, P1 ) − f (r , t , x, P2 ) | ∨ | g (r , t , x, P1 ) − g (r , t , x, P2 ) | ∨ | h(r , t , x, P1 ) − h(r , t , x, P2 ) |≤ K | P1 − P2 |, a.e.t ; (ii) μ ( r , t , x ) and

β (r , t , x) are continuous in Q such that

0 ≤ μ 0 ≤ μ ( r , t , x ) ≤ μ < ∞, 0 ≤ β ( r , t , x ) ≤ β < ∞, k 0 ≤ k ( r , t ) ≤ k ; t

∫ | |+u || < k ;

(iii) f ( r , t , x, 0) = 0, g ( r , t , x, 0) = 0,

2

3

0

3 The Main Results In this section, we will provide some theorem which are necessary for the proof of Qt convergence to the analytical Solution of this system Pt . (We only discuss the iterative scheme of the continuous-time, for iterative scheme of the discrete time, there is similar.) Theorem 3.1. Under assumptions (i)-(ii) E Proof. For (1), applying N (t ) = t

| Qt |2 =| Q0 |2 +2(1 − α ) ∫ 〈− 0

t

+2α ∫ 〈− 0

sup | Qt |2 ≤ C1T .

0 ≤t ≤T

N (t ) − λt , and Itoˆ′s formula to | Qt |2 ,

∂Qs + kdiv(Qs ∇u ) − μ ( r , s, x) Z1 ( s) + f ( r , s, x, Z1 ( s)), Qs 〉 ds ∂r

∂Qs + kdiv(Qs∇u) − μ ( r, s, x) Z 2 ( s) + f ( r, s, x, Z 2 ( s)), Qs 〉 ds ∂r

t

t

+2 ∫ d Ws + ∫ | | g (r , s, x, Z1 ) ||22 ds 0

0

t

t

0

0

+2 ∫ 〈Qs , h (r , s , x, Z1 )〉 d N s + ∫ |h ( r , s , x, Z1 ) |2 d N s

4

D. Ma and Q. Zhang

t

≤| Q0 |2 −2 ∫ 〈 0

t t ∂Qs , Qs 〉ds + 2 ∫ 〈 kdiv(Qs ∇u ), Qs 〉 ds − 2 μ0 ∫ ∫ ((1 0 0 O ∂r

−α ) Z1 ( s) + α Z 2 ( s))Qs drdxds + 2 ∫

t

∫

0 O

+ 2∫

t

∫

0 O

((1 − α ) f (r , s, x, Z1 ) + α f (r , s, x, Z 2 ))Qs drdxds

t

t

0

0 O

+ ∫ | | g (r , s, x, Z1 ) ||22 ds + 2 ∫ +2λ ∫

t

∫

g ( r , s, x, Z1 )Qs drdxd Ws

0 O

∫

h(r , s, x, Z1 )Qs drdxdN

h(r , s, x, Z1 )Qs drdxds

t t + ∫ |h(r , s, x, Z1 ) |2 dN + λ ∫ |h(r , s, x, Z1 ) |2 ds. 0

0

t

∫ 〈kdiv(Q ∇u ), Q 〉 ds ≤ k ∫ ∫ (∇Q ∇u + Q +u ) drdxds + k ∫ |Q | ds ≤ 2k ∫ | |+u || ds + 2k ∫ | | Q +u || ds + k ∫ |Q | ds ≤ 2k ∫ | |+u || ds + 2k ∫ |Q | ds ∫ | |+u || ds + k ∫ |Q

Since 2

s

0

s

t

t

2

s

0 O t

s

t

2

0

t

2

0

0

t

2

s

0

t

2

s

0

0

t

2

s

2

s

t

2

0

|2 ds.

s

0

By assumptions and the quality of operator, there exist k1 =(2k3 +1)k and k2 =2k3k such that 2

∫

t

0

t

〈 kdiv(Qs ∇u ), Qs 〉 ds ≤ k1 ∫ |Qs |2 ds + k2 . () 0

Applying (7) and −

∫

t

0

〈

t ∂Qs 1 , Qs 〉ds ≤ Aβ 2 ∫ |Qs |2 ds, we have 0 2 ∂r

t

t

0

0

t

t

0

0

| Qt |2 ≤| Q0 |2 + Aβ 2 ∫ |Qs |2 ds + k1 ∫ |Qs |2 ds + k2 + 2 μ0 ∫ |Z1 |2 ds + 2 μ0 ∫ |Z 2 |2 ds t

t

t

0 t

0

0

+2∫ | f (r, s, x, z1 (s)) |2 ds + 2∫ | f (r, s, x, z2 (s)) |2 ds + (1+ λ + μ0 )∫ |Qs |2 ds +2∫

∫

0 O t

t

t

g(r, s, x, Z1)QsdrdxdWs + ∫ || g(r, s, x, Z1)||22 ds + λ∫ |h(r, s, x, z1(s))|2 ds 0

0

 + |h(r, s, x, Z )|2 dN . +λ∫ |h(r, s, x, z1(s))|2 ds + 2∫ ∫ h(r, s, x, Z1)QdrdxdN 1 s ∫ 0

t

t

0 O

0

Hence, for any t1 ∈ [0, T ], t

E sup | Qt |2 ≤ E | Q0 |2 + k 2 + ( Aβ 2 + 1 + λ + μ 0 + k1 ) ∫ E sup | Qs |2 ds 0

0 ≤t1 ≤t

0 ≤t1 ≤t

t

t

0

0

+ (2μ0 + 3K 2 + 2 K 2λ ) ∫ |Z1 |2 ds + (2μ0 + 2 K 2 ) ∫ |Z 2 |2 ds

Convergence of the Stochastic Age-Structured Population System with Diffusion

+2E sup ∫

t

∫

0≤t1 ≤t 0 O

+2E sup ∫

t

5

g(r, s, x, Z1 )Qs drdxdWs t

∫ h(r, s, x, Z )Q drdxdN + E sup ∫ |h(r, s, x, Z ) | 1

0≤t1 ≤t 0 O

s

0≤t1 ≤t 0

2

1

dN .

By Burkholder-Davis-Gundy's inequality, there exist positive constants K1, K2 such that

E sup ∫

t

∫

t

0≤t1 ≤t 0 O

1 2

g (r, s, x, Z1 )Qs drdxd Ws ≤ 3E[sup | Qs | (∫ | | g (r, s, x, Z1 ) || ds) ] 2 2

0

0≤t1 ≤t

t t 1 1 ≤ E[sup | Qs |2 ] + K1 ∫ || g(r, s, x, Z1)||22 ds ≤ E[sup | Qs |2 ] + K2 K1 ∫ E | Z1 |2 ds, 0 0 8 0≤t1≤t 8 0≤t1≤t

In the same way, we obtain:

∫

O

E sup ∫

t

∫ h(r, s, x, Z )Q drdxdN 1

0≤t1t 0 O

s

t 1 ≤ E sup | Qs |2 + K 2 K 2 ∫ |Z1 |2 ds, byZ i ≤ sup | Qs |, i = 1, 2 0 8 0≤t1 ≤t 0≤ t1 ≤ t t 3 1 E sup | Qt |2 ≤ ( Aβ 2 + + 1λ + 5μ0 + (5 + 2 K1 + 2λ + 3K 2 ) K 2 ) ∫ E sup | Qs |2 ds + E | Q0 |2 +k2 . 0 2 8 0≤ t1 ≤ t 0 ≤t1t

At present, applying Gronwall's lemma, the proof complete. Theorem 3.2. under assumption, for each t ∈ [0, T ],

E[| Qt − Z1 (t ) |2 ] ≤ C2 +t , E[| Qt − Z 2 (t ) |2 ] ≤ C3 +t.

Proof. For arbitrary t∈[0,T], there exists k such that t ∈[k+t,(k +1)+t], so we have

| Qt − Z1(t) |2 ≤ 6+t

∫

t

k +t

|

t ∂Qs 2 | ds + 6k 2 +tk1 ∫ |Qs |2 ds + 6k 2 +tk2 k +t ∂r

t

t

+6μ2Δt ∫ |(1−α)Z1(s) +αZ2 (s)|2 ds +12λ2Δt ∫ |h(r, s, x, Z1(s))|2 ds k +t

t

k +t

t

+12Δt ∫ | f (r, s, x, Z1 (s)) | ds +12Δt ∫ | f (r, s, x, Z2 (s)) |2 ds 2

k +t

+6 | ∫

t

k +t

k +t t

 |2 . g (r , s, x, Z1 ( s))d Ws |2 +12 | ∫ h(r , s, x, Z1 ( s))dN k +t

By Burkholder-Davis-Gundy's inequality, there exist positive constants K3,K4such that

E sup | ∫ 0≤t1 ≤T

t

k +t

g (r , s, x, Z1 ( s ))d Ws |2 ≤ K 3 ∫

t

k +t

t

t

k +t

k +t

E sup | Z1 ( s ) |2 ds, 0≤t1 ≤T

E sup | ∫ h(r , s, x, Z1 ( s ))dN |2 ≤ K 4 ∫ E sup | Z1 ( s ) |2 ds, 0 ≤t1 ≤T

By

Zi ≤ sup | Qs | and assumption, have 0≤t1 ≤t

0 ≤t1 ≤T

6

D. Ma and Q. Zhang

E sup | Qt − Z1 (t ) |2 ≤ (6 K5 +t + 6k 2 +tk1 + 24μ 2 +t + 24+tK 2 + 6 K3 + 12 K 4 0 ≤t1 ≤T

+ 12λ 2 +tK 2 ) sup E | Qs |2 +6k 2 +tk2 . 0 ≤t1 ≤T

C2 = (6K5 + 6k k1 + 24μ + 24K + 6K3 +12K4 +12λ2 K2 )C1t + 6k 2 K2 , Such that 2

2

2

E[| Qt − Z1(t)|2] ≤ C2+t. (Similar method, we can obtain E[| Qt − Z 2 (t ) | ] ≤ C3 +t. ) 2

Theorem 3.3. Under assumption, for each t ∈ [0, T ], E[| P (t ) − Qt | ] ≤ C4 Δt. 2

N (t ) = N (t ) − λt and Itoˆ′s formula to | Qt |2 , t t ∂( P − Qs ) | P(t ) − Qt |2 = −2∫ 〈 Ps − Qs , s 〉ds + 2∫ 〈 Ps − Qs , kdiv(( Ps − Qs )∇u)〉 ds 0 0 ∂r

Proof. Applying

t

−2 ∫ 〈 Ps − Qs , μ (r , t , x)((1 − α )( Ps − Z1 ) + α ( Ps − Z 2 )〉 ds 0

t

+2∫ 〈Ps −Qs ,(1−α)( f (r, t, x, Ps ) − f (r, t, x, Z1)) +α( f (r, t, x, Ps ) − f (r, t, x, Z2 )〉ds 0

t

t

+∫ || g(r, t, x, Ps ) − g(r, t, x, Z1)||22 ds + 2∫ 〈Ps −Qs , g(r, t, x, Ps ) − g(r, t, x, Z1)〉dWt 0

0

t

t

+2∫ 〈Ps −Qs , h(r, t, x, Ps ) − h(r, t, x, Z1)〉d Nt + ∫ |h(r, t, x, Ps ) − h(r, t, x, Z1)|2 d Nt 0

0

t

t

0

0

t

t

0

0

≤ Aβ 2 ∫ |Ps −Qs |2 ds + k1∫ |Ps −Qs |2 ds + k2 + 2μ0 ∫ |Ps − Z1 |2 ds + 2μ0 ∫ |Ps − Z2 |2 ds t

t

+2∫ | f (r, s, x, Ps ) − f (r, s, x, z1(s)) | ds + 2∫ | f (r, s, x, Ps ) − f (r, s, x, z2 (s)) |2 ds 2

0

0

t

t

0

0 O

+(2 + λ + μ0 )∫ |Ps − Qs |2 ds + 2∫

∫ (P − Q )(g(r,t, x, P ) − g(r,t, x, Z ))drdxdW +∫ |g (r, t , x, P ) − g (r, s, x, z (s)) | ds + ∫ | | g (r, t, x, P ) − g (r, s, x, Z ) || ds +λ∫ |h(r, t, x, P ) − h(r, s, x, z (s))| ds + λ∫ |h(r, t, x, P ) − h(z (s))| ds +2∫ ∫ (h(r, t, x, P ) − h(r, s, x, Z ))(P −Q )drdxdN + ∫ |h(r, t, x, P ) − h(r, s, x, Z )| dN . t

s

s

0

t

0

t

2

2

t

1

s

s

2 2

1

s

0

t

0 O

1

s

0

1

s

s

t

2

1

s

1

s

s

2

1

s

0

Hence, for any t1 ∈ [0, T ], t

E sup | Ps − Qs |2 ≤ k 2 + ( Aβ 2 + 1 + λ + μ0 + k1 ) ∫ E sup | Ps − Qs |2 ds 0

0 ≤t1 ≤t

0 ≤t1 ≤t

t

t

0

0

+(2μ0 + 3K2 + 2K2λ)∫ |Ps − Z1 |2 ds + (2μ0 + 2K2 )∫ |Ps − Z2 |2 ds +2E sup ∫

t

∫ (P −Q )(g(r,t, x, P ) − g(r,t, x, Z ))drdxdW

0≤t1≤t 0 O

s

s

s

1

s

Convergence of the Stochastic Age-Structured Population System with Diffusion

+2E sup ∫

t

7

∫ (h(r, t, x, P ) − h(r, s, x, Z ))(P − Q )drdxdN s

0≤t1≤t 0 O

1

s

s

+E sup ∫ |h(r, t, x, Ps ) − h(r, s, x, Z1) |2 dN . t

0≤t1 ≤t 0

By Burkholder-Davis-Gundy's inequality, there exist positive constants K5, K6 such that

E sup ∫

t

∫

( Ps − Qs )( g ( r , t , x, Ps ) − g ( r , t , x, Z1 )) drdxd Ws

0 ≤t1 ≤t 0 O

t 1 ≤ E[sup | Ps − Qs |2 ] + K 2 K 5 ∫ E | ps − Z1 |2 ds, 0 8 0≤t1 ≤t

∫

O

E sup ∫ 0 ≤t1t

t

∫

0 O

( h(r , t , x, Ps ) − h( r , s, x, Z1 ))( Ps − Qs ) drdxdN

t 1 ≤ E sup | Ps − Qs |2 + K 6 K 2 ∫ E | Pt − Z1 |2 ds, 0 8 0≤t1 ≤t

Applying Theorem3.2, have E sup | Pt − Qt | ≤ C5 +t + C6 2

0≤t1 ≤t

t

∫ E sup | P − Q | 0

2

0≤t1 ≤t

t

t

dt ,

Where C5 = (4μ0 + 7 K + 4λ K + 4K5 K + 6K6 K )C2 + (4μ0 + 4K )C3 + k2 , 2

2

2

2

2

C6 = Aβ 2 + k1 + μ0 + 2 + λ + 12K 2 + 4λ K 2 + 4K5 K 2 + 6K6 K 2 . At present, applying Gronwall's lemma, the proof complete. So applying Theorem3.3, under the assumption condition, we obtain the semi-implicit Euler numerical solution Qt convergence to the analytical Solution Pt of this system.

References 1. Hernandez, G.E.: Age-density dependent population dispersal in RN. Mathematical Biosciences J. 149, 37–56 (1998) 2. Hernandez, G.E.: Existence of solutions in a population dynamic problem. J. Appl. Math. 509, 43–48 (1986) 3. Hernandez, G.E.: Localization of age-dependent ant-crowding populations. J. Q. Appl. Math. 53, 35 (1995) 4. Zhang, Q., Han, C.Z.: existence and uniqueness for a stochastic age-structured population system with diffusion. J. Science Direct 32, 2197–2206 (2008) 5. Zhang, Q., Liu, W., Nie, Z.: Existence, uniqueness and exponential stability of stochastic age-dependent population. J. Appl. Math. Comput. 154, 183–201 (2004) 6. Zhang, Q., Han, C.Z.: Convergence of numerical solutions to stochastic age-structured population system with diffusion. J. Applied Mathematics and Computation 07, 156 (2006) 7. Zhang, Q.: Exponential stability of numerical solutions to a stochastic age-structured population system with diffusion. Journal of Computational and Applied Mathematics 220, 22–33 (2008) 8. Zhang, Q., Han, C.Z.: Numerical analysis for stochastic age-dependent population equations. J. Appl. Math. Comput. 176, 210–223 (2005) 9. Gardon, A.: The Order of approximations for solutions of Ito-type stochastic differential equations with jumps. J. Stochastic Analysis and Applications 38, 753–769 (2004)

Parallel Computer Processing Systems Are Better Than Serial Computer Processing Systems Zvi Retchkiman Konigsberg Instituto Politecnico Nacional

Abstract. The main objective and contribution of this paper consists in using a formal and mathematical approach to prove that parallel computer processing systems are better than serial computer processing systems, better related to: saving time and/or money and being able to solve larger problems. This is achieved thanks to the theory of Lyapunov stability and max-plus algebra applied to discrete event systems modeled with time Petri nets. Keywords: Parallel and Serial Computer Processing Systems, Lyapunov Methods, Max-Plus Algebra, Timed Petri Nets.

1

Introduction

Serial computer processing systems are characterized by the fact of executing software using a single central processing unit (CPU) while parallel computer processing systems simultaneously use multiple CPU’s at the time. Some of the arguments which have been used to say why it is better parallel than serial are: save time and/or money and solve larger problems. Besides that there are limits to serial processing computer systems due to: transmission speeds, limits to miniaturization ans economic limitations. However we would like to be more precise and give a definitive and unquestionable formal proof to justify the claim that parallel computer processing systems are better than serial computer processing systems. The main objective and contribution of this paper consists in using a formal and mathematical approach to prove that parallel computer processing systems are better than serial computer processing systems (better related to: saving time and/or money and being able to solve larger problems). This is achieved thanks to the theory of Lyapunov stability and max-plus algebra applied to discrete event systems modeled with time Petri nets. The paper is organized as follows. Sections 2 and 3 provide the mathematical results utilized in the paper about Lyapunov theory for discrete event systems modeled with Petri nets and max-plus algebra in order to achieve its goal (for a detailed exposition see [1] and [2]). In section 4, the solution to the stability problem for discrete event systems modeled with timed Petri nets using a Lyapunov, maxplus algebra approach is given. Section 5, applies the theory presented in the S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 8–14, 2011. c Springer-Verlag Berlin Heidelberg 2011 

Parallel Computer Processing vs. Serial Computer Processing

9

previous sections to formally prove that parallel computer processing systems are better than serial computer processing systems. Finally, the paper ends with some conclusions.

2

Lyapunov Stability and Stabilization of Discrete Event Systems Modeled with Petri Nets [1]

Proposition 1. Let P N be a Petri net. P N is uniform practical stable if there exists a Φ strictly positive m vector such that Δv = uT AΦ ≤ 0

(1)

Moreover, P N is uniform practical asymptotic stable if the following equation holds Δv = uT AΦ ≤ −c(e), for c ∈ K (2) Lemma 1. Let suppose that Proposition (1) holds then, Δv = uT AΦ ≤ 0 ⇔ AΦ ≤ 0

(3)

Remark 1. Notice that since the state space of a TPN (timed Petri net) is contained in the state space of the same now not timed PN, stability of PN implies stability of the TPN. Definition 1. Let P N be a Petri net. P N is said to be stabilizable if there exists a firing transition sequence with transition count vector u such that the reachable markings M´ remain bounded. Proposition 2. Let P N be a Petri net. P N is stabilizable if there exists a firing transition sequence with transition count vector u such that the following equation holds Δv = AT u ≤ 0

3

(4)

Max-plus Algebra [2,3]

Theorem 1. The max-plus algebra max = (Rmax , ⊕, ⊗, , e) has the algebraic structure of a commutative and idempotent semiring. n×n Theorem 2. The 5-tuple n×n max = (Rmax , ⊕, ⊗, E, E) has the algebraic structure of a noncommutative idempotent semiring.

Definition 2. Let A ∈ Rn×n max and k ∈ N then the k-th power of A denoted by A⊗k is defined by: A⊗k = A ⊗ A ⊗ · · · ⊗ A, where A⊗0 is set equal to E.    k−times

10

Z.R. Konigsberg

+ + Definition 3. Let A ∈ Rn×n ∈ Rn×n = max then define the matrix A max as: A ∞  ⊗k + A . Where the element [A ]ji gives the maximal weight of any path from j k=1

to i. If in addition one wants to add the possibility of staying at a node then one must include matrix E in the definition of matrix A+ giving rise to its Kleene star representation defined by: A∗ =

∞  A⊗k .

(5)

k=0

Lemma 2. Let A ∈ Rn×n max be such that any circuit in the communication graph G(A) has average circuit weight less than or equal to . Then it holds that: A∗ =

n−1 

A⊗k .

(6)

k=0 n Definition 4. Let A ∈ Rn×n max be a matrix. If μ ∈ Rmax is a scalar and v ∈ Rmax is a vector that contains at least one finite element such that:

A⊗v =μ⊗v

(7)

then, μ is called an eigenvalue and v an eigenvector. Theorem 3. If A ∈ Rn×n max is irreducible i.e., its communication graph G(A) is strongly connected, then there exists one and only one finite eigenvalue (with possible several eigenvectors). This eigenvalue is equal to the maximal average weight of circuits in G(A): λ(A) = max

p∈C(A)

|p|w |p|1

(8)

n Theorem 4. Let A ∈ Rn×n max and b ∈ Rmax . If the communication graph G(A) has maximal average circuit weight less than or equal to e, then x = A∗ ⊗ b solves the equation x = (A ⊗ x) ⊕ b. Moreover, if the circuit weights in G(a) are negative then, the solution is unique. n Definition 5. Let Am ∈ Rn×n max for 0 ≤ m ≤ M and x(m) ∈ Rmax for −M ≤ M  m ≤ −1; M ≥ 0. Then, the recurrence equation: x(k) = Am ⊗x(k −m); k ≥

0 is called an M th order recurrence equation.

m=0

Theorem 5. The M th order recurrence equation, given by equation x(k) = M  Am ⊗ x(k − m); k ≥ 0, can be transformed into a first order recurrence m=0

equation x(k + 1) = A ⊗ x(k); k ≥ 0 provided that A0 has circuit weights less than or equal to zero.

Parallel Computer Processing vs. Serial Computer Processing

11

With any timed event Petri net, matrices A0 , A1 , ..., AM ∈ Nn ×Nn can be defined by setting [Am ]jl = ajl , where ajl is the largest of the holding times with respect to all places between transitions tl and tj with m tokens, for m = 0, 1, ..., M , with M equal to the maximum number of tokens with respect to all places. Let xi (k) denote the kth time that transition ti fires, then the vector x(k) = (x1 (k), x2 (k), ...xm (k))T , called the state of the system, satisfies the M th order M  recurrence equation: x(k) = Am ⊗ x(k − m); k ≥ 0 Now, assuming that all m=0

the hypothesis of theorem (5) are satisfied, and setting x ˆ(k) = (xT (k), xT (k − M  1), ..., xT (k − M + 1))T , equation x(k) = Am ⊗ x(k − m); k ≥ 0 can be m=0

expressed as: x ˆ(k + 1) = Aˆ ⊗ x ˆ(k); k ≥ 0, which is known as the standard autonomous equation.

4

The Solution to the Stability Problem for Discrete Event Systems Modeled with Timed Petri Nets

Definition 6. A TPN is said to be stable if all the transitions fire with the same proportion i.e., if there exists q ∈ N such that xi (k) = q, ∀i = 1, ..., n k→∞ k lim

(9)

Lemma 3. Consider the recurrence relation x(k + 1) = A ⊗ x(k), k ≥ 0, x(0) = x0 ∈ Rn arbitrary. A an irreducible matrix and λ ∈ R its eigenvalue then, lim

k→∞

xi (k) = λ, ∀i = 1, ..., n k

(10)

Now starting with an unstable T P N , collecting the results given by: proposition (2), what has just been discussed about recurrence equations for T P N and the previous lemma (3) plus theorem (3), the solution to the problem is obtained.

5

Parallel vs. Serial Computer Processing Systems

In this section, the main objective of this manuscript which consists in giving a precise and definitive answer to the question why are parallel computer processing systems preferred to serial computer processing systems (better related to: saving time and/or money and being able to solve larger problems), is presented. 5.1

Serial Computer Processing System

Consider a serial computer processing system with T P N model as depicted in Fig 1. Where the events (transitions) that drive the system are: q: a problem of size Ca has to be solved, s: the problem starts being executed by the CPU,

12

Z.R. Konigsberg

Fig. 1.

d: the problem has been solved. The places (that represent the states of the serial computer processing system) are: A: problems loading, P: the problems are waiting for a solution, B: the problem is being solved, I: the CPU of capacity Cd is idle. The holding times associated to the places A and I are Ca and Cd respectively, (with Ca > Cd). Remark 2. Notice that Ca, the size of q, is the time it takes to a problem until is completely loaded in the computer in order to be solved, larger problems will have larger Ca’ s, while Cd, the capacity of the CPU, is the time it takes to the CPU to reset. ⎡ ⎤ 0 1 0 0 The incidence matrix that represents the P N model is A = ⎣ 0 −1 1 −1 ⎦ 0 0 −1 1 Therefore since there does not exists a Φ strictly positive m vector such that AΦ ≤ 0 the sufficient condition for stability is not satisfied. Moreover, the P N (T P N ) is unbounded since by the repeated firing of q, the marking in P grows indefinitely i.e., the amount of problems that require a solution accumulate. However, by taking u = [k, k, k]; k > 0 (but unknown), we get that AT u ≤ 0. Therefore, the P N is stabilizable which implies that the T P N is stable. Now, let us proceed to determine the ⎛ exact value ⎞ of k. From the T P N model we Ca ε ε |p| obtain that: Aˆ = A∗0 ⊗ A1 = ⎝ Ca ε Cd ⎠. Therefore, λ(A) = max |p|w = 1 p∈C(A) Ca ε Cd max{Ca, Cd} = Ca. This means that in order for the T P N to be stable and work properly the speed at which the serial computer processing system works has to be equal to Ca or being more precise, that all the transitions must fire at the same speed as the problems arrive i.e., they have to be solved as soon as they are loaded into the computer which is attained by setting k = Ca. In particular, transition s which is related to the execution time of the CPU has to be fired at a speed equal to Ca.

Parallel Computer Processing vs. Serial Computer Processing

13

Summary 6. The serial computer processing system works properly if transition s fires at a speed equal to Ca which implies that the execution frequency of the CPU has to be equal to Ca. Now, if Ca increases due to the fact that the problem to be solved becomes larger then, this will result in an increment on the CPU’s execution frequency. However, there is a limit to this increment due to economical and physical limitations. One possible solution is to break this large problem into several smaller problems but this will result in larger execution times. 5.2

Parallel Computer Processing System

Consider a parallel computer processing systems with two CPU’s with T P N model as depicted in Fig 2. Where the events (transitions) that drive the system are: q: a problem of size Ca has to be solved, s1, s2: the problem starts being executed by the CPU’s, d1,d2: the problem has been solved. The places (that represent the states of the parallel computer processing system) are: A: problems loading, P: the problems are waiting for a solution, B1, B2: the problem is being solved, I1, I2: the CPU’s of capacity Cd are idle. The holding times associated to the places A and I1, I2 are Ca and Cd respectively,⎡ (with Ca > Cd). The ⎤ 0 1 0 0 0 0 ⎢ 0 −1 1 −1 0 0 ⎥ ⎢ ⎥ ⎥ incidence matrix that represents the P N model is A = ⎢ ⎢ 0 −1 0 0 1 −1 ⎥ ⎣ 0 0 −1 1 0 0 ⎦ 0 0 0 0 −1 1 Therefore since there does not exists a Φ strictly positive m vector such that AΦ ≤ 0 the sufficient condition for stability is not satisfied. Moreover, the P N (T P N ) is unbounded since by the repeated firing of q, the marking in P grows indefinitely i.e., the amount of problems that require a solution accumulate. However, by taking u = [k, k/2, k/2, k/2, k/2]; k > 0 (but unknown) we get that AT u ≤ 0. Therefore, the P N is stabilizable which implies that the T P N is stable. Now, let us proceed to determine ⎛ the exact value ⎞ of k. From the T P N Ca ε ε ε ε ⎜ Ca ε ε Cd ε ⎟ ⎜ ⎟ ∗ ˆ ⎟ model we obtain that: A = A0 ⊗ A1 = ⎜ ⎜ Ca ε ε ε Cd ⎟ ⎝ Ca 0 ε Cd ε ⎠ Ca ε ε ε Cd Therefore, λ(A) = max

p∈C(A)

|p|w |p|1

= max{Ca, Cd} = Ca. This means that in order

for the T P N to be stable and work properly the speed at which the parallel computer processing system works has to be equal to Ca or being more precise, that all the transitions must fire at the same speed as the problems arrive i.e., they have to be solved as soon as they are loaded into the computer which is attained by setting k = Ca. In particular, transitions s1 and s2 which are related to the execution time of the CPU’s have to be fired at a speed equal to Ca/2.

14

Z.R. Konigsberg

Fig. 2.

Remark 3. The previous analysis is easily extended to the case with n CPU’s, obtaining that u = [Ca, Ca/n, Ca/n, ..., Ca/n] which translates into the condition that the transitions s1,s2,...,sn, have to be fired at a speed equal to Ca/n. Summary 7. The parallel computer processing system works properly if transitions s1,s2,...,sn, fire at a speed equal to Ca/n which implies that the execution frequency of the CPU’s has to be equal to Ca/n. 5.3

Comparison

As a result of summaries (6) and( 7) the following facts are deduced: 1. (Saving time) It is possible to solve a problem of size Ca with one CPU which takes time ”Ca”, or there is the option of solving a problem of size nCa (or n problems of size Ca each one) using n CPU’s which will take the same time as with one CPU. 2. (Saving Money) In order to execute a program, there is the option of purchasing one CPU that costs ”Ca” or n CPU’s that cost ”Ca/n”. This is significant for large Ca. 3. (Solving larger problems) If Ca increases due to the fact that the problem to be solved becomes larger then this will result in an increment on the CPU’s execution frequency. As a consequence the serial computer processing option becomes expensive and/or slow. This is also true for the parallel computer processing alternative however, by distributing Ca between the n CPU’s the economical and/or time impact results to be much lower.

References 1. Retchkiman, Z.: Stability theory for a class of dynamical systems modeled with Petri nets. International Journal of Hybrid Systems 4(1) (2005) 2. Heidergott, B., Olsder, G.J., van der Woude, J.: Max Plus at Work. Princeton University Press, Princeton (2006) 3. Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronization and Linearity, Web-edition (2001)

Smooth Path Algorithm Based on A* in Games Xiang Xu and Kun Zou Department of Computer Engineering, University of Electronic Science and Technology of China Zhongshan Institute, Zhongshan, P.R. China [email protected]

Abstract. Pathfinding is a core component of many games especially real-time strategy games. The paper proposed a smooth path generation strategy based on the A * algorithm. Firstly, adopted an admissible heuristics function, and a key-point optimization is applied to make the path to be connected by a series of key points. Secondly, the paper applied Catmull-Rom splines to the key points interpolation, through add several interpolation points between the key points, make the whole path look more smooth. Experimental results show that the proposed smooth path generation algorithm improves the path of smoothness, make the pathfinding more realistic, and more suitable for use in games. Keywords: pathfinding, A* algorithm, heuristic function, smooth path, Catmull-Rom splines.

1 Introduction In games, we often use a regular grid diagram to demonstrate game map, these grids in certain proportion or resolution divide game map into small pieces of cells, each cell called a node. Based on grid game map, pathfinding's main purpose is according to different terrain and obstacles, find a shortest and lowest cost path. Many games use A* algorithm as its pathfinding strategy, such as typical RTS games and RPG games. Due to the characteristics of game software itself, its pathfinding algorithm has more request, such as searching time should be short, path found should smooth and realistic, etc. Therefore, the standard A* algorithm need to do many improvement before used in games. Aiming at the special request of pathfinding in games, this paper analyses various improvement method, and proposes a smooth path generation algorithm.

2 The Basics of A* Algorithm A* algorithm is a typical kind of heuristic algorithm in artificial intelligence. In order to understand the A* algorithm, we must first understand the state space search and heuristic algorithm. State space search is a problem solving process to look for one path from the initial state to the end state [1]. The popular spot said, because there are so many problem solving methods, and so many solving path caused by many branches, uncertain and incomplete in solving process. All these paths constitute a diagram, and this diagram is called state space diagram. The solution is actually to find a path from S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 15–21, 2011. © Springer-Verlag Berlin Heidelberg 2011

16

X. Xu and K. Zou

the starting point to the goal in this diagram. This search process is called state space search. Common state space search methods have depth first search (DFS) and breadth first search (BFS), BFS first searches the initial state layer, and then next layer until find the goal so far. DFS is according to certain order first searches a branch, and then another branch until find the goal. Breadth and depth first search have a big flaw is that they are both in a given state space exhaustion, it can be adopted when the state space is small, but will not desirable when the state space is very big, and unpredictable circumstances. They will have much lower efficiency, and even not complete pathfinding, In this case we should use heuristic pathfinding. Heuristic pathfinding will estimate each search position in the state space, and get the lowest cost of position and from this position until target. This may omit many useless path searches, improve efficiency. In heuristic pathfinding, the estimate cost of position is very important, and use different heuristic function will get different effect [2]. The heuristic function’s general form is as follows: f(n)=g(n)+h(n)

(1)

Among them, g (n) is the actual cost from the starting point to node n, and h (n) is the estimate cost from node n to the goal. Because g(n) is known, it can be calculated by reverse tracking from node n to starting point in accordance with a pointer to the parent, then accumulate all the edge cost in the path. So, heuristic function f (n)’s heuristic information relies mainly on function h (n) [3]. According to a certain known conditions of state space, heuristic function will select the node with minimum cost to search, again from this node continue to search, until you reach the goal or failure, but not expanded nodes need not search. Design of function h (n) will direct impact on whether this heuristic algorithm can become A* algorithm [4].

3 Smooth Path Design Design a path in games is more than just the application of pathfinding algorithm. It also includes several techniques for achieving more realistic looking results from pathfinding. We can use the following three main methods to improve the pathfinding algorithm: make more straight-line movement and make more smoothly movement and make more directly movement, all these optimizations bring more joyful gaming experience for gamers. And these optimizations also directly affect the implement of A* algorithm. 3.1 The Selection of Heuristic Function Function h (n) in A* algorithm usually adopts the classic Manhattan heuristic function. Namely obtain the minus of abscissa from current node to the goal, and also the minus of ordinate from current node to the goal, again both absolute values adding together. Its primary drawback is that in an eight-way pathfinder, this method is inadmissible, so it is not guaranteed to find the shortest path. Also, because it is overweighted compared to G, it will overpower any small modifiers that you try to add to the calculation of G like, say, a turning penalty or an influence map modifier. So we need to find more suitable heuristic function. And when choosing heuristic function, we still need to

Smooth Path Algorithm Based on A* in Games

17

consider calculated amount. Therefore we should take the compromise in the precise function and its calculated amount. Here we adopted an improved heuristic function shown below: h(n)=max(fabs(dest.x-current.x),fabs(dest.y-current.y))

(2)

This heuristic function can satisfy the admissible condition and guarantee to give us the shortest path from starting point to the goal. In order to improve search efficiency, we preprocess those unreachable areas in the game map. These areas possibly are separated by a bar obstacle, for example river in addition one side, also possibly be surrounded by walls, etc. For such a terrain, A* algorithm will detect all neighbour nodes around this unreachable node until failed, waste a lot of time. Through put all unreachable nodes into the unreachable list beforehand, and check whether destination is in the unreachable list before pathfinding. And for the Open table, we adopted the binary heaps to enhance the efficiency of the algorithm. The specific algorithm flow chart shown below:

Fig. 1. A* algorithm flow chart, adopted the binary heaps for the Open table, and preprocess those unreachable areas in the game map before starting searching

18

X. Xu and K. Zou

3.2 Key-Point Optimization The path generated by A* algorithm is a set of discrete node {N1 N2, N3,..., , NP-1, NP}, if a game role move along these nodes, it will encounter many twists, the path is too long, and quite time-consuming, we can adopt key-point optimization strategy, namely select limited key points to represent the whole path node set. There are many methods to select the key points, we can select each direction change point as key point, and other nodes are all omitted. Also we can calculate the original path node set{N1 N2, N3,..., , NP-1, NP}, if in any two node central segment does not have any obstacles (assume the grid map is known), then all other nodes between this two nodes, can be omitted but only this two nodes are preserved. After such processing, the path generated by A* algorithm is constituted by limited key points, the path length reduced many, the track time reduced suddenly, and convenient for further smooth path design (The experimental results is shown below) [5].

Fig. 2. Left figure shows A* algorithm generated the node sequence, and right figure shows the key-point node sequence after optimization

3.3 The Generation of Smooth Path After finished the key-point optimization, we can apply Catmull-Rom splines to the key-point interpolation, through add several interpolation points between the key points, make the whole path look more smooth. Catmull-Rom splines are a family of cubic interpolating splines formulated such that the tangent at each point Pi is calculated using the previous and next point on the spline, (Pi+1 −Pi−1). The geometry matrix is given by

p ( s ) = [1 u

u2

⎡ 0 ⎢− τ u 3 ]⎢ ⎢ 2τ ⎢ ⎣− τ

1 0

0

τ τ − 3 3 − 2τ 2 −τ τ − 2

0 ⎤ ⎡ pi−2 ⎤ 0 ⎥⎥ ⎢⎢ p i −1 ⎥⎥ − τ ⎥ ⎢ pi ⎥ ⎥ ⎥⎢ τ ⎦ ⎣ p i +1 ⎦

(3)

Catmull-Rom splines have C1 continuity, local control, and interpolation, but do not lie within the convex hull of their control points.

Smooth Path Algorithm Based on A* in Games

19

Fig. 3. A Catmull-Rom spline

Fig. 4. The effect of τ

Note that the tangent at point p0 is not clearly defined; oftentimes we set this to τ (p1 − p0) although this is not necessary for the assignment (you can just assume the curve does not interpolate its endpoints). The parameterτis known as “tension” and it affects how sharply the curve bends at the (interpolated) control points (figure 4). It is often set to 0.5 but you can use any reasonable value for this assignment. Catmull-Rom formulas required four input point coordinates, the calculation results is the point for the second point to the third point between an approximate u% places. When τ= 0.5, computation formula is as follows [6]: p ( s ) = p i − 2 * ( − 0 .5 * u + u * u − 0 .5 * u * u * u ) + p i −1 * (1 − 2 . 5 * u * u + 1 . 5 * u * u * u ) + p i * ( 0 .5 * u + 2 * u * u − 1 .5 * u * u * u )

(4)

+ p i +1 * ( − 0 . 5 * u * u + 0 . 5 * u * u * u )

Note that if u=0, the result is Pi-1; u=100, results of Pi. Smooth processing also need time consume, in order to reduce the processing time, we should as less as possible insert intermediate points under the quite satisfactory processing effect premise, this paper chooses three insert points. Respectively the choice u=0.25, 0.5 and 0.75, and produce three equal-distance points. So, every n key points, after the interpolation increased 3*(n-1) points, after smooth processing the nodes number is original (4*n-3)/n times. Because Catmull-Rom splines can only generate interpolation points between the second and the third point, so aiming at the first two key points and the last two key points, we need separately using the key point sequence (1, 1, 2, 3) and (n-2, n-1, n, n), but the other key point sequence by simply use (s-1, s, s+1, s+2). This can obtain the uniform distribution of interpolation points, specific effects are shown below.

20

X. Xu and K. Zou

Fig. 5. Catmull-Rom smooth path production. Left figure shows the key points (marked as red circle), right figure shows the smooth path.

4 Other Problems to Be Considered There are many other problems need to consider in game pathfinding [7], For instance: (1) The terrain problems. The path from A to B has two choices: one path is Steep mountain road but the distance is near, second path is smooth road but the distance is far. Sometimes we will choose the latter, but A* algorithm can only select the former. General method to solve this kind of problem is to give different terrain with different influence factor, but in order to make the choice more randomness, we can set a tolerable difference value like C, when the differentials with the mountain and the smooth road cost does not exceed C, we can allow random selection of one of the paths. For example, when the mountain road is not very steep, we can also select this path. (2) The traffic problems. For instance in a relatively narrow terrain, there is a large group of game roles requires passed by, can produce traffic congestion. It's probably best to let later roles know this situation, and dynamic increase the terrain cost, so that they can change their route. (3) March algorithm. In a real-time strategy game, if we need simultaneously to move many game roles, we will call A* algorithm to calculate many times, the time-consuming unbearable. The solution has two kinds: a kind of simple method is to all the cells pressed into the queue, carries on the pathfinding in order. This can quickly get response, but each unit can not move together, some roles can first move, behind of slowly catch up with. Another common way is in the army, random select a role as a guide, just once pathfinding, other roles as long as the tailgate afterward can. (4) Other moving units. Due to the map still exists other moving units, their positions are not fixed, so it is very difficult to calculate one path that can avoid other units at the beginning. The most simple and feasible method is another collision detection, after the collision to search path or direct selection to the right or left movement. (5) The number of pathfinders. In many real-time strategy games, also exist such problems: need to allow multiple units to move together. In order to reduce pathfinding time, must reduce the number of pathfinders, namely a March unit's population upper limit.

Smooth Path Algorithm Based on A* in Games

21

5 Conclusion This paper analyzed the A* standard algorithm, and proposed one kind of improved strategy. The algorithm adopted an admissible heuristics function, and performed key-point optimization on the node sequence, in view of the key-point sequence, the paper realized a kind of smooth path generation based on the Catmull-Rom splines. Generated by the search path can be reflected in the game actual path effect, and embodies the certain intelligence and humanization. But in execution efficiency, take sacrifices the storage space and the CPU time as the price. Future game need more intelligences, more user-friendly game roles, therefore, hoped that can have better algorithms to solve problems in game pathfinding.

References 1. Tao, Z.H., Hang, C.Y.: Path Finding Using A* Algorithm. Micro Computer Information 23(17), 238–240 (2007) 2. Lester, P.: A* pathfinding for beginners (2005), http://www.policyalmanac.org/games/aStarTutoria.lhtm 3. Heping, C., Qianshao, Z.: Applicaion And Implementation of A*Agorithms in the Game Map Pathfinding. Computer Applications and Software 22(12), 118–120 (2005) 4. Lester, P.: Using Binary Heaps in A* Pathfinding (2003), http://www.policyalmanac.org/games/binaryHeaps.htm 5. Wei, S., Zhengda, M.: Smooth path design for mobile service robots based on improved A* algorithm. Journal of Southeast University (Natural Science Edition) 40sup(I) (September 2010) 6. Deloura, M.: Game Programming Gems. Charles River Media, Inc., London (2000) 7. Higgins Daniel, F.: Pathfinding Design Architecture. In: AI Game Programming Wisdom. Charles River Media, London (2002)

The Features of Biorthogonal Binary Poly-scale Wavelet Packs in Bidimensional Function Space Zhihao Tang* and Honglin Guo Department of Fundamentals, Henan Polytechnic Institute, Nanyang 473009, China [email protected]

Abstract. Wavelet analysis has become a developing branch of mathematics for over twenty years. In this paper, the notion of orthogonal nonseparable bivariate wavelet packs, which is the generalization of orthogonal univariate wavelet packs, is proposed by virtue of analogy method and iteration method. Their biorthogonality traits are researched by using time-frequency analysis approach and variable separation approach. Three orthogonality formulas regarding these wavelet wraps are obtained. Moreover, it is shown how to draw new orthonormal bases of space L2 ( R 2 ) from these wavelet wraps. A procedure for designing a class of orthogonal vector-valued finitely supported wavelet functions is proposed by virtue of filter bank theory and matrix theory. Keywords: Nonseparable, binary wavelet packs, wavelet frame, Bessel sequence, orthonormal bases, time-frequency analysis approach.

1 Introduction and Notations The main advantage of wavelet packs is their time-frequency localization property. Construction of wavelet bases is an important aspect of wavelet analysis, and multiresolution analysis method is one of importment ways of constructing various wavelet bases. There exist many kinds of scalar scaling functions and scalar wavelet functions. Although the Fourier transform has been a major tool in analysis for over a century, it has a serious laking for signal analysis in that it hides in its phases information concerning the moment of emission and duration of a signal. Wavelet analysis [1] has been developed a new branch for over twenty years. Its applications involve in many areas in natural science and engineering technology. The main advantage of wavelets is their time-frequency localization property. Many signals in areas like music, speech, images, and video images can be efficiently represented by wavelets that are translations and dilations of a single function called mother wavelet with bandpass property. Wavelet packets, owing to their good properties, have attracted considerable attention. They can be widely applied in science and engineering [2,3]. Coifman R. R. and Meyer Y. firstly introduced the notion for orthogonal wavelet packets which were used to decompose wavelet components. Chui C K.and Li Chun L.[4] generalized the concept of orthogonal wavelet packets to the case of non-orthogonal wavelet packets so that wavelet packets can be employed in *

Corresponding author.

S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 22–28, 2011. © Springer-Verlag Berlin Heidelberg 2011

The Features of Biorthogonal Binary Poly-scale Wavelet Packs

23

the case of the spline wavelets and so on. Tensor product multivariate wavelet packs has been constructed by Coifman and Meyer. The introduction for the notion on nontensor product wavelet packs attributes to Shen Z [5]. Since the majority of information is multidimensional information, many researchers interest themselves in the investigation into multivariate wavelet theory. But, there exist a lot of obvious defects in this method, such as, scarcity of designing freedom. Therefore, it is significant to investigate nonseparable multivariate wavelet theory. Nowadays, since there is little literature on biorthogonal wavelet wraps, it is neces-sary to investigate biorthogonal wavelet wraps . In the following, we introduce some notations. Z and Z + denote all integers and all nonnegative integers, respectively. R denotes all real numbers. R 2 denotes the 2dimentional Euclidean space. L2 ( R 2 ) denotes the square integrable function space. 2 ω − iω 2 Let x = ( x1 , x2 ) ∈ R , ω = (ω1 , ω2 ) ∈ R 2 , k = ( k , k ) ∈ Z , z = e , z2 = e 2 . The inner product for any functions ( x ) and ( x ) ( ( x ), ( x) ∈ L2 ( R 2 )) and the Fourier transform of ( x ) are defined, respectively, by −i

2

1

=∫

,

( x) ( x ) dx,

R2

2

2

1

(ω ) = ∫

R2

( x) e − iω ⋅ x dx,

where ω ⋅ x = ω1 x1 + ω2 x2 and ( x) denotes the complex conjugate of ( x ) . Let R and C be all real and all complex numbers, respectively. Z and N denote, respectively, all integers and all positive integers. Set Z + = {0} ∪ N , a, s ∈ N as well as a ≥ 2 By 2 algebra theory, it is obviously follows that there are a elements d 0 , d1 , , d a 2 −1 in 2 Z + = {( n1 , n2 ) : n1 , n2 ∈ Z + } such that Z 2 = ∪ d ∈Ω ( d + mZ 2 ) ; (d1 + mZ 2 ) ∩ (d2 + mZ 2 ) = φ , where Ω 0 = {d 0 , d1 , , d a 2 −1} denotes the aggregate of all the different representative elements in the quotient group Z 2 /(mZ 2 ) and order d 0 = {0} where {0} is the null element of Z and d1 , d 2 denote two arbitrary distinct elements in Ω0 .Let Ω = Ω 0 − {0} and Ω, Ω 0 to be two index sets. Define, By L2 ( R 2 , C s ) , we denote the set of all vector-valued functions L2 ( R 2 , C s ) := { ( x) , = (h1 ( x)), h2 ( x), , hu ( x))T : hl ( x) ∈ L2 ( R 2 ), l = 1, 2, , s} ,where T means the transpo2 2 s -se of a vector. For any ∈ L ( R , C ) its integration is defined as follows T ∫ R2 ( x)dx = (∫ R2 h1 ( x)dx, ∫ R2 h2 ( x)dx, , ∫ R2 hs ( x)dx) . 0

2

+

Definition 1. A sequence {

n

( y ) n∈Z 2 ⊂ L2 ( R 2 , C s )} is called an orthogonal set, if

〈

n

,

v

〉 = δ n ,v I s , n, v ∈ Z 2 ,

(1)

where I s stands for the s × s identity matrix and δ n , v , is generalized Kronecker symbol, i.e., δ n ,v = 1 as n = v and δ n ,v = 0 , otherwise.

2 The Bivariate Multiresolution Analysis 2

2

Firstly, we introduce multiresolution analysis of space L ( R ). Wavelets can be constructed by means of multiresolution analysis. In particular, the existence

24

Z. Tang and H. Guo

theorem[8] for higher-dimentional wavelets with arbitrary dilation matrice has been given. Let h( x) ∈ L ( R ) satisfy the following refinement equation: 2

2

f ( x) = m2 ⋅ ∑ k∈Z 2 bk f (mx − k )

(2)

where {b(n)}n∈Z 2 is real number sequence which has only finite terms.and f ( x) is called scaling function. Formula (1) is said to be two-scale refinement equation. The frequency form of formula (1) can be written as

f (ω ) = B ( z1 , z2 ) f (ω m),

(3)

where

B( z1 , z2 ) =

∑ b(n , n ) ⋅ z1n ⋅ z2 n . 1

1

( n1 , n2 )∈Z

Define a subspace

2

(4)

2

2

X j ⊂ L2 ( R 2 ) ( j ∈ Z ) by V j = closL2 ( R2 ) m j f ( m j x − k ) : k ∈ Z 2 .

(5)

Definition 2. We say that f ( x) in (2) generate a multiresolution analysis {V j } j∈Z of

L2 ( R 2 ) , if the sequence {V j } j∈Z defined in (4) satisfy the following properties: (i) V j ⊂ V j +1 , ∀ j ∈ Z ; (ii) ∩ V j = {0}; ∪ V j is dense in L2 ( R 2 ) ; (iii) ψ ( x) ∈ V j∈Z j∈Z ⇔ ψ (mx) ∈ Vk +1 , ∀ k ∈ Z (iv) the family { f (m j x − n) : n ∈ Z 2 } forms a Riesz basis for the spaces V j . Let

Yk (k ∈ Z ) denote the complementary subspace of V j in V j +1 , and ass-ume

that there exist a vector-valued function G ( x) = {g1 ( x), g 2 ( x), a Riesz basis for

, g m2 −1 ( x)} constitutes

Yk , i.e., Y j = closL2 ( R2 ) g λ: j , n : λ = 1, 2,

, m 2 − 1; n ∈ Z 2 ,

where j ∈ Z , and g λ : j , k ( x) = m j / 2 g λ ( m j x − k ), λ = 1, 2, dition (5), it is obvious that g1 ( x ), g 2 ( x ), (λ ) n

(6)

, m 2 − 1; k ∈ Z 2 . Form con-

, g m2 −1 ( x ) are in Y0 ⊂ X 1. Hence

there exist three real number sequences {q }(λ ∈ Δ = {1, 2,

, m 2 − 1}, n ∈ Z 2 ) such

that

gλ ( x) = m 2 ⋅ ∑ qk( λ ) f (mx − k ),

(7)

k∈Z 2

Formula (7) in frequency domain can be written as

gλ (ω ) = Q ( λ ) ( z1 , z2 ) f (ω a ), λ = 1, 2,

, m 2 − 1.

(8)

The Features of Biorthogonal Binary Poly-scale Wavelet Packs

where the signal of sequence {qk( λ ) }(λ = 1, 2,

∑

Q ( λ ) ( z1 , z2 ) =

25

, m 2 − 1, k ∈ Z 2 ) is

q((nλ ),n ) ⋅ z1 ⋅ z2 . n1

( n1 , n2 )∈Z

1

n2

(9)

2

2

A bivariate function f ( x ) ∈ L (R ) is called a semiorthogonal one, if 2

2

f (⋅), f (⋅ − k ) = δ 0, k , n ∈ Z 2 .

(10)

We say G ( x ) = {g1 ( x), g 2 ( x), , g m2 −1 ( x)} is anorthogonal bivariate vectorvalued wavelets associated with the scaling function f ( x ) , if they satisfy:

f (⋅), gν (⋅ − k ) = 0 , ν ∈ Δ, k ∈ Z 2 ,

(11)

g λ (⋅), gν (⋅ − n) = δ λ ,ν δ 0,n , λ , ν ∈ Δ, n ∈ Z 2

(12)

3 The Traits of Nonseparable Bivariate Wavelet Packs To construct wavelet packs, we introduce the following notation: a = 3, h0 ( x ) = f ( x), hν ( x ) = gν ( x), b( 0) (n) = b(n), b(ν ) (n) = q (ν ) (n), whereν ∈ Δ We are now in a position of introducing orthogonal bivariate nonseparable wavelet wraps. Definition 3. A family of functions { hmk +ν ( x) : n = 0,1, 2,

3, ⋅⋅⋅, ν ∈ Δ } is called a

nonseparable bivariate wavelet packs with respect to an orthogonal scaling function Λ 0 ( x) , where Λ mk +ν ( x) = ∑ n∈Z 2 b(ν ) (n) Λ k (mx − n),

(13)

whereν = 0,1, 2, 3. By taaking the Fourier transform for the both sides of (12), we have h nk +ν (ω ) = B (ν ) ( z1 , z2 ) ⋅ h k (ω 2 ) .

(14)

where B (ν ) ( z1 , z2 ) = B (ν ) (ω / 2 ) = Lemma 1. [6]. Let φ ( x ) ∈ L2 (R 2 ). Then

φ ( x)

∑ | φ ( ω + 2k π ) |

2

∑ bν

( )

k ∈Z

(k ) z1k1 z2k2

(15)

2

is an orthogonal one if and only if

=1 .

(16)

k∈Z2

Lemma 2. Assuming that f ( x ) is an semiorthogonal scaling function. B ( z1 , z2 ) is the symbol of the sequence {b( k )} defined in (3). Then we have

Π = B ( z1 , z 2 ) + B (− z1 , z 2 ) + B ( z1 , − z 2 ) + B ( − z1 , − z 2 ) 2

2

2

2

(17)

26

Z. Tang and H. Guo

∑

f ( x) is an orthogonal bivariate function, then

Proof. If

2

k∈Z

2

f (ω + 2kπ ) =1 .

Therefore, by Lemma 1 and formula (2), we obtain that

∑ | B (e

1=

− i (ω1 2+ k1π )

, e − i ( ω2

2 + k 2π )

) ⋅ f ((ω1 , ω2 ) 2 + (k1 , k2 )π ) |2

k∈Z 2

=| B( z1 , z2 ) ∑ k ∈Z 2 f (ω + 2kπ ) |2 + | B( − z1 , z2 ) ⋅ ∑ k ∈Z 2 f (ω + 2kπ + (1,0)π ) |2 + | B( z1 , − z2 ) ⋅ ∑ k ∈Z 2 f (ω + 2kπ + (0,1)π ) |2 + | B(− z1 , − z2 ) ⋅ ∑ k ∈Z 2 f (ω + 2kπ + (1,1)π ) |2

= B ( z1 , z2 ) + B ( − z1 , z2 ) + B ( z1 , − z2 ) + B ( − z1 , − z2 ) 2

2

2

2

This complete the proof of Lemma 2. Similarly, we can obtain Lemma 3 from (3), (8), (13). Lemma 3. If ψ ν ( x ) (ν

= 0,1, 2,3 ) are orthogonal wavelet functions associated with

h( x) . Then we have

∑

{ B (( −1) z1 , ( −1) z 2 ) B (( −1) z1 , ( −1) z 2 ) + B (λ )

1 j =0

j

(ν )

j

(ν )

⋅ B (( −1)

j +1

j

j

z1 , ( −1) z2 )}: = Ξ λ , μ j

(λ )

(( −1)

j +1

z1 , ( −1) z 2 ) j

= δ λ ,ν , λ ,ν ∈ {0,1, 2, 3}.

(18)

n ∈ Z + , expand it by

For an arbitrary positive integer

n = ∑ j =1ν j 4 j −1 , ν j ∈ Δ = {0,1, 2,3} . ∞

(19)

n ∈ Z + and n be expanded as (17). Then we have

Lemma 4. Let

∞

h n (ω ) = ∏ B j =1

(ν j )

(e −iω1 / 2 , e− iω12 / 2 )h 0 ( 0 ) . j

j

Lemma 4 can be inductively proved from formulas (14) and (18). Theorem 1. For

n ∈ Z + , k ∈ Z 3 , we have hn (⋅), hn (⋅ − k ) = δ 0,k .

(20)

Proof. Formula (20) follows from (10) as n=0. Assume formula (20) holds for the

0 ≤ n < 4r0 ( r0 is a positive integer). Consider the case of 4 r0 ≤ n < 4r0 +1 . For ν ∈ Δ , by induction assumption and Lemma 1, Lemma 3 and Lemma 4, we have case of

( 2π ) =

2

2

hn (⋅), hn (⋅ − k ) = ∫ 2 h n (ω ) ⋅ exp {ikω} dω R

4 π ( j1 +1) 4 π ( j2 +1)

∑ ∫ j∈Z

2

4π

= ∫0

∫

4π j1

∫

4π

0

B

B (ν ) ( z1 , z2 ) ⋅ h[ n / 8] (ω / 2) ⋅ eikω dω 2

4 π j2

(ν )

(z , z ) ∑| h 2

1

2

j ∈Z

2

[n / 8]

(

ω 2

+ 2π j ) | ⋅ e d ω 2

ik ω

The Features of Biorthogonal Binary Poly-scale Wavelet Packs

=

4π

∫ ∫ 0

4π 0

2

B (ν ) ( z1 , z 2 , z 3 ) ⋅ e ik ω d ω =

2π

∫ ∫ 0

2π

0

27

Π ⋅ e ik ω d ω = δ o , k

Thus, we complete the proof of theorem 1. Theorem 2. For every

k ∈ Z 2 and m, n ∈ Z + , we have

hm (⋅), hn (⋅ − k ) = δ m ,nδ 0,k .

(21)

，

m = n (20) follows from Theorem 1.As m ≠ n and m, n ∈ Ω0 , the result (20) can be established from Theorem 2, where Ω0 = {0,1, 2,3} . In what follows, assuming that m is not equal to n and at least one of {m, n} doesn’t belong to Ω0 , rewrite m , n as m = 4 m1 +λ1 , n = 4n1 + μ1 , where m1 , n1 ∈ Z + , and λ1 , μ1 ∈ Ω 0 . Proof. For the case of

Case 1. If

m1 = n1 , then λ1 ≠ μ1 . By (17), formulas (21) follows, since

(2π )2 hm (⋅), hn (⋅ − k ) = ∫ R2 h4 m1 + λ1 (ω )h 4 n1 + μ1 (ω ) ⋅ exp{ikω}d ω

= ∫ [0,4π ]2 B ( λ1 ) ( z1 , z2 )

1 = (2π )2

∫

[0,2π ]2

∑ hm (ω 1

s∈Z

2 + 2sπ ) ⋅ h m1 (ω 2 + 2sπ ) B ( μ1 ) ( z1 , z2 ) ⋅ eikω d ω

2

Ξ λ1 , μ1 ⋅ exp{ikω} dω = O.

Case 2. If m1 ≠ n1 we order m1 = 4m2 + λ2 , n1 = 4n2 + μ 2 , where m2 , n2 ∈ Z + , and λ2 , μ2 ∈ Ω 0 . If m2 = n2 , then λ2 ≠ μ 2 . Similar to Case 1, we have (21) follows. That is to say, the proposition follows in such case. As m2 ≠ n2 , we order m2 = 2m3 + λ3 , n2 = 2n3 + μ3 , once more, where m3 , n3 ∈ Z + , and λ3 , μ3 ∈ Ω0 . Thus, after taking finite steps (denoted by r ), we obtain mr , nr ∈ Ω0 , and λr , μ r ∈ Ω0 . If α r = β r , then λr ≠ μ r . Similar to Case 1, (21) holds. If α r ≠ β r , Similar to Lemma 1, we conclude that

1 ikω hm (⋅), hn (⋅ − k ) = dω 2 h 4 m1 + λ1 (ω ) h 4 n1 + μ1 (ω ) ⋅ e 2 r (2π ) ∫ R r 1 = { B ( λι ) (ω / 2ι )} ⋅ O ⋅{∏ B ( μι ) (ω / 2ι )} ⋅ eikω d ω = O. 2 ∫ [0,2r +1 π ]2 ∏ (2π ) ι =1 ι =1 Theorem 3. If {Gβ ( x), β ∈ Z +2 } and {Gβ ( x), β ∈ Z +2 } are vector-valued wavelet packs with respect to a pair of biorthogonal vector-valued scaling functions G0 ( x) and G0 ( x) , then for any α , σ ∈ Z +2 , we have

Gα (⋅), Gσ (⋅ − k ) = δα ,σ δ 0,k I s , k ∈ Z 2 .

(22)

28

Z. Tang and H. Guo

References 1. Telesca, L., et al.: Multiresolution wavelet analysis of earthquakes. Chaos, Solitons & Fractals 22(3), 741–748 (2004) 2. Iovane, G., Giordano, P.: Wavelet and multiresolution analysis: Nature of ε∞  Cantorian space-time. Chaos, Solitons & Fractals 32(4), 896–910 (2007) 3. Zhang, N., Wu, X.: Lossless Compression of Color Mosaic Images. IEEE Trans. Image Processing 15(16), 1379–1388 (2006) 4. Chen, Q., et al.: A study on compactly supported orthogonal vector-valued wavelets and wavelet packets. Chaos, Solitons & Fractals 31(4), 1024–1034 (2007) s 5. Shen, Z.: Nontensor product wavelet packets in L2 ( R ) . SIAM Math. Anal.~26(4), 1061--1074 (1995)

6. Chen, Q., Qu, X.: Characteristics of a class of vector-valued nonseparable higherdimensional wavelet packet bases. Chaos, Solitons & Fractals 41(4), 1676–1683 (2009) 7. Chen, Q., Qu, X.: Characteristics of a class of vector-valued nonseparable higherdimensional wavelet packet bases. Chaos, Solitons & Fractals 41(4), 1676–1683 (2009) 8. Chen, Q., Huo, A.: The research of a class of biorthogonal compactly supported vectorvalued wavelets. Chaos, Solitons & Fractals 41(2), 951–961 (2009)

The Traits of Dual Multiple Ternary Fuzzy Frames of Translates with Ternary Scaling Functions* ShuKe Zhou1,** and Qingjiang Chen2 1

Dept. of Math. & Physics, Henan University of Urban Construction, Pingdingshan 467036 2 School of Science, Xi'an University of Architecture & Technology, Xi’an 710055, P.R. China [email protected]

Abstract. The rise of frame theory in appled mathematics is due to the flexibility and redundancy of frames. Structured frames are much easier to construct than Structured orthonormal bases. In this work, the notion of the ternary generalized multiresolution structure (TGMS) of subspace L2 ( R 3 ) is proposed, which is the generalization of the ternary frame multiresolution analysis. The biorthogonality character is characterized by virtue of iteration method and variable separation approach. The biorthogonality formulas concerning these wavelet packages are established.The construction of a TGMS of Paley-Wiener subspace of L2 ( R 3 ) is studied. The pyramid decomposition scheme is obtained based on such a TGMS and a sufficient condition for its existence is provided. A procedure for designing a class of orthogonal vector-valued finitely supported wavelet functions is proposed by virtue of multiresolution analysis method. Keywords: Dual pseudoframes, trivariate wavelet transform, Haar wavelet, Bessel sequence, dual ternary fuzzy frames, time-frequency representation.

1 Introduction Every frame(or Bessel sequence) determines an synthesis operator, the range of which is important for a lumber of applications. The main advantage of wavelet function is their time-frequency localization property. Construction of wavelet functions is an important aspect of wavelet analysis, and mul-tiresolution analysis approach is one of importment ways of designing all sorts of wavelet functions. There exist a great many kinds of scalar scaling functions and scalar wavelet functions. Although the Fourier transform has been a major tool in analysis for over a century, it has a serious laking for signal analysis in that it hides in its phases information concerning the moment of emission and duration of a signal. The frame theory has been one of powerful tools for researching into wavelets. Duffin and Schaeffer introduced the notion of frames *

**

Foundation item: The research is supported by Natural Scientific Foundation of Sh-aanxi Province (Grant No:2009J M1002), and by the Science Research Foundation of Education Department of Shaanxi Provincial Government (Grant No:11JK0468). Corresponding author.

S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 29–35, 2011. © Springer-Verlag Berlin Heidelberg 2011

30

S. Zhou and Q. Chen

for a separable Hilbert space in 1952. Later, Daubechies, Grossmann, Meyer, Benedetto, and Ron revived the study of frames in[1,2], and since then, frames have become the focus of active research, both in theory and in applications, such as signal processing, image processing and sampling theory. The rise of frame theory in applied mathematics is due to the flexibility and redundancy of is due to the flexibility and redundancy of frames, where robust-ness, error tolerance and noise suppression play a vital role [3,4]. The concept of frame multiresolution analysis (FMRA) as described in [2] generalizes the notion of MRA by allowing non-exact affine frames. Inspired by [2] and [5], we introduce the notion of a trivariate generalized multiresolution structure(TGMS) of L2 ( R 3 ) , which has a pyramid decomposition scheme. It also lead to new constructions of affine frames of L2 ( R 3 ) . Sampling theorems play a basic role in digital signal processing. They ensure that continuous signals can be represented and processed by their discrete samples. The classical Shannon Sampling Theorem asserts that band-limited signals can beexactly represented by their uniform samples as long as the sampling rate is not less than the Nyquist rate. Wavelet wraps, owing to their good properties, have attracted considerable attention. They can be widely applied in science and engineering. Since the majority of information is multidimensional information, many researchers interest themselves in the investigation into multivariate wavelet theory. But, there exist a lot of obvious defects in this method, such as, scarcity of designing freedom. Therefore, it is significant to study nonseparable multivariate wavelet theory. Nowadays, since there is little literature on biorthogonal wavelet wraps, it is necessary to research biorthogonal wavelet wraps.

2 Notations and the Dual Ternary Pseudoframes of Translate We start from some notations. Z and Z + denote all integers and all nonnegative integers, respectively. R denotes all real numbers. R3 denotes the 3-dimentional Euclidean space. Let L2 ( R 3 ) be the square integrable function space on R 3 . Set and Z 3 = {( z1 , z2 , z3 ) : zr ∈ Z , r = 1, 2,3}, Z +3 = {{( z1 , z2 , z3 ) : : zr ∈ z+ , r = 1, 2,3}. Let U be a separable Hilbert space and Ω is an index set. We say that a sequence {λν }v∈Ω ⊆ U is a frame for U if there exist positive real constrants L1 , L2 such that

∀

∈ U , L1

2

≤ ∑ v∈Ω |

, λv | ≤ L2 2

2

,

(1)

A family {λν }v∈Ω ⊆ U is a Bessel sequence if (only) the upper inequality of (1) holds. If only for all ∈ Γ ⊂ U , the upper inequality of (1) follows, the sequence {λν }v∈Ω ⊆ U is a Bessel sequence with respect to (w.r.t.) Γ . If a family {λv } ⊂ U is * a frame for U , there exists a dual frame {λv } such that

∀ β ∈U

， β = ∑ β,λ v∈Ω

v

λv* = ∑ β , λv* λv . v∈Ω

(2)

The Traits of Dual Multiple Ternary Fuzzy Frames of Translates

31

For a sequence c = {c(v)} ∈ 2 ( Z ) , we define its discrete-time Fourier transform as the function in space L2 (0,1)3 as follows Fc ( γ ) = C ( γ ) = ∑ u∈Z 3 c ( u )e −2π iuγ

，

(3)

Note that the discrete-time Fourier transform is 1-periodic. Let τ v g ( x ) stand for integer translates of a function g ( x) ∈ L2 ( R 3 ) , i.e., (τ va g )( x) = g ( x − va) , and g n ,va

= 2 n g (2n x − va ) , where a is a positive real constant number. Let ϕ ( x) ∈ L ( R 2

3

)

and let V0 = span{Tv : v ∈ Z } denote a closed subspace of L ( R ) . Assume that 2

2

∧

3

Φ (γ ) := ∑ v∈Z 3 | ϕ ( γ + v ) |2 ∈ L∞ [ 0,1] . In [ 3] , the sequence {τ vϕ ( x )}v is a frame for 3

and only if there exist positive constants

L1 ≤ Φ (ω ) ≤ L2

V0 if

L1 and L2 such that

，a.e., ω ∈ [0,1] \ N = {ω ∈ [0,1] : Η(ω ) = 0} . 3

3

(4)

We begin with introducing the concept of pseudoframes of translates. Definition 1. Let {τ saφ , s ∈ Z 3 } and {τ sa φ , s ∈ Z } be two sequences in L2 ( R 3 ) . Let Y be a closed subspace of L2 ( R 3 ) . We say {τ vaφ , v ∈ Z 3 } forms an affine pseudoframe for Y with respect to {τ va φ , v ∈ Z 3 } if 3

∀ h( x) ∈ Y , h( x) = ∑ v∈Z h,τ va φ τ vaφ ( x ) 3

Define an operator K : Y →

2

( Z 3 ) by

∀ h( x) ∈ Y , and define another operator

S:

(5)

2

Kh = { h,τ vaφ } ,

(6)

( Z 2 ) → Y such that

∀ c = {c(u )} ∈ 2 ( Z 3 ) , Sc = ∑ u∈Z c(u ) τ ua φ. 3

(7)

Theorem 1. Let {τ vaφ }v∈Z 3 ⊂ L2 ( R 3 ) be a Bessel sequence with respect to the subspace Y ⊂ L2 ( R 3 ) , and {τ va φ}v∈Z 3 is a Bessel sequence in L2 ( R 3 ) . Assume that K be defined by (7), and S be defined by (8). Assume P is a projection from L2 ( R 3 ) onto Y . Then {τ vaφ }v∈Z 3 is pseudoframes of translates for the subspace Y with re-spect to {τ va φ}v∈Z 3 if and only if

KSP = P .

(8)

Proof. The convergence of all summations of (7) and (8) follows from the assumptions that the family {τ vaφ}v∈Z 3 is a Bessel sequence with respect to the subspace Y , and he family {τ va φ}v∈Z 3 is a Bessel sequence in L2 ( R 3 ) with which the proof of the theorem is direct forward.

32

S. Zhou and Q. Chen

We say that a trivariate generalized multiresolution structure (TGMS){Vn , f ( x), f ( x)} of L2 ( R 3 ) is a sequence of closed linear subspaces {Vn }n∈Z of L2 ( R 3 ) and two

Vn ⊂ Vn +1 , n ∈ Z ; (ii) ∩ V = {0} ; ∪ V is dense in L ( R ) ; (iii) g ( x) ∈ Vn if and only if g (2 x) ∈ Vn+1 ∀n ∈ Z . (iv) ϕ ( x ) ∈ V0 implies τ vaϕ ( x ) ∈ V0 , for v ∈ Z ; (v) {τ vaφ }v∈Z forms pseud-oframes of elements f ( x) , f ( x) ∈ L2 ( R 3 ) such that (i) 2

n∈ Z

n∈ Z

n

3

n

3

translates for with respect to {τ va φ}v∈Z 3 .

Proposition 1[3]. Let f ( x) ∈ L2 ( R 3 ) satisfy | f | a.e. on a connected neighbourhood of 0 in [− 12 , 21 )3 , and | f | = 0 a.e. otherwise. Define Λ ≡ {ω ∈ R 2 : | f (ω ) | ≥ C > 0}, and ∧ V0 = PWΛ = {φ ∈ L2 ( R 3 ) : supp(φ ) ⊆ Λ} . Then for arbitraryfun. f ( x) ∈ L2 ( R 3 ) , {τ v f : v ∈ Z 3 } is pseudoframes of trananslates for V0 with respect to {τ v f : v ∈ Z 3 } if and only if ∧

∧

f (ω ) f (ω ) χ Λ ( ω ) = χ Λ (ω ) a. e. ,

(9)

where χ Λ is the characteristic function on Λ . Moreover, if f (ω ) is the above conditions then {τ v f : v ∈ Z 3 } and {τ v f : v ∈ Z 3 } are a pair of commutative pseudoframes of translates for V0 , i. .e.

，

h ( x ) = ∑ k ∈ Z 3 h , τ k f τ k f ( x ) = ∑ k ∈Z 2 h , τ k f τ k f ( x ) .

∀ h( x) ∈ V0 ,

Proposition 2[3]. Let {τ va f }v∈Z 3 be pseudoframes of translates for

{τ va f }v∈Z 3 . Define

Vn by Vn ≡ {h( x) ∈ L2 ( R 3 ) : h( x / 2n ) ∈ V }

Then { f n , va }v∈Z 3 is an affine pseudoframe for

(10)

V0 with respect to

，n∈Z ,

(11)

Vn with respect to { f n ,va }v∈Z . 3

The filter functions associated with a TGMS are presented as follows. Define filter functions

D0 (ω )

B 0 (ω ) = ∑ s∈Z 3 b0 ( s ) e

and −2π isω

D 0 (ω )

by

D0 (ω ) = ∑ s∈Z 3 d 0 ( s ) e −2π isω

and

of the sequences d 0 = {d 0 ( s)} and d 0 = {d 0 ( s )} , resp-

ectively, wherever the sum is defined. Let

{b0 (v)} be such that D0 (0) = 2 and

B0 (ω ) ≠ 0 in a neighborhoood of 0. Assume also that D0 (ω ) ≤ 2 . Then there exists f ( x) ∈ L2 ( R 2 ) (see ref.[3]) such that f ( x) = 2 ∑ s∈Z 3 d0 ( s ) f (2 x − sa ) .

(12)

There exists a scaling relationship for f ( x) under the same conditions as that of for a sequence d 0 , i.e.,

f ( x) = 2 ∑ s∈Z 3 d 0 ( s) f (2 x − sa) .

d0

(13)

The Traits of Dual Multiple Ternary Fuzzy Frames of Translates

33

3 The Traits of Nonseparable Trivariate Wavelet Packages Denoting by G0 ( x) = F ( x ), Gμ ( x ) = Ψ μ ( x ), G0 ( x ) = F ( x ), Gμ ( x) = Ψ μ ( x), Qk(0) = Ω k , (μ) (μ) (μ ) (μ) 3 3 Qk = Bk , Qk(0) = Ω k , Qk = Bk , μ ∈ Γ, k ∈ Z , M = 4 I v . For any α ∈ Z + and the given vector-valued biorthogonal scaling functions G0 ( x) and G0 ( x) , iteratitively define, respectively,

Gα ( x) = G2σ + μ ( x) = Gα ( x) = G2σ + μ ( x) =

∑ Q μ Gσ (2 x − k ),

μ ∈ Γ0 ,

(14)

∑ Q μ Gσ (2 x − k ),

μ ∈ Γ0 .

(15)

( ) k

k ∈Z

3

( ) k

k ∈Z

3

where σ ∈ Z +3 is the unique element such that α = 4σ + μ , μ ∈ Γ 0 follows. Lemma 1[4]. Let F ( x), F ( x) ∈ L2 ( R 3 , C v ). Then they are biorthogonal if and only if

∑ Fˆ (γ + 2kπ ) Fˆ (γ + 2kπ )

k∈Z

*

= Iv .

(16)

3

Definition 2. We say that two families of vector-valued functions {G2σ + μ ( x ), σ ∈ Z +3 , μ ∈ Γ 0 } and {G2σ + μ ( x), σ ∈ Z +3 , μ ∈ Γ 0 } are vector-valued wavelet packets with respect to a pair of biorthogonal vector-valued scaling functions G0 ( x) and G0 ( x) , resp., where G2σ + μ ( x) and G2σ + μ ( x) are given by (14) and (15), respectively. Applying the Fourier transform for the both sides of (14) and (15) yields, resp.,

Gˆ 2σ + μ (2γ ) = Q ( μ ) (γ )Gˆσ (γ ), μ ∈ Γ 0 , ˆ ˆ G2σ + μ (2γ ) = Q ( μ ) (γ )Gσ (γ ),

μ ∈ Γ0 ,

(17) (18)

Lemma 2[6]. Assume that Gμ ( x), Gμ ( x) ∈ L ( R , C ), μ ∈ Γ are pairs of biorthogonal vector-valued wavelets associated with a pair of biorthogonal scaling functions G0 ( x) and G0 ( x) . Then, for μ ,ν ∈ Γ 0 , we have 2

Q μ ((γ + 2 ρπ ) / 2)Q ν ∑ ρ ( )

∈Γ0

( )

3

v

((γ + 2 ρπ ) / 2)* = δ μ ,ν I v .

(19)

Theorem 2[8]. Assume that {Gβ ( x), β ∈ Z +3 } and {Gβ ( x), β ∈ Z +3 } are vectorvalued wavelet packets with respect to a pair ofbiorthogonal vector-valued functions G0 ( x) and G0 ( x) , respectively. Then, for β ∈ Z +3 , μ , v ∈ Γ 0 , we have

[Gα (⋅), Gα (⋅ − k )] = δ 0, k I v , k ∈ Z 3 . [G2 β + μ (⋅), G2 β +v (⋅ − k )] = δ 0, k δ μ ,ν I v , k ∈ Z 3 .

(20) (21)

34

S. Zhou and Q. Chen

Theorem 3. If {Gβ ( x), β ∈ Z +3 } and {Gβ ( x), β ∈ Z +3 } are vector-valued wavelet wraps with respect to a pair of biorthogonal vector scaling functions G0 ( x) and G0 ( x) , then for any α ,σ ∈ Z +3 , we have

[Gα (⋅), Gσ (⋅ − k )] = δα ,σ δ 0, k I v , k ∈ Z 3 .

(22)

Proof. When α = σ , (22) follows by Theorem 2. as α ≠ σ and α , σ ∈ Γ 0 , it follows from Theorem 1 that ((22)) holds, too. Assuming that α is not equal to β , as well as at least one of {α ,σ } doesn’t belong to Γ 0 , we rewrite α , σ as α = 2α1 + ρ1 , σ = 2σ 1 + μ1 , where ρ1 , μ1 ∈ Γ 0 . Case 1. If α1 = σ 1 , then ρ1 ≠ μ1 . (22) follows by virtue of (17), (18) as well as Lemma 1 and Lemma 2, i.e.,

ˆ (2π )3 [Gα (⋅), Gσ (⋅ − k )] = ∫ 3 Gˆ 2α1 + ρ1 (γ )G2σ1 + μ1 (γ )* ⋅ exp{ik ⋅ γ }d γ R

=∫

[0,2 π ]3

δ ρ , μ I v ⋅ exp{ik ⋅ γ }d γ = O. 1

1

Case 2. If α1 ≠ σ 1 , order α1 = 2α 2 + ρ 2 , σ 1 = 2σ 2 + μ 2 , where α 2 , σ 2 ∈ Z +3 , and

ρ 2 , μ2 ∈ Γ0 . If α 2 = σ 2 , then ρ 2 ≠ μ2 . Similar to Case 1, (22) follows. As α 2 ≠ σ 2 , order α 2 = 2α 3 + ρ3 , σ 2 = 2σ 3 + μ3 , where α 3 , σ 3 ∈ Z +3 , ρ3 , μ3 ∈ Γ 0 . Thus, taking finite steps (denoted by κ ), we obtain α κ ∈ Γ 0 , and ρκ , μκ ∈ Γ 0 . ˆ 8π 3 [Gα (⋅), Gσ (⋅ − k )] = ∫ Gˆα (γ )Gσ (γ )* ⋅ eik ⋅γ d γ R3

=∫

R

=∫

ˆ Gˆ 2α1 + λ1 (γ )G2 β1 + μ1 (γ )* ⋅ exp{ik ⋅ γ }d γ 3 κ

κ

([0,2⋅2 π ]

3

{∏Q ( ρl ) (γ / 2l )} ⋅ O ⋅ {∏ l =1Q ( μl ) (γ / 2l )}* ⋅ exp{−ik ⋅ γ }d γ = O. κ

l =1

Therefore, for any α , σ ∈ Z +3 , result (22) is established.

4 Conclusion The construction of a TGMS of Paley-Wiener subspace of L2 ( R 3 ) is studied. The pyramid decomposition scheme is obtained based on such a TGMS. The biorthogonality formulas concerning these wa-velet packages are established.

References 1. Telesca, L., et al.: Multiresolution wavelet analysis of earthquakes. Chaos, Solitons & Fractals 22(3), 741–748 (2004) 2. Iovane, G., Giordano, P.: Wavelet and multiresolution analysis: Nature of ε ∞  Cantorian space-time. Chaos, Solitons & Fractals 32(4), 896–910 (2007)

The Traits of Dual Multiple Ternary Fuzzy Frames of Translates

35

3. Li, S., et al.: A theory of generalized multiresolution structure and pseudoframes of translates. Fourier Anal. Appl. 7(1), 23–40 (2001) 4. Chen, Q., et al.: A study on compactly supported orthogonal vector-valued wavelets and wavelet packets. Chaos, Solitons & Fractals 31(4), 1024–1034 (2007) s 5. Shen, Z.: Nontensor product wavelet packets in L2 ( R ) . SIAM Math. Anal. 26(4), 1061–1074 (1995) 6. Chen, Q., Qu, X.: Characteristics of a class of vector-valued nonseparable higherdimensional wavelet packet bases. Chaos, Solitons & Fractals 41(4), 1676–1683 (2009) 7. Chen, Q., Wei, Z.: The characteristics of orthogonal trivariate wavelet packets. Information Technology Journal 8(8), 1275–1280 (2009) 8. Chen, Q., Shi, Z.: Biorthogonal multiple vector-valued multivari-ate wavelet packets associa-ted with a dilation matrix. Chaos, Solitons & Fractals 35(3), 323–332 (2008)

The Characteristics of Multiple Affine Oblique Binary Frames of Translates with Binary Filter Banks YongGan Li* Office of Financial affairs, Henan Quality Polytechnic, Pingdingshan 467000, P.R. China [email protected]

Abstract. Frame theory has been the focus of active research for twenty years, both in theory and applications. In this paper, the notion of the bivariate generalized multiresolution structure (BGMS) of subspace L2 ( R 2 ) , which is the generalization of frame multiresolution analysis, is proposed. The biorthogonanality traits on wavelet wraps are researched by using time-frequency analysis approach and variable separation approach. The construction of a BGMS of Paley-Wiener subspace of L2 ( R 2 ) is studied. The pyramid decomposition scheme is obtained based on such a GMS and a sufficient condition for its existence is provided. A procedure for designing a class of orthogonal vectorvalued finitely supported wavelet functions is proposed by virtue of filter bank theory and matrix theory. Keywords: Affine pseudoframes, bivariate wavelet wraps, wavelet frame, Bessel sequence, orthonormal bases, time-frequency analysis approach.

1 Introduction and Notations The main advantage of wavelet function is their time-frequency localization property. Construction of wavelet functions is an important aspect of wavelet analysis, and multiresolution analysis approach is one of importment ways of designing all sorts of wavelet functions. There exist a great many kinds of scalar scaling functions and scalar wavelet functions. Although the Fourier transform has been a major tool in analysis for over a century, it has a serious laking for signal analysis in that it hides in its phases information concerning the moment of emission and duration of a signal. The frame theory has been one of powerful tools for researching into wavelets. Duffin and Schaeffer introduced the notion of frames for a separable Hilbert space in 1952. Later, Daubechies, Grossmann, Meyer, Benedetto, Ron revived the study of frames in[1,2],and since then, frames have become the focus of active research, both in theory and in applications, such as signal processing, image processing and sampling theory. The rise of frame theory in applied mathematics is due to the flexibility and redundancy of is due to the flexibility and redundancy of frames, where robustness, error toleranceand noise suppression play a vital role [3,4]. The concept of frame *

Corresponding author.

S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 36–41, 2011. © Springer-Verlag Berlin Heidelberg 2011

The Characteristics of Multiple Affine Oblique Binary Frames

37

multiresolution analysis (FMRA) as described in [2] generalizes the notion of MRA by allowing non-exact affine frames. However, subspaces at different resolutions in a FMRA are still generated by a frame formed by translates and dilates of a single function. This paper is motivated from the observation that standard methods in sampling theory provide examples of multiresolution structure which are not FMRAs. Inspired by [2] and [5], we introduce the notion of a bivariate generalized multiresolution structure(BGMS) of L2 ( R 2 ) , which has a pyramid decomposition scheme. It also lead to new constructions of affine frames of L2 ( R 2 ) . Wavelet wraps, owing to their good properties, have attracted considerable attention. They can be widely applied in science and engineering. Since the majority of information is multidimensional information, many researchers interest themselves in the investigation into multivariate wavelet theory. But, there exist a lot of obvious defects in this method, such as, scarcity of designing freedom. Therefore, it is significant to investigate nonseparable multivariate wavelet theory. Nowadays, since there is little literature on biorthogonal wavelet wraps, it is necessary to investigate biorthogonal wavelet wraps . In the following, we introduce some notations. Z and Z + denote all integers and all nonnegative integers, respectively. R denotes all real numbers. R 2 denotes the 2dimentional Euclidean space. L2 ( R 2 ) denotes the square integrable function space. 2 − iω 2 Let x = ( x1 , x2 ) ∈ R , ω = (ω1 , ω2 ) ∈ R 2 , k = ( k1 , k 2 ) ∈ Z 2 , z1 = e − iω 2 , z2 = e 2 . The inner product for any functions ( x ) and ( x ) ( ( x ), ( x) ∈ L2 ( R 2 )) and the Fourier transform of ( x ) are defined, respectively, by

,

=∫

R2

( x) ( x ) dx,

(ω ) = ∫

R2

( x) e −2π iω ⋅ x dx,

where ω ⋅ x = ω1 x1 + ω2 x2 and ( x) denotes the complex conjugate of ( x) . Let R and C be all real and all complex numbers, respectively. Z and N denote, respectively, all integers and all positive integers. Set Z + = {0} ∪ N , m, s ∈ N as well as m ≥ 2 By 2 algebra theory, it is obviously follows that there are m elements λ0 , λ1 , , λm2 −1 in 2 2 Z +2 = {(n1 , n2 ) : n1 , n2 ∈ Z + } such that Z = ∪ d ∈Ω (λ + mZ ) ; 2 2 (λ1 + mZ ) ∩ (λ2 + mZ ) = φ , where Λ 0 = {λ0 , λ1 , , λm2 −1} denotes the aggregate of all the different representative elements in the quotient group Z 2 /(mZ 2 ) and order λ = {0} where {0} is the null element of Z and λ1 , λ2 denote two arbitrary distinct elements in Λ 0 .Let Λ = Λ 0 − {0} and Λ, Λ 0 to be two index aggregates. By L2 ( R 2 , C s ) , we denote the set of vector-valued functions L2 ( R 2 , C s ) := { ( x) = (h1 ( x), h2 ( x), , hu ( x))T : hl ( x) ∈ L2 ( R 2 ), l = 1, 2, , s} ,where T means the transpose of a vector. 0

2

+

0

Definition 1. A sequence {

n

( x) n∈Z 2 ⊂ L2 ( R 2 , C s )} is called an orthogonal set, if

〈

n

,

v

〉 = δ n ,v I s , n, v ∈ Z 2 ,

(1)

where I s stands for the s × s identity matrix and δ n ,v , is generalized Kronecker symbol, i.e., δ n ,v = 1 as n = v and δ n ,v = 0 , otherwise.

38

Y. Li

Let W be a separable Hilbert space and Λ is an index set. We recall that a sequence {ξ v : v ∈ Z } ⊆ W is a frame for H if there exist positive real numbers C , D such that ∀ η ∈W , C η

2

≤ ∑ v∈Λ | η , ξv | ≤ D η 2

2

,

(2)

A sequence {ξ v : v ∈ Z } ⊆ W is a Bessel sequence if (only) the upper inequality of (2) holds. If only for all ∈ Γ ⊂ W , the upper inequality of (2) holds, the sequence {ξ v } ⊆ W is a Bessel sequence with respect to (w.r.t.) Γ . If { f v } is a frame, there * exists a dual frame { f v } such that

∀ ϒ ∈W

， ϒ = ∑ ϒ, f v∈Λ

For a sequence c = {c(v)} ∈

2

v

f v* = ∑ ϒ, f v* f v .

(3)

v∈Λ

( Z ) , we define its discrete-time Fourier transform as

the function in L2 (0,1) 2 by Fc (ω ) = C (ω ) = ∑ v∈Z 2 c ( v )e −2π ixω dx

，

Note that the discrete-time Fourier transform is 1-periodic. Let

(4)

Tvφ ( x ) stand for

φn,va

integer translates of a function φ ( x) ∈ L ( R ) , i.e., (Tvaφ )( x) = φ ( x − va) , and 2

2

= 4 φ (4 x − va ) , where a is a positive real constant number. Let ( x) ∈ L ( R 2 ) n

n

2

and let V0 = span{Tv : v ∈ Z 2 } be a closed subspace of L2 ( R 2 ) . Assume that ∧

(ω + v ) | ∈ L [0,1] . In [5] , the sequence {Tv ( x)}v is a frame for V0 if and only if there exist positive constants L1 and L2 such that Η (ω ) := ∑ v |

2

L1 ≤ Η (ω ) ≤ L2

∞

2

，a.e., ω ∈ [0,1] \ N = {ω ∈[0,1] : Η(ω) = 0} . 2

2

(5)

2 The Bivariate Affine Pseudoframes of Translate We begin with introducing the concept of pseudoframes of translates. Definition 1. Let {Tva ϒ, v ∈ Z 2 } and {Tva ϒ, v ∈ Z 2 } be two sequences in L2 ( R 2 ) . Let U be a closed subspace of L2 ( R 2 ) . We say {Tva ϒ, v ∈ Z 2 } forms an affine pseudoframe for U with respect to {Tva ϒ, v ∈ Z 2 } if

∀ f ( x) ∈ U , f ( x) = ∑ v∈Z f , Tva ϒ Tva ϒ( x ) 2

Define an operator K : U →

2

( Z 2 ) by

∀ f ( x) ∈U , and define another operator

S:

(6)

2

Kf = { f , Tva ϒ } ,

( Z 2 ) → W such that

(7)

The Characteristics of Multiple Affine Oblique Binary Frames

39

∀ c = {c(k )} ∈ 2 ( Z 2 ) , Sc = ∑ v∈Z c(v) Tva ϒ.

(8)

2

Theorem 1. Let {Tva ϒ}v∈Z 2 ⊂ L2 ( R 2 ) be a Bessel sequence with respect to the subspace U ⊂ L2 ( R 2 ) , and {Tva ϒ}v∈Z 2 is a Bessel sequence in L2 ( R 2 ) . Assume that K be defined by (7), and S be defined by(8). Assume P is a projection from L2 ( R 2 ) onto U . Then {Tva ϒ}v∈Z 2 is pseudoframes of translates for U with respect to {Tva ϒ}v∈Z 2 if and only if

KSP = P .

(9)

Proof. The convergence of all summations of (7) and (8) follows from the assumptions that the family {Tva ϒ}v∈Z 2 is a Bessel sequence with respect to the subspace Ω , and he family {Tva ϒ}v∈Z 2 is a Bessel sequence in L2 ( R 2 ) with which the proof of the theorem is direct forward.

We say that a bivariate generalized multiresolution structure (BGMS) {Vn , f ( x),

f ( x)} of L2 ( R 2 ) is a sequence of closed linear subspaces {Vn }n∈Z of L2 ( R 2 ) and two

Vn ⊂ Vn +1 , n ∈ Z ; (ii) ∩ V = {0} ; ∪ V is dense in L ( R ) ; (iii) h ( x ) ∈Vn if and only if h ( 4 x ) ∈Vn +1 ∀n ∈ Z . (iv) g ( x) ∈ V0 implies Tva g ( x) ∈ V0 , for v ∈ Z ; (v) {Tva f }v∈Z forms pseudoframes of elements f ( x) , f ( x) ∈ L2 ( R 2 ) such that (i) 2

n∈ Z

n∈ Z

n

2

n

2

translates for with respect to {Tva f , v ∈ Z } . 2

( )

Proposition 1[6]. Let f ∈ L R satisfy | f | a.e. on a connected neighbourhood of 0 in [− 12 , 12 )2 , and | f | = 0 a.e. otherwise. Define ∧ 2 2 Λ ≡ {ω ∈ R 2 : | f (ω ) | ≥ C > 0}, and V0 = PWΛ = {φ ∈ L ( R ) : supp(φ ) ⊆ Λ} . 2

Then for

2

f ∈ L2 ( R 2 ) , {Tv f : v ∈ Z 2 } is pseudoframes of trananslates for V0 with

respect to {Tv f : v ∈ Z 2 } if and only if ∧

∧

h (ω ) f ( ω ) χ Λ ( ω ) = χ Λ ( ω ) a. e. , where

χΛ

is the characteristic function on

(11)

Λ . Moreover, if f (ω ) is the above

conditions then {Tv f : v ∈ Z 2 } and {Tv f : v ∈ Z 2 } are a pair of commutative pseudo-

，

frames of translates for V0 , i. .e.

∀ Γ( x) ∈ V0 ,

Γ( x) = ∑ k∈Z 2 Γ, Tk f Tk f ( x) = ∑ k∈Z 2 Γ, Tk f Tk f ( x) .

Proposition 2[5]. Let {Tva f }v∈Z 2 be pseudoframes of translates for

{Tva f }v∈Z 2 . Define

Vn by Vn ≡ {ϒ( x) ∈ L2 ( R 2 ) : ϒ( x / 4 n ) ∈ V }

，n∈Z ,

(12)

V0 with respect to (13)

40

Y. Li

Then, { f n , va }v∈Z 2 is an affine pseudoframe for

Vn with respect to { f n ,va }v∈Z . 2

The filter functions associated with a BGMS are presented as follows. Define filter functions

D0 (ω )

and

B 0 (ω ) = ∑ s∈Z 2 b0 ( s) e

−2π iω

D0 (ω ) = ∑ s∈Z 2 d 0 ( s ) e−2π iω

D 0 (ω ) by

and

of the sequences d 0 = {d 0 ( s )} and d 0 = {d 0 ( s )} , res-

pectively, wherever the sum is defined. Let

{b0 (v)} be such that D0 (0) = 2 and

B0 (ω ) ≠ 0 in a neighborhoood of 0. Assume also that D0 (ω ) ≤ 2 . Then there exists f ( x) ∈ L2 ( R 2 ) (see ref.[3]) such that f ( x) = 2∑ s∈Z 2 d 0 ( s ) f (4 x − sa ) .

(14)

f ( x) under the same conditions as that of

There exists a scaling relationship for d0 for a seq. d 0 , i.e.,

f ( x) = 2∑ s∈Z 2 d0 (v) f (4 x − sa) .

(15)

3 The Traits of Nonseparable Bivariate Wavelet Packs To construct wavelet packs, we introduce the following notation: a = 3, h0 ( x ) = f ( x),

hν ( x ) = gν ( x), b( 0) (n) = b(n), b(ν ) (n) = q (ν ) (n), whereν ∈ Δ We are now in a position of introducing orthogonal bivariate nonseparable wavelet wraps. Definition 3. A family of functions { hmk +ν ( x ) : n = 0,1, 2,

3, ⋅⋅⋅, ν ∈ Δ } is called a

nonseparable bivariate wavelet packs with respect to an orthogonal scaling function

Λ 0 ( x) , where Λ mk +ν ( x ) = ∑ n∈Z 2 b(ν ) (n) Λ k (mx − n), whereν

(16)

= 0,1, 2, 3. By taaking the Fourier transform for the both sides of (12), we have h nk +ν (ω ) = B (ν ) ( z1 ,2 z ) ⋅ h k (ω B

( z1 , z2 ) = B (ω / 2 ) = ∑ b

(ν )

(ν )

2). (ν )

(17) k1 1

(k ) z z

k2 2

(18)

k ∈Z 2

Lemma 1[6]. Let φ ( x) ∈ L2 (R 2 ). Then

φ ( x)

∑ | φ ( ω + 2k π ) |

2

k∈Z

is an orthogonal one if and only if

=1 .

(19)

2

Lemma 2[6]. Assuming that f ( x) is an semiorthogonal scaling function. B ( z1 , z2 ) is the symbol of the sequence {b(k )} defined in (3). Then we have

Π = B ( z1 , z 2 ) + B (− z1 , z 2 ) + B ( z1 , − z 2 ) + B ( − z1 , − z 2 ) 2

2

2

2

(20)

The Characteristics of Multiple Affine Oblique Binary Frames

ψν ( x) ( ν = 0,1, 2,3 )

Lemma 3[8].

If

associated with

h( x ) . Then we have

∑

1 j =0

are orthogonal binary wavelet functions

{ B (( −1) z1 , ( −1) z 2 ) B (( −1) z1 , ( −1) z 2 ) + B (λ )

j

(ν )

⋅ B (( −1) Lemma 4[6]. Let

(ν )

j

j +1

j

j

z1 , ( −1) z 2 )}: = Ξ λ , μ j

41

(λ )

(( −1)

j +1

z1 , ( −1) z 2 ) j

= δ λ ,ν , λ ,ν ∈ {0,1, 2, 3}.

(21)

n ∈ Z + and n be expanded as (17). Then we have ∞

h n (ω ) = ∏ B j =1

(ν j )

(e −iω1 / 2 , e − iω12 / 2 )h 0 ( 0 ) . j

j

Lemma 4 can be inductively proved from formulas (14) and (18). 3 Theorem 1[6]. For n ∈ Z + , k ∈ Z , we have

hn (⋅), hn (⋅ − k ) = δ 0,k . Theorem 2[7]. For every

(22)

k ∈ Z 2 and m, n ∈ Z + , we have

hm (⋅), hn (⋅ − k ) = δ m, nδ 0, k .

(23)

References 1. Telesca, L., et al.: Multiresolution wavelet analysis of earthquakes. Chaos, Solitons & Fractals 22(3), 741–748 (2004) 2. Iovane, G., Giordano, P.: Wavelet and multiresolution analysis: Nature of ε ∞  Cantorian space-time. Chaos, Solitons & Fractals 32(4), 896–910 (2007) 3. Li, S., et al.: A theory of generalized multiresolution structure and pseudoframes of translates. Fourier Anal. Appl. 7(1), 23–40 (2001) 4. Chen, Q., et al.: A study on compactly supported orthogonal vector-valued wavelets and wavelet packets. Chaos, Solitons & Fractals 31(4), 1024–1034 (2007) s 5. Shen, Z.: Nontensor product wavelet packets in L2 ( R ) . SIAM Math. Anal. 26(4), 1061–1074 (1995) 6. Chen, Q., Qu, X.: Characteristics of a class of vector-valued nonseparable higherdimensional wavelet packet bases. Chaos, Solitons & Fractals 41(4), 1676–1683 (2009) 7. Chen, Q., Wei, Z.: The characteristics of orthogonal trivariate wavelet packets. Information Technology Journal 8(8), 1275–1280 (2009) 8. Yang, S., Cheng, Z., Wang, H.: Construction of biorthogonal multiwavelets. J. Math. Anal. Appl. 276(1), 1–12 (2002)

Generation and Characteristics of Vector-Valued Quarternary Wavelets with Poly-scale Dilation Factor* Ping Luo1,** and Shiheng Wang2 1

2

Department of Fundamentals, Henan Polytechnic Institute, Nanyang 473009, China Department of Computer Science, Nanyang Agricultural College, Nanyang 473003, China [email protected]

Abstract. Wavelet analysis has become a developing branch of mathematics for over twenty years. In this paper, the notion of orthogonal nonseparable bivariate wavelet packs, which is the generalization of orthogonal univariate wavelet packs, is proposed by virtue of analogy method and iteration method. Their biorthogonality traits are researched by using time-frequency analysis approach and variable separation approach. Three orthogonality formulas regarding these wavelet wraps are obtained. Moreover, it is shown how to draw new orthonormal bases of space L2 ( R 4 ) from these wavelet wraps. A procedure for designing a class of orthogonal vector-valued finitely supported wavelet functions is proposed by virtue of filter bank theory and matrix theory. Keywords: Nonseparable, quarternary wavelet packs, frames of transform, Bessel sequence, Gabor frames, time-frequency analysis approach.

1 Introduction and Notations There is a large diversity of research fields where Gabor systems and frames play a role. The main advantage of wavelet packs is their time-frequency localization property. Construction of wavelet bases is an important aspect of wavelet analysis, and multiresolution analysis method is one of importment ways of constructing various wavelet bases. There exist many kinds of scalar scaling functions and scalar wavelet functions. Although the Fourier transform has been a major tool in analysis for over a century, it has a serious laking for signal analysis in that it hides in its phases information concerning the moment of emission and duration of a signal. Wavelet analysis [1] has been developed a new branch for over twenty years. Its applications involve in many areas in natural science and engineering technology. The main advantage of wavelets is their time-frequency localization property. Many signals in areas like music, speech, images, and video images can be efficiently represented by wavelets that are translations and dilations of a single function called mother wavelet with bandpass property. They can be widely applied in science and engineering [2,3]. * Foundation item: The research is supported by Natural Scientific Foundation of Shaanxi Province (Grant No:2009J M1002), and by the Science Research Foundation of Education Department of Shaanxi Provincial Government (Grant No:11JK0468). ** Corresponding author. S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 42–48, 2011. © Springer-Verlag Berlin Heidelberg 2011

Generation and Characteristics of Vector-Valued Quarternary Wavelets

43

Coifman R. R. and Meyer Y. firstly introduced the notion for orthogonal wavelet packets which were used to decompose wavelet components. Chui C K.and Li Chun L.[4] generalized the concept of orthogonal wavelet packets to the case of nonorthogonal wavelet packets so that wavelet packets can be employed in the case of the spline wavelets and so on. Tensor product multivariate wavelet packs has been constructed by Coifman and Meyer. The introduction for the notion on nontensor product wavelet packs attributes to Shen Z [5]. Since the majority of information is multidimensional information, many researchers interest themselves in the investigation into multivariate wavelet theory. Therefore, it is significant to investigate nonseparable multivariate wavelet theory. In the following, we introduce some notations. Z and Z + denote all integers and all nonnegative integers, respectively. R denotes all real numbers. R 2 denotes the 4dimentional Euclidean space. L2 ( R 4 ) denotes the square integrable function space. 4 x = ( x1 , x2 , x3 , x4 ) ∈ R , ω = (ω1 , ω2 , ω3 , ω4 ) ∈ R 4 , k = (k , k , k , k ) ∈ Z , Let ωl z =e , i = 1, 2, 3, 4 . The inner product for any functions ψ ( x ) and ( x ) and the Fourier transform of ψ ( x ) are defined, respectively, by 4

1

−i

2

3

4

2

l

ψ,

= ∫ 4 ψ ( x) ( x ) dx, ψ (ω ) = ∫ 4 ψ ( x) e − iω ⋅ x dx, R

R

where ω ⋅ x = ω1 x1 + ω2 x2 + ω3 x3 + ω4 x4 and ( x) denotes the complex conjugate of ( x) . Set Z + = {0} ∪ N , m, s ∈ N as well as m ≥ 2 By algebra theory, it is obviously 4 follows that there are m elements d 0 , d1 , , d m4 in Z +4 = {(n1 , n2 , n3 , n4 ) : nl ∈ Z + } nl ∈ Z + } such that Z = ∪ (d + mZ ) ; (d1 + mZ 4 ) ∩ (d 2 + mZ 4 ) = φ , where Ω 0 = {d 0 , d1 , , d m4 −1} denotes the aggregate of all the different representative elements in the quotient group Z 4 /(mZ 4 ) and order d 0 = {0} where {0} is the null element of Z and d1 , d 2 denote two arbitrary distinct elements in Ω0 .Let Ω = Ω 0 − {0} and Ω, Ω 0 to be two index sets. Define, By L2 ( R 4 , C s ) , we denote the set of all vector-valued functions L2 ( R 4 , C s ) := { ( x) , = (h1 ( x)), h2 ( x), , hu ( x))T : hl ( x ) ∈ L2 ( R 2 ), l = 1, 2, , s} ,where T means the transpose of a vector. For any ∈ L2 ( R 4 , C s ) its integration is defined as follows 4

4

d ∈Ω 0

4

+

∫

R4

( x)dx = ( ∫ R4 h1 ( x)dx, ∫ R4 h2 ( x)dx,

, ∫ R4 hs ( x)dx )T .

Definition 1. A sequence {

n

( y ) n∈Z 4 ⊂ L2 ( R 4 , C s )} is called an orthogonal set, if

〈

n

,

v

〉 = δ n ,v I s , n, v ∈ Z 4 ,

(1)

2 The Bivariate Multiresolution Analysis 2

4

Firstly, we introduce multiresolution analysis of space L ( R ) . Wavelets can be constructed by means of multiresolution analysis. In particular, the existence

44

P. Luo and S. Wang

theorem[8] for higher-dimentional wavelets with arbitrary dilation matrice has been given. Let ϕ ( x ) ∈ L2 ( R 4 ) satisfy the following refinement equation:

ϕ ( x) = m 4 ⋅ ∑ k∈Z bkϕ (mx − k ) ,

(2)

4

where {b(n)}n∈Z 4 is real number sequence which has only finite terms.and ϕ ( x ) ∈ L2 ( R 4 ) is called scaling function. Formula (1) is said to be two-scale refinement equation. The frequency form of formula (1) can be written as

ϕ (ω ) = B( z1 , z2 , z3 , z4 ) ϕ (ω m), B( z1 , z2 , z3 , z4 ) = ∑ b( n) ⋅ z1n ⋅ z2 n ⋅ z3n ⋅ z4 n 1

n∈Z

3

2

4

(3) .

(4)

2

Define a subspace V j ⊂ L ( R ) ( j ∈ Z ) by 2

4

V j = closL2 ( R4 ) m 2ϕ ( m j x − k ) : k ∈ Z 4 .

(5)

Definition 2. We say that ϕ ( x) in (2) generate a multiresolution analysis {V j } j∈Z of

L2 ( R 4 ) , if the sequence {V j } j∈Z defined in (4) satisfy the following properties: 2 4 (i) V j ⊂ V j +1 , ∀ j ∈ Z ; (ii) ∩ V j = {0}; ∪ V j is dense in L ( R ) ; (iii) h( x) ∈ Vk j∈Z j∈Z j ⇔ h( mx) ∈ Vk +1 , ∀ k ∈ Z (iv) the family {ϕ (m x − n) : n ∈ Z 4 } forms a Riesz basis for the spaces V j .

Let U k (k ∈ Z ) denote the complementary subspace of V j in V j +1 , and assume

that there exist a vector-valued function Φ ( x) = {ϕ1 ( x), ϕ 2 ( x),

, ϕ m4 −1 ( x)} constitutes

a Riesz basis for U k , i.e., U j = closL2 ( R4 ) ϒ λ : j , n : λ = 1, 2,

, m4 − 1; n ∈ Z 4 ,

where j ∈ Z , and ϕ λ : j , k ( x ) = m j / 2ϕ λ ( m j x − k ), λ = 1, 2,

dition (5), it is obvious that ϕ1 ( x ), ϕ 2 ( x),

(6)

, m 4 − 1; k ∈ Z 4 . Form con-

, ϕ m4 −1 ( x) are in U 0 ⊂ X 1. Hence there

(λ ) n

exist three real number sequences {q }(λ ∈ Δ = {1, 2,

, m 4 − 1}, n ∈ Z 4 ) such that

ϒλ ( x) = m 4 ⋅ ∑ qk( λ )ϕ (mx − k ),

(7)

k∈Z 4

Formula (7) in frequency domain can be written as

ϒ λ (ω ) = Q ( λ ) ( z1 , z2 , z3 , z4 ) f (ω m), λ = 1, 2, (λ ) k

where the signal of sequence {q (λ )

, m 4 − 1.

}(λ = 1, 2,

, m − 1, k ∈ Z ) is

∑q

⋅ z1n ⋅ z2 n ⋅ z3n ⋅ z4 n .

Q ( z1 , z2 , z3 , z4 ) =

n∈Z

4

(λ ) n

4

1

(8)

4

2

3

4

(9)

Generation and Characteristics of Vector-Valued Quarternary Wavelets

45

A bivariate function ϕ ( x ) ∈ L (R ) is called a semiorthogonal one, if 2

4

ϕ (⋅), ϕ (⋅ − k ) = δ 0, k , n ∈ Z 4 .

(10)

We say G ( x) = {ϒ1 ( x), ϒ 2 ( x), , ϒ m4 −1 ( x)} is an orthogonal binary vector-valued wavelets according to the scaling function ϕ ( x) ∈ L2 (R 4 ) ,if they satisfy:

ϕ (⋅), ϒν (⋅ − k ) = 0 , ν ∈ Δ, k ∈ Z 4 ,

(11)

ϒ λ (⋅), ϒν (⋅ − n) = δ λ ,ν δ 0, n , λ , ν ∈ Δ, n ∈ Z 4

(12)

3 The Traits of Nonseparable Bivariate Wavelet Packs Denoting by H 0 ( x) = F ( x), H μ ( x) = Ψ μ ( x), H 0 ( x) = F ( x), (0) k

Q

(μ ) k

= Pk , Q

(μ ) k

=B

(0) k

, Q

= Pk Q

(μ ) k

H μ ( x) = Ψ μ ( x),

= B , μ ∈ Γ, k ∈ Z . Let us order S = 4 I v . (μ) k

4

For any α ∈ Z and the given vector-valued biorthogonal scaling functions G0 ( x) 4 +

and G0 ( x ) , iteratively define, respectively,

H α ( x) = H 4σ + μ ( x) = H α ( x) = H 4σ + μ ( x) =

∑ Q μ Hσ (4 x − k ),

(13)

∑Qμ

(14)

( ) k

k∈Z 4

( ) k

H σ (4 x − k ).

k∈Z 4

where μ ∈ Γ 0 , σ ∈ Z +4 is the unique element such that α = 4σ + μ , μ ∈ Γ0 holds. Lemma 1[4]. Let F ( x ), F ( x ) ∈ L2 ( R 4 , C v ). Then they are biorthogonal ones if and only if

∑ Fˆ (γ + 2kπ ) Fˆ (γ + 2kπ )

*

= Iv .

(15)

k∈Z 4

Definition 4. We say that two families of vector functions {H 4σ + μ ( x ), σ ∈ Z +4 , μ ∈ Γ 0 }

and {H 4σ + μ ( x), σ ∈ Z +4 , μ ∈ Γ 0 } are vector wavelet packets with respect to a pair of biorthogonal vector scaling functions H 0 ( x) and H 0 ( x) , respectively, where

H 4σ + μ ( x) and H 4σ + μ ( x) are given by (13) and (14), respectively. Taking the Fourier transform for the both sides of (13) and (14) yields, resp.,

H 4σ + μ (γ ) = Q ( μ ) (γ / 4) H σ (γ / 4), μ ∈ Γ 0 , H 4σ + μ (4γ ) = Q ( μ ) (γ ) H σ (γ ), μ ∈ Γ 0 , 1 Q ( μ ) (γ ) = 4 ∑ Qk( μ ) ⋅ exp{−ik ⋅ γ }, μ ∈ Γ 0 4 k∈Z 4

(16) (17) (18)

46

P. Luo and S. Wang

Q ( μ ) (γ ) =

1 44

∑Qμ

( ) k

k∈Z

⋅ exp{−ik ⋅ γ }, μ ∈ Γ 0 .

(19)

4

Lemma 2[7]. Assume that H μ ( x), H μ ( x) ∈ L ( R ) , μ ∈ Γ are pairs of biorthogonal vector-valued wavelets associated with a pair of biorthogonal scaling functions H 0 ( x) and H 0 ( x) . Then, for μ , λ ∈ Γ 0 , we have 4 n

2

∑ρ

∈Γ0

Q ( μ ) ((γ + 2 ρπ ) / 4)Q ( λ ) ((γ + 2 ρπ ) / 4)* = δ μ ,λ I v

(20)

Theorem 1. Assume that {H β ( x), β ∈ Z +4 } and {H β ( x), β ∈ Z +4 } are vector-

valued wavelet packets with respect to a pair of biorthogonal vector-valued functions

H 0 ( x) and H 0 ( x) , respectively. Then, for β ∈ Z +4 , μ , v ∈ Γ 0 , we have [ H 4 β + μ (⋅), H 4 β +v (⋅ − k )] = δ 0,k δ μ ,ν I v , k ∈ Z 4 .

(21)

Proof. Since the space R 4 has the following partition: R 4 = ∪ u∈Z ([0, 2π ]4 + 2uπ ) , and 4

([0, 2π ]4 + 2π u1 ) ∩ ([0, 2π ]4 + 2π u2 ) = Ø, where u1 ≠ u2 , u1 , u2 ∈ Z 4 ,then 1 [ H 4 β + μ (⋅), H 4 β +ν (⋅ − k )] = H 4 β + μ (γ ) H 4 β +ν (γ )* ⋅ exp{ik ⋅ γ }d γ 4 ∫R 4 (2π )

=

1 (2π )

4

∫

Rs

Q ( μ ) (γ / 4) H β (γ / 4) H β (γ / 4)*Q (ν ) (γ / 4)* eik ⋅γ d γ

2

= ( ) 4 ∑ ∫[0,2π ] + 2 kπ Q ( μ ) (γ ) H β (γ ) H β (γ )* ⋅ Q (ν ) (γ )* ei 4 k ⋅γ d γ π k ∈Z 4

4

=

1 (2π )

4

∫

[0,8π ]4

Q ( μ ) (γ / 4)Q (ν ) (γ / 4)* eik ⋅γ d γ =

1 (2π )4

∫

[0,2π ]4

δ μ ,v I u ei 4⋅γ d γ = δ 0, k δ μ , v I v .

This completes the proof of Lemma 1. Theorem 2. If {H β ( x), β ∈ Z + } and {H β ( x), β ∈ Z + } are vector wavelet packets 4

4

with respect to a pair of biorthogonal vector-valued scaling functions H 0 ( x) and

H 0 ( x) , then for any α ,σ ∈ Z +4 , we have [ Hα (⋅), H σ (⋅ − k )] = δα ,σ δ 0, k I v , k ∈ Z 4 .

(28)

Proof. When α = σ ,(28) follows by Lemma 3. as α ≠ σ and α , σ ∈ Γ 0 , it follows from Lemma 4 that (28) holds, too. Assuming that α is not equal to β , as well as at least one of {α ,σ } doesn’t belong to Γ0 , we rewrite α , σ as α = 4α1 + ρ1 , σ = 4σ 1 + μ1 , where ρ1 , μ1 ∈ Γ 0 . Case 1. If α1 = σ 1 , then ρ1 ≠ μ1 . (28) follows by virtue of (24), (25) as well as Lemma 1 and Lemma 2,i.e.,

Generation and Characteristics of Vector-Valued Quarternary Wavelets

(2π ) 4 [ H α (⋅), H σ (⋅ − k )] =

=∫

∫

R4

H 4α1 + ρ1 (γ ) H 4σ1 + μ1 (γ )* ⋅ exp{ik ⋅ γ }d γ

Q ( ρ ) (γ / 4){ ∑ H α (γ / 4 + 2uπ ) ⋅ H α (γ / 4 + 2uπ )* }Q ( μ ) (γ / 4)* ⋅ e ik ⋅γ d γ 1

[0,8 π ]

4

47

1

1

u∈ Z

=∫

[0,2π ]4

1

s

δ ρ , μ I v ⋅ exp{ik ⋅ γ }d γ = O. 1

1

Case 2. If α1 ≠ σ 1 , order α1 = 4α 2 + ρ 2 , σ 1 = 4σ 2 + μ 2 , where α 2 , σ 2 ∈ Z +s , and ρ 2 , μ2 ∈ Γ0 . Provided that α 2 = σ 2 , then ρ 2 ≠ μ2 . Similar to Case 1, (28) can be established. When α 2 ≠ σ 2 ,we order α 2 = 4α 3 + ρ3 , σ 2 = 4σ 3 + μ3 , where α 3 ,σ 3 ∈ Z +4 , ρ3 , μ3 ∈ Γ 0 . Thus, after taking finite steps (denoted by κ ), we obtain α κ ∈ Γ0 , and ρκ , μκ ∈ Γ0 . If α κ = σ κ , then ρκ ≠ μκ . Similar to the Case 1, (28) follows. If α κ ≠ σ κ , then it gets from (12)-(15) that

16π 4 [ H α (⋅), H σ (⋅ − k )] = ∫ 4 H α (γ ) H σ1 (γ )* ⋅ eik ⋅γ d γ R

= ∫ 4 H 4α1 + λ1 (γ ) H 4 β1 + μ1 (γ )* ⋅ exp{ik ⋅ γ }d γ = R κ κ γ γ γ γ = ∫ κ {∏Q ( ρ ) ( l )}{ ∑ H ακ ( l + 2uπ ) H σ κ ( l + 2uπ )*}{∏Q ( μ ) ( l )}* ⋅ eik ⋅γ d γ l

[0,2⋅4 ]4

l

4

l =1

u∈Z 4

4

4

l =1

4

γ κ γ )} ⋅ O ⋅{∏ l =1Q ( μ ) ( l )}* ⋅ exp{−ik ⋅ γ }d γ = O. l 4 4 l =1 4 Therefore, for any α , σ ∈ Z + , result (28) is established. =∫

κ

κ

([0,2⋅ 4 π ]

4

{∏Q ( ρl ) (

l

4 Conclusion The concept of biorthogonal vector four-dimensional wavelet packets was introduced. Three biorthogonality formulas with respect to these wavelet packets are obtained. The direct decomposition of space L2 ( R 4 ) n is proposed by constructing a series of subspaces of the wavelet wraps.

References 1. Telesca, L., et al.: Multiresolution wavelet analysis of earthquakes. Chaos, Solitons & Fractals 22(3), 741–748 (2004) 2. Iovane, G., Giordano, P.: Wavelet and multiresolution analysis: Nature of ε∞ Cantorian space-time. Chaos, Solitons & Fractals 32(4), 896–910 (2007) 3. Zhang, N., Wu, X.: Lossless Compression of Color Mosaic Images. IEEE Trans. Image Processing 15(16), 1379–1388 (2006) 4. Chen, Q., et al.: A study on compactly supported orthogonal vector-valued wavelets and wavelet packets. Chaos, Solitons & Fractals 31(4), 1024–1034 (2007) s 5. Shen, Z.: Nontensor product wavelet packets in L2 ( R ) . SIAM Math. Anal. 26(4), 1061–1074 (1995)

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6. Chen, Q., Qu, X.: Characteristics of a class of vector-valued nonseparable higherdimensional wavelet packet bases. Chaos, Solitons & Fractals 41(4), 1676–1683 (2009) 7. Chen, Q., Wei, Z.: The characteristics of orthogonal trivariate wavelet packets. Information Technology Journal 8(8), 1275–1280 (2009) 8. Chen, Q., Huo, A.: The research of a class of biorthogonal compactly supported vectorvalued wavelets. Chaos, Solitons & Fractals 41(2), 951–961 (2009)

A Kind of New Strengthening Buffer Operators and Their Applications Ran Han* and Zheng-peng Wu College of Science, Communication University of China, Beijing 100024, China [email protected], [email protected] Abstract. Under the axiomatic system of buffer operator in grey system theory, two new strengthening buffer operators are established, which have been based on strictly monotone function. Meanwhile, the characters and the inherent relation among them are studied. The problem that there are some contradictions between quantitative analysis and qualitative analysis in pretreatment for vibration data sequences is resolved effectively. A practical example shows theirs validity and practicability. Keywords: Grey system, Monotone increasing function, Buffer operator, Strengthening buffer operator, Fixed point.

1 Basic Concept Definition 1. Assume that the sequence of data representing a system’s behavior is given, X=(x(1),x(2),…,x(n)) then (1)X is called a monotonously increasing sequence if ∀k = 1,2, " , n − 1 ,

x(k ) < x(k + 1) . (2) X is called a monotonously decreasing sequence, if ∀k = 1,2, " , n − 1 ,

x(k ) > x(k + 1) .

(3) X is called a vibration sequence if k , k ′ ∈{1,2, " , n − 1} , x(k ) < x(k + 1) and

x(k ) > x(k + 1) . And M= max x ( k ) , m= min x (k ) , then M-m is called the amplitude of X. 1≤ k ≤ n

1≤ k ≤ n

Definition 2. Assume that X is a sequence of raw data, D is an operator worked on X, and the sequence, obtained by having D worked on X, is denoted as X=(x(1)d,x(2) d,…,x(n) d). Then D is called a sequence operator, and XD is the first order sequence worked on by the operator D. Sequence is referred to as operator. A sequence operator can be applied as many times as needed. It can obtain a second order sequence, even order sequence, they can be denoted as XD , " , XD . 2

r

Axiom 1. Axiom of Fixed Points. Assume that X is a sequence of raw data and D is a sequence operator, then D must satisfy x(n) d= x(n). *

Corresponding author.

S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 49–54, 2011. © Springer-Verlag Berlin Heidelberg 2011

50

R. Han and Z.-p. Wu

In the effect of the Axiom of Fixed Points which limit the sequence operator, x(n) in the raw data is never changed. Axiom 2. Axiom on Sufficient Usage of Information. When a sequence operator is applied, all the information contained in each datum x(k), (k=1,2,…,n)of the sequence X of the raw data should be sufficiently applied, and any effect of each entry x(k), (k=1,2,…,n) should also be directly reflected in the sequence worked on by the operator. The Axiom 2 limits any sequence operator which should be defined without the sequence of raw data and based on the information in the sequence we have kept. Axiom 3. Axiom of Analytic Representations. For any x(k)d, (k=1,2,…,n) can be described with a uniform and elementary analytic representation in x(1),x(2),…,x(n). The Axiom 3 requires procedures which the operator is applied to the raw data clearly, standardized, and as simple as possible, in order to calculate and makes calculation easier on the computer. Definition 3 All sequence operators, satisfying these three axioms, are called buffer operators, XD is called buffer sequence. Definition 4 Assume X is a sequence of raw data, D is an operator worked on X, when X is a monotonously increasing sequence, a monotonously decreasing sequence or a vibration sequence, if the buffer sequence XD increases or decreases more rapidly or vibrate with a bigger amplitude than the original sequence X, the buffer operator D is termed as a strengthening operator. Theorem 1 (1) When X is a monotonously increasing sequence, XD is a buffer sequence, then D is a strengthening operator ⇔ x( k ) d ≤ x ( k ) ( k = 1, 2,3,..., n ) ; (2)When X is a monotonously decreasing sequence, XD is a buffer sequence, then D is a strengthening operator ⇔ x( k ) d ≥ x( k ) ( k = 1,2,3,..., n) ; (3)When X is a monotonously vibration sequence and D is a strengthening operator, XD is a buffer sequence ,then

max x(k ) ≥ max x( k )d , min x(k ) ≤ min x( k )d . 1≤ k ≤ n

1≤ k ≤ n

1≤ k ≤ n

1≤ k ≤ n

That is, the data in a monotonously increasing sequence shrink when a strengthening operator is applied and data in a monotonously decreasing sequence expand when a strengthening operator is applied[1].

2 Study on a Kind of New Strengthening Buffer Operators Theorem 2. Assume X=(x(1),x(2),…,x(n)) is a sequence of raw data, x(k)>0, (k=1,2,…,n), fi is a strictly monotonic function and fi>0,i=1,2,…,n, gi is its inverse function, and XD1=(x(1)d1,x(2) d2,…,x(n) dn),

⎡ ⎛ f (x (k )) ⎞ n − k +1 ⎤ k = 1, " , n , ⎥, ⎟ x(k ) d 1 = g i ⎢ f i ( x(k ))⎜⎜ i f i (x (n )) ⎟⎠ ⎢ ⎥ ⎝ ⎣ ⎦ 1

when X is a monotonously increasing sequence, a monotonously decreasing sequence or a vibration sequence, D1 is a strengthening buffer operator.

A Kind of New Strengthening Buffer Operators and Their Applications

51

Proof: It is easily proved that D1 satisfies the three axioms of buffer operator, so D1 is a buffer operator. We prove that D1 is strengthening buffer operator. (1) When X is a monotonously increasing sequence, because 0≤x(k)≤…≤x(n) and fi is a strictly monotonic increasing function and fi>0,i=1,2,…,n, 0≤fi (x(k))≤…≤fi (x(n)) ,

[ (( (( )))) ] fi x k fi x n

1 n − k +1

[ (( (( )))) ]

≤ 1 , f i ( x(k ))

1 n − k +1

fi x k fi x n

≤ f i ( x(k )) .

And because gi is inverse function of fi,

[ (( (( )))) ]

x(k )d1 = g i ⎧⎨ f i ( x(k )) ⎩

1 n − k +1

fi x k fi x n

⎫ ≤ g [ f x(k )] = x(k ) . ⎬ i i ⎭

According to theorem 1, D1 is a strengthening buffer operator. (2)When X is a monotonously decreasing sequence, because x(k)≥…≥x(n)>0, fi is a strictly monotonic increasing function and fi>0,i=1,2,…,n,, fi (x(k))

≥…≥fi (x(n))>0,

[ (( (( )))) ]

1 n −k +1

fi x k fi x n

[ (( (( )))) ]

≥ 1 , f i ( x(k ))

fi x k fi x n

1 n − k +1

≥ f i ( x(k )) ,

and gi is inverse function of fi ,so

[ (( (( )))) ]

x(k )d1 = g i ⎧⎨ f i ( x(k )) ⎩

fi x k fi x n

1 n − k +1

⎫ ≥ g [ f x(k )] = x(k ) . ⎬ i i ⎭

According to theorem 1, D1 is a strengthening buffer operator. (2) When X is a vibration sequence, let x( k ) = max x(i ), x( h) 1≤i ≤ n

= min x (i ), 1≤ i ≤ n

∈{1,2, … (k)≥ (1),…, ( ) (h)≤ (1),…, ( ) (k)≥ ( ), (h)≤ ( ),

for i , n},we have x x x n ; x x x n ,because x x n x x n fi is a strictly monotonic increasing function, f i > 0 , f i ( x(k )) ≥ f i ( x(n )), f i ( x(h )) ≤ f i ( x (n )) ,

[ (( (( )))) ] ≥ 1, [ (( (( )))) ] ≤ 1 , f (x(k ))[ (( (( )))) ] ≥ f (x(k )) f (x (h ))[ (( (( )))) ] ≤ f ( x(h )) , and g is inverse function of f , therefore x(k )d = g ⎧⎨ f ( x(k ))[ (( (( )))) ] ⎫⎬ ≥ g [ f x(k )] = x(k ) , ⎩ ⎭ x(h )d = g ⎧⎨ f ( x(h ))[ (( (( )))) ] ⎫⎬ ≤ g [ f x(h )] = x(h ) ⎩ ⎭ 1 n − k +1

fi x k fi x n

fi x h fi x n

fi x k fi x n

i

1 n − k +1

i

,

1 n − k +1

fi x h fi x n

i

1 n − k +1

i

i

1

i

i

fi x k fi x n

1

i

i

fi x h fi x n

i

1 n − k +1

i

i

i

i

1 n − k +1

max{x(k )d 1 } ≥ max{x(k )} 1≤ k ≤ n

and

1≤ k ≤ n

min{x(k )d 1 } ≤ min{x(k )} .According to theorem 1, D1 is a strengthening

1≤ k ≤ n

1≤ k ≤ n

buffer operator.

X = ( x(1), x(2), " , x(n)) is a sequence of raw data, x(k ) > 0, k = 1,2," , n , f i is a strictly increasing monotonic function and f i > 0, i = 1,2, ", n , g i is its

Theorem 3. Assume

inverse function, and XD 2 = ( x(1)d 2 , x(2) d 2 , " , x( n)d 2 ) is its buffer sequence,

52

R. Han and Z.-p. Wu

⎡ f ( x (k )) n ⎛ f ( x(i )) ⎞ n − i +1 ⎤ ⎜⎜ i ⎟⎟ ⎥, k = 1, " , n , x( k )d 2 = g i ⎢ i ∑ n − k + 1 ⎢ ⎥ i = k ⎝ f i ( x (n )) ⎠ ⎣ ⎦ 1

when X is a monotonously increasing sequence or a monotonously decreasing sequence, D2 is a strengthening buffer operator.

D2 satisfies the three axioms of buffer operator, so D2 is a buffer operator. We prove that D2 is strengthening buffer operator. Proof: It is easily proved that

(1)When X is a monotonously increasing sequence, because 0≤x(k)≤…≤x(n)and fi is a strictly increasing monotonic function and

[ (( (( )))) ] ≤ 1 , f ( x(k )) 1 ( ( )) ( ( )) [ [ ≤ 1, ≤ f ( x (k )) , and ∑ ∑ ( ( )) ] ( ( )) ] n − k +1 n − k +1

f i > 0, i = 1,2," , n , 0 ≤ f i ( x(k )) ≤ " ≤ f i ( x(n )) , n

i =k

fi x i fi x n

n

1 n − i +1

i

i=k

fi x i fi x n

fi x k fi x n

1 n −k +1

1 n − i +1

i

because g i is inverse function of fi,

⎧ f ( x(k )) n f i ( x (i )) n −1i +1 ⎫ x(k )d 2 = g i ⎨ i ∑ fi ( x(n )) ⎬⎭ ≤ g i [ f i x(k )] = x(k ) . ⎩ n − k + 1 i =k According to theorem 1, D2 is a strengthening buffer operator.

[

]

We can follow the example of the proof of theorem 2, when X is a monotonously decreasing sequence, D2 is a strengthening buffer operator.

3 Case Study We take Per capita power consumption (Unit :kwh) for example to prove the effect of strengthening buffer operator in this paper on GM(1,1) prediction model. Per capita power consumption of China from 2000 to 2006 is chosen as a sequence of raw data (Table1). Table 1. Per capita power consumption(Unit: kwh)

Year Percapita power consumpti on

2000

2001

2002

2003

2004

2005

2006

132.4

144.6

156.3

173.7

190.2

216.7

249.4

At present, China is on economic development of the process of industrialization and more dependent on electric power. From 2000 to nowadays, China overcame the effects of Asian financial crisis, the government adopted a proactive fiscal policy and

A Kind of New Strengthening Buffer Operators and Their Applications

53

prudent monetary policy for economic growth, injecting vigor, cause the power demand increased sharply. Therefore, this article takes China's per capita power consumption of raw data sequence from 2000 to 2005 as modeling data, the data in 2006 as a model test data. Calculating per capita consumption growth power is as follows: 9.215%, 8.091%, 11.132%, 9.499, 13.933%, 15.090%, an annual average increase rate of 9.468%. We test the primary data sequence for quasi-smooth, when t ≥ 2003 , its smooth ratio is 0.401,0.313, 0.272, 0.246 which is in (0,0.5).Its smooth ratio is decreasing and satisfies quasi-smooth. Therefore a accumulation of the primary data sequence has quasi-exponential law. But it is clear that the former has a little slow growth rate, the latter has fast growth rate. Therefore, it is best to smooth the original data sequence to weaken the impact of the shock disturbance and highlight the laws of the data. Take

f i ( x ) = x 2 , g i ( x ) = x 0.5 , i = 1,2," , n to construct the buffer operator,

and apply the second order buffer operator which in this article to strengthen the raw data. Then we build the prediction model and compare with the raw data sequence (Table 2). The GM(1,1) model directly built by the raw data sequence without buffer operator is

xˆ (2000 + t ) = 1317.848e0.102t − 1185.448 . The GM(1,1) model built by the strengthening buffer operator sequence XD1 is

xˆ ( k + 1) = 654.7787e 0.1549 k − 544.7797 . The GM(1,1) model built by the strengthening buffer operator sequence XD2 is

xˆ ( k + 1) = 801.78e 0.139 k − 686.784 . Table 2. Case 3 GM(1,1) model

Sequence

X

Predictive value 236.831

Predictive relative error(%) 5.040

XD12

238.2672

4.53

XD22

241.23

3.28

From the table 2, all the predictive relative error of the strengthening buffer sequences which are applied by the second order buffer operator D1 and D2 are smaller than the predictive relative error of the raw data sequences in the model. The predictive relative error applied with D2 is smaller, and its predictor is 241.23 and closer to the observation 249.4. Its one step predictive error is only 3.28%, namely the predictive accuracy is the highest.

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4 Conclusion Based on the recent literature, two new strengthening buffer operators are established, which have been based on strictly monotone function. The example shows that the predictive accuracy increases by the strengthening buffer sequences which are applied by the second order buffer operator D1 and D2 .

References 1. Liu, S.-f., Dang, Y.-g., Fang, Z.-g.: Grey system theory and its application. Science Press, Beijing (2004) 2. Liu, S.-f.: The three axioms of buffer operator and their applications. The Journal of Grey System 3(1), 39–48 (1991) 3. Liu, S.-f.: Buffer operator and its application. Theories and Practices of Grey System 2(1), 45–50 (1992) 4. Liu, S.-f.: The trap in the prediction of a shock disturbed system and the buffer operator. Journal of Huazhong University of Science and Technology 25(1), 25–27 (1997) 5. Dang, Y.-g., Liu, S.-f., Liu, B., et al.: Study on the strengthening buffer operators. Chinese Journal of Management Science 12(2), 108–111 (2004)

Infrared Target Detection Based on Spatially Related Fuzzy ART Neural Network BingWen Chen1, WenWei Wang1, and QianQing Qin2 1

College of Electronic Information, Wuhan University, 430079 Wuhan, China 2 State Key Laboratory for Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, 430079 Wuhan, China [email protected]

Abstract. In order to solve the ghosts, the halo effect as well as the lower signal-to-noise ratio problems more effectively, this paper presents a spatially related fuzzy ART neural network. We introduce a laterally-inspirited learning mode into the background modeling stage. At first, we combine the regionbased feature with the intensity-based feature to train the spatially related fuzzy ART neural network by the laterally-inspirited learning mode. Then two spatially related fuzzy ART neural networks are configured as master-slave pattern to build the background models and detect the infrared targets alternately. Experiments have been carried out and the results demonstrate that the proposed approach is robust to noise, and can eliminate the ghosts and the halo effect effectively. It can detect the targets effectively without much more post-process. Keywords: infrared target detection, laterally-inspirited learning mode, spatially related fuzzy ART neural network, master-slave pattern.

1 Introduction Visual surveillance is a hot topic in computer vision. In recent years, thanks to the improvement of infrared technology and the drop of its cost, the thermal imagery has been widely used in the surveillance field. In comparison with visible imagery, adopting the thermal imagery has many benefits, such as all-day working ability, robust to light change, no shadow, etc. However, the thermal imagery has its own difficulties, such as a lower signal-to-noise ratio, a lower resolution, an uncalibrated white-black polarity change as well as the “halo effect” which appears around the very hot or cold objects [1]. The comparison between visible and thermal imagery can be learned from Lin’s review [2]. Many detection approaches have been proposed for thermal imagery. Some approaches base on the target’s appearance [3-5]; some approaches base on the target’s motion [1, 6]. A fuzzy ART neural network approach is proposed to detect the targets in our previous works [7]. It is capable of learning the total number of categories adaptively and stable, but it has some shortages. It does not take advantage of the strong spatial correlation of thermal imagery. It only exploits the time domain information, and S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 55–61, 2011. © Springer-Verlag Berlin Heidelberg 2011

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utilizes only one simple feature to build the background models; moreover its background model updating strategy is not comprehensive. In this paper, we exploit and combine the spatial information with the temporal information, and propose a novel infrared target detection approach based on spatially related fuzzy ART neural network. The remainder of this paper is described as follows: Sec. 2 introduce the laterally-inspirited learning mode and the master-slave working pattern, and present the detection approach comprehensively. The experimental evaluation of our approach is described in Sec. 3. Finally Sec. 4 includes conclusions and further research directions.

2 The Spatially Related Fuzzy ART Neural Network The infrared target detection approach using spatially related fuzzy neural network can be divided into two stages: background modeling stage; target detection and model updating stage. The whole detection schema is the same for each location of infrared image. To facilitate the interpretation, we present the approach for one location pixel in the following. 2.1 Background Modeling Stage Several frames ahead (Ft, t=0…N) are allocated to build the background model, so we can obtain the sample data like formula 1. [ f 0 ( i , j ), ..., f t ( i , j ), ..., f N ( i , j )] .

(1)

The background modeling stage includes two sub-stages: initial background model sub-stage and build background model sub-stage. Initial Background Model Sub-stage. This stage is focused on the fuzzy ART neural network initialization and sample data filter out. A fuzzy ART neural network is initialized and trained with the sample data. Taking advantage of the strong spatial correlation of infrared image, we combine the regionbased feature that the median of pixel’s neighbor with the intensity-based feature that the pixel’s own intensity to train the fuzzy ART neural network (like formula 3). Pt (i, j ) = [ f t (i, j ); Median{ f t (i ± k , j ± k ) | k = 0,1,..., ( M − 1) / 2}] .

(2)

P ( i , j ) = [ P0 (i , j ),..., Pt ( i , j ),..., PN ( i , j )] .

(3)

Where, M denotes the size of pixel’s neighbor, P (i,j) denotes the input feature vector. After training, we can obtain each category’s information, such as category’s appearing frequency and each sample data’s category, and then filter each category out according to its properties. If a category belongs to the background, then its appearing frequency must be bigger than others or a threshold value, so we set a threshold θ, if the category’s frequency larger than θ, then this category belongs to background, or else, this category belongs to non-background and must be deleted. In the same way, the sample data are filtered out according to its category. By means of

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filtering, the proposed approach can suit for the real situation that the foreground objects appear during the background modeling period. Build Background Model Sub-stage. This stage is focused on the fuzzy ART neural network retrained and background model refined. Owing to the thermal imagery mechanism, the thermal image has a strong spatial correlation in a local region, so the background models should also reflect this property. In order to exploit and fuse the spatial and temporal correlation effectively, we introduce the laterally-inspirited learning mode as follow: According to the category of Pt(i,j), for each location, its initialized fuzzy ART neural network and those initialized fuzzy ART neural networks in its neighbor are retrained, like formula 4.

⎧ {FART(i ± k, j ± k) | k = 0,1,...,(R −1) / 2} are retrained with P (i, j), ⎨ ⎩nothing, t

if Pt (i, j) belongs to background if Pt (i, j) belongs to foreground.

(4)

Where, R denotes the size of laterally-inspirited region, FART (i,j) denotes the fuzzy ART neural network of location (i,j). The fuzzy ART neural network is retrained by the laterally-inspirited learning mode can associate the background models with the spatial information. Furthermore, it combines with the region-based feature can exploit the spatial correlation information effectively, and make the detection more stable and robust to noise. We name this type of neural network as spatially related fuzzy ART neural network. 2.2 Target Detection and Model Updating Stage The approach in reference [7] only updates the background model when the current pixel belongs to the background model, but do not consider the updating method when the current pixel belongs to the non-background model. Therefore, when this approach works for a long time, the background model may be not fit the actual situation, and the false objects, often referred to as ghosts [8] will be turn out. For example, the still background objects start moving, or the moving foreground objects stop for a long time. These may be lead to ghost appearance, and the performance may also worsen. In order to solve this problem, two spatially related fuzzy ART neural networks are configured as master-slave pattern to detect the infrared targets alternately. Here, we suppose that: A, B denote the two spatially related fuzzy ART neural networks; T denotes the alternation time interval; Wj denotes the weights of category j; β denotes the learning rate parameter. Then the target detection and model updating strategy as follow: 1) In the background modeling stage, A is established as the master network and trained with the sample data follow the strategy in background modeling stage; 2) In target detection stage, A is utilized to detect the targets. If the new input pixel can find the matched category and the resonance happen, then the corresponding category’s weights are updated like formula 5. At the same time, B is established as the slave network and trained with current data follow the strategy in background modeling stage.

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W jt+ 1 = β ( P t ( i , j ) ^ W jt ) + ( 1 -β ) W jt .

(5)

3) When the alternation time interval T passes, the slave network B turn to be master network and begin to detect the targets, and the master network A is reset and established as slave network. When each alternation time interval T passes, the master network and the slave network swap.

3 Experimental Results To evaluate the performance of the proposed approach, we tested the approach with the OTCBVS Benchmark Dataset Collection [9]. Some example images of five original thermal sequences are shown in the first row of Fig. 1. We can see from these images that the thermal images have a lower resolution; the “halo effect” appears around objects; the target’s temperature is not always higher than environment’s. In our experimental evaluation, all detection results obtained by any approach are the raw detection results without the post-processing. In order to quantitatively measure the performance of the detection results, we manually segment the person regions and take it as the ground truth to compare with the automatic detection results. Some silhouettes segmented are shown in the second row of Fig. 1. We compare the spatially related fuzzy ART neural network (SRFART) with three other approaches: the single weighted Gaussian approach (SWG) [1]; the codebook approach (CB) [10] and the fuzzy ART neural network (FART) [7]. The single weighted Gaussian approach is an improved version of single Gaussian model. The codebook approach is one of prominent detection approaches. About parameter setting, for single weighted Gaussian approach, we set the stand deviation σˆ = 5 , the detection threshold T=8; for codebook approach, we set the light control parameters α=0.7, β=1.3; for FART approach and the proposed approach, we set the choice parameter α=0, the learning rate parameter β=1, the vigilance parameter ρ=0.85 for seq.1, 2, 5, ρ=0.75 for seq.3, 4; for the proposed approach, we set the alternation time interval T=10s, the size of pixel’s neighbor M=3, the size of laterallyinspirited region R=3. We show the silhouette results obtained by each detection approaches on representative images (from the five sequences) in Fig. 1. From the third row, we can see that the detection results obtained by codebook approach have much noise, and even exist the ghosts like the seq. 4, 5. From the fourth row, we can see that the detection results obtained by single weighted Gaussian approach also have much noise, and the halo effect is very clear which severely impairs the detection results like the sequences 3, 4. From the fifth row, we can see that the detection results obtained by fuzzy ART neural network approach have some noise, like seq. 1, and cannot detect the targets effectively when the contrast between target and environment is low, like seq. 5. In contrast, from the last row, we can see that the detection results obtained by the proposed approach have less noise and do not exist the ghosts; the proposed approach is able to detect most portions of people’s bodies even when the contrast between target and environment is low; the proposed approach can eliminate most of halo and extract the people effectively.

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Fig. 1. Visual comparisons of detection results of the four approaches across different images

Fig. 2. Visual comparison of detection results between whether to utilize the master-slave pattern

Fig. 2 shows the visual comparison of detection results between whether to utilize the master-slave pattern. Image (a) in Fig. 2 is one example image of the background sample frames. We can see that there is a person standing in front of door and the door is open. After a while, the person goes into the house and the door is closed, and we can see it from the image (b). Image (c) shows the ground truth detection result for image (b). From the image (d) we can see that: when we do not utilize the masterslave pattern, the detection result has two ghosts (false object) obviously. In contrast,

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when we do utilize the master-slave pattern, the background model can reflect the real situation as accurately as possible, and can eliminate the ghosts effectively. In order to evaluate the detection results objectively and quantitatively, the F1 metric [8] is adopted. The F1 metric, also known as Figure of Merit or F-measure, is the weighted harmonic mean of precision and recall, as defined in formula 6. Recall, also known as detection rate, gives the percentage of detected true positives as compared to the total number of true positives in the ground truth; Precision, also known as positive prediction rate, gives the percentage of detected true positives as compared to the total number of detected items. Higher value of F1 means a better detection performance. F1=

2 * R e c a ll* P r e c is io n R e c a ll+ P r e c is i o n

(6)

The F1 measurements of each approach for five sequences in the Fig. 1 are shown in table 1. Comparing the measurements in table 1, we can see that: Due to the presence of halo and ghosts, the F1 values of single weighted Gaussian and codebook are poor, and due to the lower contrast between target and environment, the F1 values of the fuzzy ART neural network is low, like seq. 5. In contrast, the spatially related fuzzy ART neural network approach performs well all the same even when the halo and ghosts exist. Table 1. The F1 measurements of four approaches for five sequences Detection approach SWG CB FART SRFART

Seq.1 0.709 0.678 0.702 0.772

Seq.2 0.668 0.600 0.815 0.858

Seq.3 0.593 0.655 0.829 0.834

Seq.4 0.528 0.536 0.833 0.881

Seq.5 0.626 0.343 0.566 0.756

Generally speaking, the proposed approach is robust to noise and can reflect the real situation as accurately as possible. It can eliminate the halo and the ghosts effectively and detect the targets effectively without much more post-process.

4 Conclusions We presented a novel infrared target detection approach based on spatially related fuzzy ART neural network. It associates the background models with the spatial information by the laterally-inspirited learning mode, and it reflects the real situation as accurately as possible by adopting the master-slave working pattern. The approach can detect targets more stable and robust to noise. It can eliminate the halo and ghosts effectively and detect the targets effectively without much more post-process. In the implementation of the proposed approach, we do not take account of the processing speed. In the end, the detection speed of the proposed approach is only 2 fps. So we are going to study how to speed the model up in the future.

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Acknowledgments. This research was supported in part by the Natural Science Foundation of Hubei Province of China No. 2009CDA141.

References 1. Davis, J.W., Sharma, V.: Background subtraction in thermal imagery using contour saliency. International Journal of Computer Vision 71, 161–181 (2007) 2. Lin, S.-S.: Review: Extending visible band computer vision techniques to infrared band images. Technical Report, University of Pennsylvania (2001) 3. Li, Z., Bo, W., Ram, N.: Pedestrian detection in infrared imaged based on local shape features. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1–8. IEEE Press, Minneapolis (2007) 4. Kai, J., Michael, A.: Feature based person detection beyond the visible spectrum. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 30–33. IEEE Press, Miami (2009) 5. Stephen, O.H., Ambe, F.: Detecting People in IR Border Surveillance Video Using Scale invariant image moments. In: Proceedings of SPIE, pp. 73400L-1–6. SPIE, Orlando (2009) 6. Fida, E.B., Thierry, B., Bertrand, V.: Fuzzy Foreground Detection for Infrared Videos. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pp. 1–6. IEEE Press, Anchorage (2008) 7. Chen, B., Wang, W., Qin, Q.: Infrared target detection based on Fuzzy ART neutral network. In: The Second International Conference on Computational Intelligence and Natural Computing, pp. 240–243. IEEE Press, Wuhan (2010) 8. Lucia, M., Alfredo, P.: A self-organizing approach to background subtraction for visual surveillance applications. IEEE Transactions on Image Processing 17, 1168–1177 (2008) 9. Object Tracking and Classification in and Beyond the Visible Spectrum, http://www.cse.ohio-state.edu/otcbvs-bench/ 10. Kyunqnam, K., Chalidabhonqse, T.H., David, H., et al.: Real-time foreground-background segmentation using codebook model. Image Segmentation 11, 172–185 (2005)

A Novel Method for Quantifying the Demethylation Potential of Environmental Chemical Pollutants Yan Jiang1,2,3 and Xianliang Wang2,* 1

College of Oceanography and Environmental Science, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China 2 State Key Laboratory of Environmental Criteria and Risk Assessment, Chinese Research Academy of Environmental Sciences, Beijing 100012, People’s Republic of China Tel.: +86-10-84916422; Fax: +86-10-84916422 [email protected] 3 College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, People’s Republic of China Abstract. We developed a novel method to quantify the demethylation epigenetic toxicity of pollutants. A hyper-methylated pEGFP-C3 plasmid eukaryotic expression vector was constructed and used to evaluate the epigenetic toxicity of aquatic pollutants samples from polluted coastal waters of Tianjin, China. The methylated pEGFP-C3 plasmid was transfected into HepG2 cells and incubated with 5-aza-2-deoxycytidine at various concentrations. The HepG-2 cell line reporter gene vector was used to assess the epigenetic toxicity of heavy metal extracts from polluted marine waters, and shellfish samples. Results indicated that the demethylation ability of 5-aza-dC at doses between 0.0008 and 0.1 µM could be quantitatively detected. Nine of the 19 aquatic samples showed strong demethylation ability at values between 0.0064 and 0.0387 µM 5-AZA equivalents. A GFP reporter gene vector with a hypermethylated CMV promoter was constructed, and a relatively sensitive response relationship between GFP gene expression and 5-AZA dose was observed, providing a novel method for quantifying the demethylation ability of pollutants. Keywords: Demethylation, Epigenetic toxicity, Cellular test system, Enhanced green fluorescent protein, Aquaculture.

1 Introduction Before chemical compounds can be used, evaluation of their safety is essential. Currently, assessment of contamination of genetic material, such as DNA mutation, focuses mainly on detecting direct damage. However, many pollutants that may be deemed safe by existing safety evaluation protocols can have indirect effects. For example, pollutants may cause DNA methylation and activate other epigenetic mechanisms that can lead to slight changes in long-term biological traits but may not cause genetic mutations, chromosomal aberrations, and other genetic damage [1; 2] *

Corresponding author.

S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 62–71, 2011. © Springer-Verlag Berlin Heidelberg 2011

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Low-dose pollutant exposure can cause DNA methylation changes, and long-term, mild, recurring methylation changes can begin the process of damaging health. Thus, developing a tool to assess indirect damage is essential. [3; 4] Currently, research of the epigenetic toxicity of environmental pollutants is in its infancy. [5] Some researchers have proposed the use of serum biochemical analysis, histopathological evaluation, and analysis of the DNA methylation status in healthy animals treated with chronic and/or sub-chronic pollutant exposure in vivo. [6; 7] However, such a detection system has not been developed, and there is an urgent need to establish a stable and practical detection method to evaluate the epigenetic toxicity of pollutants. [8; 9] Our approach to quickly evaluate epigenetic toxicity of pollutants was to use the non-toxic enhanced green fluorescent protein (EGFP) to build a screening system, which we call quantifying the demethylation potential (QDMP) . The basic principles of the system are as follows: DNA methylase Mss.I was used to artificially modify commercial green fluorescent protein plasmid (EGFP) in vitro so that the EGFP gene promoter was highly methylated. [10] The methylated EGFP plasmid was transfected into the human hepatoma cell line HepG-2, and successfully transfected HepG-2 cells were selected through several rounds of screening of plate cultures. [11] The selected HepG-2 cells that contained the methylated EGFP plasmid were used as the target and confirm the demethylation matter arsenate, hydralazine as a positive control. Next, the heavy metals Cd and Ni, atmospheric particulate matter extracts, and contaminated shellfish samples extracts in a gradient of doses were tested using the screening system. [12; 13; 14] For each pollutant type, gene promoter methylation, cell expression of EGFP mRNA, and green fluorescence intensity of EGFP in cells were measured to establish a linear relationship between these parameters and pollutant dose. In this way, the level of EGFP gene promoter methylation was linked to the level of cell green fluorescence intensity, so that the former could be calculated from the latter. Thus, the lower the level of methylation of the EGFP gene promoter of cultured cells, the stronger the green fluorescence. [15] When pollutants and HepG-2 cells transfected with hypermethylated pEGFP-C3 were co-cultured, those with a strong DNA demethylation function exhibited significantly enhanced green fluorescence; such pollutants were considered to have low-methylation epigenetic toxicity. [16] In contrast, the lack of a significant increase in green fluorescence was indicative of weak DNA demethylation potential, and such compounds were deemed to have low-methylation epigenetic toxicity (Fig. 1).

Fig. 1. Chemical contaminants demethylation principles of epigenetic toxicity. A: Positive test substance, B: control.

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2 Materials and Methods 2.1 Preparation of the Methylated C3 Plasmid DNA (500 ng) digested with the restriction enzymes MspΙ and HpaΙΙ was treated with sodium bisulfite as previously reported (Gonzalgo, 2002; Xianliang, 2006). [17] 2.2 Cell Culture and C3 Plasmid Transfection The human liver cancer cell line HepG-2 was purchased from the China Type Culture Collection (Chinese Academy of Medical Sciences, Beijing, China) and grown in DMEM supplemented with 10% fetal bovine serum (FBS; JRH Bioscience, San Antonio, TX, USA) and 1% NEAA (Chinese Academy of Medical Sciences). A 599 kb CMV promoter of the pEGFP-C3 plasmid was transfected into HepG-2 cells using the FuGENE HD transfection reagent (F. Hoffmann-La Roche Ltd, Basel, Switzerland). This reagent (3, 4, 5, 6, or 7 μl) was directly pipetted into the medium containing the diluted pEGFP-C3 DNA (0.02 μg/μl) without allowing contact with the walls of the plastic tubes. The transfection reagent:DNA complex was incubated for 15 min at room temperature, and then the transfection complex was added to the cells in a drop-wise manner. The wells were swirled to ensure distribution over the entire plate surface. Cell growth was best and the fluorescence was brightest when 7 μl of the transfection reagent were used. 2.3 Treatment with 5-aza-dC Cells were seeded at a density of 3×105 cells/10 cm dish on day 0. On day 1, the medium was changed to one containing 5-aza-dC (Amresco Co Ltd, Amresco, ,USA), which was freshly dissolved in PBS and filtered through a 0.2μm filter. On day 4, cells were harvested and genomic DNA was obtained by serial extraction with phenol/chloroform and ethanol precipitation. Total RNA was extracted by ISOGEN kit (QIAGEN Co Ltd, Germany) HepG-2 cells, into which the hypermethylation EGFP-C3 plasmid gene had been introduced by homologous recombination, were exposed to heat at 37 °C for 6 h on days 2, 3, and 4 because this treatment effectively induced higher expression of hypermethylated EGFP-C3 mRNA. 2.4 Quantification of Methylation of the CMV Promoter In the next step, competent bacteria were prepared using E. coli strain DH5α. After the PCR product was purified, it was linked with the pGEM2T vector, transformed into the competent bacteria, and screened via blue-white screening. The product was separated by 1.5% agarose gel electrophoresis and photographed using a gel imaging system camera. Positive clones were screened by PCR amplification, and bacteria were cultured with shaking at 37 °C, and confirmed using universal primer T7 and SP6 sequencing by the Beijing Genomics Institute of Bacteria Biotechnology Company.

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2.5 Relative Quantification of CMV Promoter by Real-Time PCR To determine the quantitative PCR amplification efficiency, a PCR standard curve was generated. Samples of extracted cDNA were used to create a series of solutions representing 1/8, 1/4, 1/2, 1, and 1.5 times the cDNA content of β1, β2, and β actin cDNA, and these samples were analyzed using TaKaRa's sybrr premix Ex Taq™ quantitative , (TaKaRa' Ltd, Tokyo,Japan) PCR kit. In a 12.5 μl reaction system, in which the first three times higher than the concentration of cDNA diluted liquid samples, the following components (1 μl) were added. Components are composed of 1.5 times the liquid content (1.5 μl) is added, adding sybr quantitative PCR and reagent mixture (6.25 μl), the upstream and downstream primer mix solution (0.25 μl), the final complement to 12.5 μl with additional ultrapure water of sterilization. The PCR reaction conditions were as follows: predenaturation at 94°C for 10 s; denaturation at 94 °C for 45 s; annealing at 57 °C for 45 s; extension at 72 °C for 45 s; 95 °C for 1 min post-production thermal melting point curve to determine the amplification product specificity. Real-time PCR analysis was performed using a Stratagene MX3100 (USA). Glyceraldehyde-3-phosphate dehydrogenase (GAPDH) was used to normalize the mRNA expression levels. Primer sequences were as follows: EGFP-C3, 5'TAATGGGAGTTTGTTTTGGTATT-3' (sense), 5'-TTA TAC TCC AAC TTA TAC CCC AAA A-3' (antisense); and GAPDH, 3'-AGGTGAAGGTCGGAGTCAACG-3' (sense), 5' -AGGGGTCATTGATGGCAACA-3' (antisense). For both genes, PCR was performed for 40 cycles with annealing at 60 °C. Primer sequences and PCR conditions for other genes have been reported. 2.6 Detection of EGFP Protein by Fluorescent Imaging and Flow Cytometry Prior to flow cytometry and fluorescent imaging analysis, cells were washed with PBS and fixed with 4% paraformaldehyde. Cells were assayed using a Beckman Coulter Altra flow cytometer. Cells were placed into 24-well. After cells digestion, adding 1 ml PBS puff uniform each sample was placed into a 1.5 ml tube and centrifuged at 2000 g to remove the supernatant. Next, 1 ml PBS was added, followed by washing and centrifugation. 1 ml PBS re-suspended cell pellet, gently puff uniform, through 300 mesh filter to the stream-specific tube, into flow cytometry and to detect positive EGFP fluorescent percentage. 2.7 Analysis of Environmental Samples Using the QDMP Fish and shellfish samples were collected from the coast of Tianjin and washed, dried, and digested as previously reported (C. Brunori 2006) [18] Laid in Tianjin coastal sampling sites from north to south, a total of eleven sampling points, including two breeding base for seawater samples, seven samples for the marine capture fishing port, two sampling points for the seafood wholesale market, are important local sea Product supply base. Representative to collect the local residents like to eat fish, shellfish, eight varieties, a total of forty samples of the same species in each location, as close to the size of individual samples collected. For each sample type, the digestion solution was transferred carefully to a 1.5 ml polypropylene centrifuge tube. The sample was adjusted to a concentration of 1 g/L, and the test liquid was assayed for HepG-2 cell EGFP green fluorescence using the QDMP.

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3 Results and Discussion 3.1 Preparation of Methylated C3 Plasmid Unmethylated plasmids that were digested with HpaΙΙ and MspΙ and methylated plasmids digested with MspI exhibited the "smearing" phenomenon on gels, whereas methylated plasmids digested with HpaΙΙ showed a single band, suggesting the plasmid that treatment by methylation in the methylation status. 3.2 Exposure Optimization To establish a sensitive detection system, we first searched for an endogenous methylated CMV promoter in HepG2 cells that would show a sensitive response to 5AZA-dC. The CMV promoter of pEGFP-C3 was found to be methylated in HepG2 cells, and no mRNA expression was detected by RT-PCR before the addition of 5AZA-dC. By treating cells with very low doses (0.02–0.1 μM) of 5-aza-dC for 30 h, expression of pEGFP-C3 was restored. Next, expression levels were quantitatively analyzed by quantitative real-time RT-PCR, and CMV of pEGFP-C3 was found to be sensitively and abundantly expressed. 3.3 The Demethylation Potential of 5-aza-dC and Its Effect on EGFP Expression Methylation of the CMV promoter. The sequencing results were compared with the sequence of the C3 plasmid EGFP gene CpG island DNA from the GenBank database. Our sequence showed no mutation and the CpG cytosines were not modified by sodium bisulfite; these results indicated that the HepG2 cell lineC3 plasmid EGFP gene sub-CpG islands were methylated. Figure 2 shows

Fig. 2. Representative treated with sodium bisulfite sequencing group and the untreated group. Note: the picture in the sequence of the nine target sites in the CG, the figure shows the sites of which three from the methylation sites, six loci from the non-methylated sites; the chart below shows nine loci in the methylation status of all.

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are presentative sequence diagram and pattern of methylation, which showed that methylation of plasmid construction management is better, the plasmid EGFP gene promoter methylation after treatment with 90.4% of CG sites was methylated, and hypermethylation status of the EGFP gene promoter was quantitatively expressed. Relative quantification of EGFP expression. Quantitative real-time PCR detection results map: real-time fluorescence detection of fluorescence were displayed all sorts of quality control within the 22 cycles are significantly enhanced, showing all sorts of quality control have been good amplification. The fluorescence signal was stable, and a moderate elevation gradient was observed (Fig. 3). Real-time quantitative PCR results showed that the EGFP expression vector and the relative amount of 5-aza-dC exhibited a good dose-response relationship: y = 37.022x + 0.3087; R2 = 0.9821.

Fig. 3. Real-time quantitative PCR detection of cell sample results and dissolution curves of PCR products

Fluorescence of the EGFP protein. An obvious fluorescence gradient was detected in samples cultured with different doses of 5-aza-dC for 30 h; as the 5-aza-dC dose increased, the fluorescence intensity also increased. Flow cytometry results revealed a

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good linear relationship between the 5-aza-dC dose gradient and the mean fluorescence intensity :y = 10.402x + 6.0334; R2 = 0.829. This result shows that eukaryotic cells containing the green fluorescent protein gene vector exhibit a methylation-sensitive response relationship. The intensity of cellular green fluorescence can be used as an indicator of the presence of certain pollutants. [18] 3.4 Performance of the QDMP The sensitivity of the assay system was tested by adding various doses of 5-aza-dC to it. Demethylation of the promoter CMV of pEGFP-C3 was clearly observed at doses of 0.1μM or higher. Furthermore, fluorescence intensity was induced in a dose-dependent manner. We also examined the appearance of green fluorescence before and after the addition of 5-aza-dC. Under a fluorescence microscope, significant fluorescence was observed in the cytoplasm after the addition of 0.0008 μM of 5-aza-dC.

Fig. 4. Green fluorescent flow cytometry results

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3.5 The Application of QDMP for Aquatic Sampling To aquatic mixture for detection, according to the above method based on fluorescence flow cytometry to the relationship between methylation capacity equation, there are eight positive, obtained demethylation potential of the port of Tianjin Tanggu is strongest, equally 8 times the minimum concentration of 5-aza-dC dose toxicity. The epigenetic toxicity of polluted marine samples collected from the coast of Tianjin was evaluated based on the HepG-2 cell line reporter gene vector (i.e., the QDMP). The heavy metal extracts from aquatic samples were prepared and then cocultured with the test system. The demethylation potential of the samples was quantified relative to the corresponding equivalent of 5-aza-dC. EGFP fluorescence was quantified using microscopy. Demethylation by 5-aza-dC could be successfully detected in a quantitative manner at doses between 0.0008 and 0.1µM. Nine of the 19 aquatic samples had a relatively strong demethylation ability, with values ranging from 0.0064 to 0.0387µM 5-aza-dC equivalents. (Fig. 4)

4 Conclusions A system to assay for demethylating agents was established using the promoter CMV of the plasmid pEGFP-C3. To our knowledge, this is the first assay system that uses an endogenous CMV promoter. Methylation stability is high for endogenous sequences , and CGIs(CpG islands)in promoter regions have higher fidelity than those outside [19]. Mechanisms involved in the maintenance and monitoring of the methylated status of CGIs are expected to function well for the promoter CMV of the plasmid pEGFP-C3. Therefore, accurate estimation of epimutagens should be possible with this system. To establish a sensitive detection system, it was necessary to use a methylated promoter CMV that responds to low doses of demethylating agents (i.e., 5-aza-dC). In this study, we used the CMV promoter of the plasmid pEGFP-C3 because our recent studies [20] identified it as one that met the requirements. The CMV was demethylated by 5-AZA-dC at doses as low as 0.01 μM in parental HepG2 cells. This represents high sensitivity, considering that laboratory use of 5-aza-dC is between 0.1 and 10 μM. Demethylation of the CMV and expression of the introduced EGFP were observed after addition of 5-aza-dC at doses of 0.1 μM or higher. Fluorescence of the EGFP product was detected by fluorescence microscopy after addition of 1 μM 5AZA-dC. In summary, we established a detection system for demethylating agents using an endogenous promoter CMV; this system is expected to allow accurate detection of epimutagens. Acknowledgements. This work was supported by the National Nature Science Foundation of China(No. 20907047) and National Nonprofit Institute Research Grant of CRAES (No. 2008KYYW05).

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References 1. Eriksen, T.A., Kadziola, A., Larsen, S.: Binding of cations in Bacillus subtilis 2.

3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17.

18.

phosphoribosyldiphosphate synthetase and their role in catalysis. Protein Sci. 11, 271–279 (2002) Zoref, E., Vries, A.D., Sperling, O.: Mutant feedback-resistant phosphoribosylpyrophosphate synthetase associated with purine overproduction and gout. Phosphoribosylpyrophosphate and purine metabolism in cultured fibroblasts. J. Clin. Invest. 56, 1093–1099 (1975) Becker, M.A., Smith, P.R., Taylor, W., Mustafi, R., Switzer, R.L.: The genetic and functional basis of purine nucleotide feedback-resistant phosphoribo sylpyro phosphate synthetase superactivity. J. Clin. Invest. 96, 2133–2141 (1995) Reichard, J.F., Schnekenburger, M., Puga, A.: Long term low-dose arsenic exposure induces loss of DNA methylation. Biochem. Biophys. Res. Commun. 352, 188–192 (2007) Olaharski, A.J., Rine, J., Marshall, B.L., et al.: The flavoring agent dihydrocoumarin reverses epigenetic silencing and inhibits sirtuin deacetylases. PLoS Genet. 1(6), e77 (2005) Birnbaum, L.S., Fenton, S.E.: Cancer and developmental exposure to endocrine disruptors. Environ. Health Perspect. 111, 389–394 (2003) Salnikow, K., Zhitkovich, A.: Genetic and epigenetic mechanisms in metal carcinogenesis and cocarcinogenesis: nickel, arsenic, and chromium. Chem. Res. Toxicol. 21, 28–44 (2008) Tang, W.Y., Newbold, R., Mardilovich, K., et al.: Persistent hypomethylation in the promoter of nucleosomal binding protein 1 (Nsbp1) correlates with overexpression of Nsbp1 in mouse uteri neonatally exposed to diethylstilbestrol or genistein. Endocrinology 149, 5922–5931 (2008) Reik, W., Dean, W., Walter, J.: Epigenetic reprogramming in mammalian development. Science 293, 1089–1093 (2001) Bombail, V., Moggs, J.G., Orphanides, G.: Perturbation of epigenetic status by toxicants. Toxicol. Lett. 149, 51–58 (2004) Feil, R.: Environmental and nutritional effects on the epigenetic regulation of genes. Mutat. Res. 600, 46–57 (2006) Wu, C., Morris, J.R.: Genes, genetics, and epigenetics: a correspondence. Science 293, 1103–1105 (2001) Feinberg, A.P., Ohlsson, R., Henikoff, S.: The epigenetic progenitor origin of human cancer. Nat. Rev. Genet. 7, 21–33 (2006) Suzuki, M.M., Bird, A.: DNA methylation landscapes: provocative insights from epigenomics. Nat. Rev. Genet. 9, 465–476 (2008) Barreto, G., Schaefer, A., Marhold, J., et al.: Gadd45α promotes epigenetic gene activation by repair-mediated DNA demethylation. Nature 445, 671–675 (2007) Wade, P.A., Archer, T.K.: Epigenetics: environmental instructions for the genome. Environ. Health Perspect. 114, A140–A141 (2006) Schmelz, K., Sattler, N., Wagner, M., et al.: Induction of gene expression by 5-aza-2’deoxycytidine in acute myeloid leukemia (AML) and myelodysplastic syndrome (MDS) but not epithelial cells by DNA-methylation-dependent and -independent mechanisms. Leukemia 19, 103–111 (2005) Olaharski, A.J., Rine, J., Marshall, B.L., et al.: The flavoring agent dihydrocoumarin reverses epigenetic silencing and inhibits sirtuin deacetylases. PLoS Genet. 1, e77 (2005)

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19. Appanah, R., Dickerson, D.R., Goyal, P., et al.: An unmethylated 3’ promoter-proximal region is required for efficient transcription initiation. PLoS Genet. 3, e27 (2007)

20. Okochi-Takada, E., Ichimura, S., Kaneda, A., et al.: Establishment of a detection system 21. 22. 23. 24.

for demethylating agents using an endogenous promoter CpG island. Mutat. Res. 568, 187–194 (2004) Wang, X., et al.: High-throughput assay of DNA methylation based on methylationspecific primer and SAGE. Biochem. Biophys. Res. Commun. 341, 749–754 (2006) Brunori, C., Ipolyi, I., Massanisso, P., Morabito, R.: New Trends in Sample Preparation Methods for the Determination of Organotin Compounds in Marine Matrices. Handbook Environment Chemistry, Part O(5), 51–70 (2006) Barreto, G., Schaefer, A., Marhold, J., et al.: Gadd45α promotes epigenetic gene activation by repair-mediated DNA demethylation. Nature 445, 671–675 (2007) Cheetham, S., Tang, M.J., Mesak, F., et al.: SPARC promoter hypermethylation in colorectal cancers can be reversed by 5-aza-2’deoxycytidine to increase SPARC expression and improve therapy response. Br. J. Cancer 98, 1810–1819 (2008)

Study of Quantitative Evaluation of the Effect of Prestack Noise Attenuation on Angle Gather Junhua Zhang, Jing Wang, Xiaoteng Liang, Shaomei Zhang, and Shengtao Zang College of Geo-Resource and Information, China University of Petroleum, Qingdao, 266555

Abstract. With the increasing refinement of exploration activity, the information of different angles has to be used in the extraction of prestack attributes and prestack inversion. For large angle gather, denoising methods should be different from conventional prestack noise attenuation methods used in CSP and CMP gathers. We established specific theoretical model and quantitatively study the effect of prestack noise attenuation on angle gather. Through the research, we draw the following conclusions: 1) The antinoise ability of angle gather stacking is much better than that of CMP stacking. 2) The larger the apparent velocity of coherent noise is, the more seriously the effect of it on angle gather. Otherwise it is opposite. 3) Surface wave has strong energy and low frequency and a small amount of residues will affect the angle gather largely. After suppressing surface wave, there will be energy loss in near angle gather, which should be compensated. Keywords: angle gather, random noise, coherent noise, surface wave, quantitative evaluation of noise attenuation.

1 Introduction Seismic data preserved-amplitude processing is crucial to achieve reliable prestack characteristic and prestack inversion results. Although the importance of AVO forward and inversion technique has been shown by Jonathan E et al. (2006) [1] and Heidi Anderson Kuzma et al. (2005) [2], there will be false abnormal responses in AVO analysis because of disadvantages of prestack noise attenuation and energy compensation methods which easily lead to energy inconsistencies of near and far offset. Then the following inversion and attribute analysis processing will appears multi-solutions, making inversion results not able to truly reflect subsurface lithology and physical property changes, which is not conducive to lithologic inversion, reservoir prediction and fluid discrimination, as for example in Gary Mavko et al.(1998) [3], Yinbin Liu et al.(2003) [4] and Fatti J L et al.(1994) [5]. The information of different angles has to be used in the extraction of prestack attributes and prestack inversion. For big angle gather, because of the characteristics of angle gather and the impact of oil/gas AVO and random noise and coherent noise, the denoising methods may be different from conventional methods used in CSP and CMP gathers. As the real data is complex and direct quantitative evaluation is not S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 72–77, 2011. © Springer-Verlag Berlin Heidelberg 2011

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easy to achieve, it has both necessity of theoretical study and feasibility of achieving that we establish specific theoretical model and quantitatively study the effect of prestack noise attenuation on angle gather. Now we will carry out quantitative evaluation for random noise, coherent noise and surface wave always occurring in real data.

2 Quantitative Evaluation of the Effect of Random Noise Attenuation on Angle Gathers We established the horizontal layer model (Figure 1). Figure 1, left shows velocity model and right shows the shot gather. The model has 7 layers and the fourth layer is used for quantitative evaluation. The number of total traces is 241, sampling points is 3001, sampling interval is 2ms and trace offset is 50m. While establishing model, we applied time-varying wavelet and the frequency from the first layer to the seventh layer becomes smaller. Based on this model, we added random noise of signal to noise ratio(S/N) of 3, 2, 1 and 0.5, then we made quantitative evaluation of the effect of random noise attenuation. By experiment, we found that adding random noise with S/N of 3 and 2 is similar to that of 1. In the paper, for the limitation of space, we discuss the S/N 1 directly.

Fig. 1. Horizontal layer model. Left) Velocity model. Right) Shot gather.

Figure 2, left, shows CMP gather before and after NMO and the angle gather without random noise and with noise of signal to noise ratio of 1 and 0.5. While S/N is 1, random noise affects event slightly and AVO phenomenon can be obviously observed. After attenuating noise, a small amount of residual random noise has a little effect on events. While S/N is 0.5, random noise affects event very largely and we can’t observe AVO phenomenon. After attenuating noise, a great amount of residual random noise affects angle gather to a large extent. In order to see more obvious changes in angle gather, we selectively stacked 1º13º, 13º-26º and 27º-39º of the angle gather into one trace and repeated it five times (Figure 3). When S/N is 1, AVO phenomenon of small reflection efficient has been difficult to distinguish visually, the middle angle information in the deeper formation has been largely polluted and far angle information has been also greatly affected. After attenuating noise, the middle angle gather in the deeper formation has been

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clearly improved, but the distortion of the contaminated weak signal slightly increased. When S/N is 0.5, only strong reflection information can be distinguished. Far angle, middle angle and near angle gather are all greatly polluted. After attenuating noise, the angle gather section has been greatly improved entirely, but the contaminated events of far angle, middle angle and near angle gather have not.

Fig. 2. Stacking angle gather. From left to right: without random noise; before attenuating random noise (S/N=1), after attenuating random noise(S/N=1), before attenuating random noise(S/N=0.5) and after attenuating random noise(S/N=0.5).

Fig. 3. Stacking angle gather. From left to right: without random noise; before attenuating random noise (S/N=1), after attenuating random noise, before attenuating random noise(S/N=0.5) and after attenuating random noise.

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Fig. 4. Comparison of AVA curve of the fourth layer. Left) S/N=1. Right) S/N=0.5.

Figure 4 shows comparison of AVA curve of the fourth layer when S/N is 1 and 0.5. We can find that as signal to noise ratio decreases, the jitter phenomenon is more and more serious and the reflection coefficient occurred a very significant jump.

3 Quantitative Evaluation of the Effect of Coherent Noise Attenuation on Angle Gathers Based on the horizontal layer model (Figure 1), we adding coherent noise to carry out quantitative evaluation (Figure 5). Figure 5, left, shows coherent noise has a certain degree of cross with effective wave, and the apparent velocity changes a little on angle gather section. Right shows coherent noise has almost been eliminated, only a small amount of which is remained because of edge effect. Figure 5 right shows that inphase property of coherent noise has almost disappeared, but some glitches occur on stacking angle gather section. After attenuating coherent noise there is little effect of coherent noise on angle gather section, but there are residual locally, especially in big angle traces.

Fig. 5. From left to right: CMP gather after NMO and angle gather before and after attenuating coherent noise, stacking angle gather before and after attenuating coherent noise

Fig. 6. Comparison of AVA curve of the fourth layer

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Figure 6 shows comparison of AVA curve of the fourth layer before and after attenuating coherent noise. We can clearly observe the main interference point of coherent noise, and the reflection coefficient has a very significant jump.

4 Quantitative Evaluation of the Effect of Surface Wave Attenuation on Angle Gathers We carry out finite difference surface wave simulation through programming. A free surface boundary condition for numerical simulation of Rayleigh wave is built by using 2×10 order high precision staggered grids finite difference when the source is at the free surface. PML absorbing boundary condition is applied to eliminate boundary reflections, as shown in Xu Yixian et al. (2007) [6] and Saenger E H et al. (2004) [7]. Then we add surface wave into the horizontal layer model (Figure 1) and get the results shown in Figure7. From Figure 7 we can see that surface wave has almost been eliminated, only a small amount of which is remained. Figure 7 right shows that surface wave largely affects angle gather because of strong energy. The events in near angle gather can hardly be distinguished, but middle angle and far angle gather are less affected. After attenuating surface wave, there is little effect of surface wave on angle gather section. However, because surface wave has strong energy and low frequency, a small amount of residue affects the angle gather largely, especially affects the near offset. Figure 8 shows the comparison of AVA curve of the fourth layer. We find that the effect of surface wave mainly occurs in near offset and reflection coefficient has a slight jump after attenuating it.

Fig. 7. From left to right: CMP gather after NMO and angle gather before and after attenuating surface wave, stacking angle gather before and after attenuating surface wave

Fig. 8. Comparison of AVA curve of the fourth layer

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5 Conclusions Through the analysis, we draw the following conclusions: 1) Through adding random noise of different signal to noise ratio and attenuating it, we find that the anti-noise ability of angle gather stacking is much better than that of stacking CMP gather. For original angle gather, the threshold of signal to noise ratio is 1. For angle gather stacking, the minimum signal to noise ratio can be up to 0.5. 2) Through adding random noise of different signal to noise ratio and attenuating it, we find that the effect of coherent noise on angle gather is smaller than that on CMP gather. After attenuating coherent noise, stacking angle gather became better and only a small amount of high frequency glitches are remained. The larger the apparent velocity of coherent noise is, the more seriously the effect of it on angle gather. 3) Based on attenuating surface wave, we find that surface wave has strong energy and low frequency and a small amount of residue will affect the angle gather largely. As surface wave mainly occurs in near offset, the effect of it on near offset than that on far offset. After suppressing surface wave, there will be energy loss in near angle gather, which should be compensated.

References 1. Downton, J.E., Ursenbach, C.: Linearized amplitude variation with offset (AVO) inversion with supercritical angles. Geophysics 71(5), E49–E55 (2006) 2. Kuzma, H.A., Rector, J.W.: The zoeppritz equations, information theory and support vector machines. In: SEG/Houston 2005 Annual Meeting, pp. 1701–1705 (2005) 3. Mavko, G., Mukerji, T.: A rock physics strategy for quantifying uncertainty in common hydrocarbon indicators. Geophysics 63(6), 1997–2008 (1998) 4. Liu, Y., Schmitt, D.R.: Amplitude and AVO responses of a single thin bed. Geophysics 68(4), 1161–1168 (2003) 5. Fatti, J.L., Vail, P.J., Smith, G.C., et al.: Detection of gas in sandstone reservoirs using AVO analysis: A case seismic case history using the Geostack technique. Geophysics 59(5), 1362–1376 (1994) 6. Xu, Y., Xia, J., Miller, R.D.: Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach. Geophysics 72(5), 147–153 (2007) 7. Saenger, E.H., Bohlen, T.: Finite difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid. Geophysics 69, 583–591 (2004)

A User Model for Recommendation Based on Facial Expression Recognition Quan Lu1, Dezhao Chen2, and Jiayin Huang2 1

Center for Studies of Information Resources, Wuhan University, Wuhan, China 2 Research Center for China Science Evaluation, Wuhan University, Wuhan, China [email protected], [email protected], [email protected]

Abstract. User modeling is crucial in recommendation system. By analyzing users’ behavior, gathering users’ interest information, the user model can well express the users’ need. While users’ emotion is also good expression of users’ needs. A user model based on facial expression recognition is built. The user model is on the base of traditional user model, affective function α is provided to reflect user’s emotion. As well as the user model updating times t is also discussed in the model. And then a discussion of the updating strategy and application in recommendation is had. Keywords: user model, recommendation, facial expression recognition, emotion recognition.

1 Introduction User modeling is crucial in personalized services such as recommendation system. Personalized service provides the necessary information to the user who needs it, and the user model expresses the user’s need. User modeling firstly needs getting user’s characteristics, such as user’s profile, user’s behavior when browsing and clicking. Some researchers have researched in user modeling and provided some methods. Topic method uses some topics to express user’s interest and this method has applied to My Yahoo system[1]. Bookmark and keywords are also used to express user’s interest, some systems such as SiteSeer[2], WebWatcher[3] use the method to recommend users interesting information. To reflect user’s semantic information, ontology technology has adopted to the user model. Researchers build domain ontology to mine user’s interesting information, and track user’s interest shifting[4]. User’s interest shifts as the environment changes. User model should reflect the dynamic interest. Document[5] founds three hierarchical user model and uses weight function to update user’s interesting information. Users’ emotion may also express users’ interest, while user models at present don’t contain the emotion. A user model based on facial expression recognition is built in this paper. This model avoids the fault that a user clicked some information while he isn’t interested in them. S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 78–82, 2011. © Springer-Verlag Berlin Heidelberg 2011

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Section 1 makes an introduction to this paper, Section 2 provides the user model and section 3 gives the method to build the user model, including t and α value. Section 4 introduces updating mechanism of this model. Section 5 makes a summary of this paper.

2 User Modeling Based on Facial Expression Recognition When a user browses one topic of information, a reaction may be made to it, then he may click it, restore it, and his expression may also reflect his interest. In the user model, emotion is important to obtain user’s interest. User’s emotion can be recognized and expressed, document [6] has researched in the affective visualization, document [7] has built affective model for emotion. The user model built in this paper is combined with affective model, and user’s emotion is added to the model. This user model can be described as:

U = {T , C , W , t , α }

(1)

Ti = {(C , W ), t i , α i }

(2)

(C ,W ) = {(C1 ,W1 ), (C 2 , W2 ),.....(C n ,Wn )}

(3)

Formula (1) describes the elements of the model. In this model, T are the topics which the user is interested in. C represents the characteristic of the topic. W represents the weight related to the characteristic, t represents the times user model changes. α is an affective function reflecting the user emotion by facial expression recognition, α is emotional sign of interest degree. Every topic Ti has a series of (C, W) and ti, α i. Very active

Very negative

Very possitive

Very passive Fig. 1. Continuous emotion space

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The result affective function α computes is based on coordinate which decides the emotion of expression. User’s emotion is continuous and can be classified by two dimensions[8] , as it shows in firgure1. User’s emotion may be mapped in this space. In this model, it needn’t recognize the emotion, but the degree of the emotion, such as positive or negative. Because in recommendation system, t just knows whether user is interested in the topic or not, and to what degree. So it should map the emotion to the emotion dimension, then it may use 0 to express very negative emotion, and 1 very positive emotion, the value 0.5 just represents neutral.

3 User Model Building Building the user model, topics are from the information user browsed, such as the web page user clicked. Every topic has some characteristics and related weight, as it shows in Formula (2) and (3). Ci represents the characteristic of each topic and is decided by the statistics. Wi represents the weight of the characteristic and reflects the importance of the Ci. Ci is also expressed by ontology. Wi is decided by TF/IDF formula, TFIDF(ti) = TF(ti)*log(n/DF(ti))

(4)

TF(ti) represent times item ti occurs in the document, and DF(ti) represents the amount of documents. t represents the times topic T updates, it reflects the frequency user’s interest updates. If t is increasing, it may infer that the user is paying more attention to the topic. At the same time the t should include the time t value changes. The times user model updates and the time it updates will be referred when the model eliminates topics. To decide the topics, it should also gather user’s facial expression which is used to analyze user’s emotion. Affective function α computes user’s emotion, which is a sign of user’s interest. Affective function α computes user’s emotional value between 0 and 1, and the value represents different emotion in different degree. Affection computing depends on the facial expression recognition, and decides the emotion by analyzing the facial characteristics such as structure of face, shape and so on[9]. And then the function will compute the emotion, the result is presented as potential value. It is decided by expression potential formula.

K (e, Ex) =

1 1 + a || e − Exc || 2

(5)

||●||2 represents the distance norm of expression, a is a constant, controls the fading speed of basic expression Ex’s potential. Exc represents the center of expression. Expression potential can be used to classify expression, and then the expression should be mapped to the emotion value, its value is between 0 and 1. Affective function α can express user’s real emotion, avoiding the problem of traditional model that user clicks a web page or browses one topic information while he isn’t interested in it.

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4 User Model Updating and Shifting Mechanism After building the user model, it will update in the following learning. It includes two aspects: one is to update the model, such as to adjust the value Ci, Wi, α , t , and to add new topic. Another is to eliminate some topics when the storage is not so large. When a topic is in the user model, the user model will compute similarity based on SVM theory. And then the user model will be updated including the Ci, Wi, α , t. Every time the topic updates, the t related to the topic will be added: t = t+1

(6)

When a new topic is found, it will be added to the model, if the storage can store the topic, or some topics will be eliminated. Elimination strategy includes α , and t. If the emotion value according to α is low, it indicates that user’s interest is not so strong. So the topic will have priority of elimination. If the t is low, it may indicate that this topic is not updated constantly and user doesn’t browse this topic information. This topic may be context information and user does not pay attention to this topic after a period of time. From the point view of information lifecycle management theory, the topic has no value or low value[10]. So the topic is chosen to eliminate. If t is high but the time model updates is long, it means the model doesn’t update for a long time, so the topic may be out of date, and the topic can be chosen. The long time is relative. When deciding which topic to be eliminated, the user model should consider t and α , and adjust the user model to reflect user’s interest.

5 Summary This paper makes an introduction to our user model, and details the processing and application of the user model. In the personalized service, users’ emotion is also important, while the user model doesn’t express it. So the model takes advantage of the facial expression recognition and builds a user model which contains users’ emotion. This model utilizes affective function α , as a sign of users’ interest. At the same time, this model provides t to reflect the frequency of the topic changes. t is also used to decide whether the user pays attention to the topic or not. t and α values are important in updating the model and recommending information to users. As users require more accurate information and affective computing develops, user model will adopt the emotion factor and applied to the recommendation system. Acknowledgement. This research is supported by National Natural Science Foundation of China (No: 70833005), the MOE Project of Key Research Institute of Humanities and Social Sciences at Universities (No: 2009JJD870002) and Education Ministry's Humanities & Social Sciences Program (No: 09YJC870020).

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References 1. Ying, X.: The Research on User Modeling for Internet Personalized Services. National University of Defence Technology (2003) 2. Rucker, J., Polanco, M.J.: Siteseer: Personalized Navigation for the Web. Communications of the ACM 40(3), 73–75 (1997) 3. Joachims, T., Freitag, D., Mitchell, T.: WebWatcher:A tour guide for the world wide web. In: Artificial Intelligence, Japan (August 1998) 4. Yan, D., Liu, M., Xu, Y.: Toward Fine-rained User Preference Modeling Based on Domain Ontology. Journal of the China Society for Scientific and Technical Information 29(3), 442–443 (2010) 5. Li, S.: The Representation and Update for User Profile in Personalized Service. Journal of the China Society for Scientific and Technical Information 29(1), 67–71 (2010) 6. Zhang, S., Huang, Q.: Affective Visualization and Retrieval for Music Video. IEEE Transactions on Multimedia 12(6) (2010) 7. Qin, Y., Zhang, X.: A HMM-Based Fuzzy Affective Model For Emotional Speech Synthesis. In: 2nd International Conference on Signal Processing Systems (2010) 8. Li, J.: Study on Mapping Method of Image Features and Emotional Semantics. Taiyuan University of Technology (2008) 9. Skelley, J.P.: Experiments in Expression Recognition. Master’s thesis, Massachusetts Institute of Technology, EECS (2005) 10. Rief, T.: Information lifecycle management. Computer Technology Review 23(8), 38–39 (2003)

An Improved Sub-Pixel Location Method for Image Measurement Hu Zhou, Zhihui Liu, and Jianguo Yang College of Mechanics Engineering, Donghua University, Shanghai, China {tigerzhou,liuzhihui,jgyangm}@dhu.edu.cn Abstract. Sub-pixel edge location is an effective way to improve measurement accuracy. For the deficiency of traditional gaussian interpolation algorithm for sub-pixel location, this paper presents an improved algorithm: After obtaining pixel-accuracy edge of object to be measured by LoG operator, use Hough transform to get the curve slope of the boundary line and the normal of the corresponding point on the edge; Weighing Lagrange interpolation in the normal direction is performed to obtain the gray values in the direction of the gradient under new coordinate system; finally, perform sub-pixel relocation in the gradient direction based on the gaussian interpolation algorithm. Experimental results show that the improved method can get much better precision than the traditional algorithm. Keywords: Sub-Pixel Edge Detection, Precision Measurement, Machine Vision, Lagrange Interpolation.

1 Introduction Machine vision based measurement is one of the key technologies in manufacturing. It can perform shape and dimensional inspection to ensure that they lie within the required tolerances [1]. In order to improve the precision of image measurement, many scholars have put forth some effective sub-pixel location algorithm [2-6]. As the actual measurement system should meet the requirements of precision, efficiency and reliability, it should not be too complicated. Interpolation based sub-pixel edge detection method has been widely used in practice for its fast calculation speed. It uses the interpolation function to restore onedimensional continuous light intensity approximately. However, for the discrete distribution of pixel points in the digital image, the traditional gaussian interpolation algorithms are only available in horizontal, vertical and diagonal direction (± 45 °, ± 135 °). Certain errors will be generated inevitably for the location of arbitrary edge direction. Consequently, it is necessary to improve the algorithm to improve the sub-pixel location precision.

2 Principle and Problem 2.1 Sub-Pixel Location Algorithm Based on Gaussian Interpolation The distribution of gray values on the object edges for general image is shown in Fig. 1(a), and the distribution of the gray value difference is shown in Fig.1(b). S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 83–92, 2011. © Springer-Verlag Berlin Heidelberg 2011

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(b)

(a)

Fig. 1. Distribution of gray values and difference

The position on the maximum of the gray value difference is the boundary to discriminate the background and the object. Because the integration effects and optical diffraction effects of optical components, as well as the aberrations of the optical system, the change of the gray values become gradient style in the image, which should be drastic changes in reality. Classic edge extraction principle considers that the maximum difference presents the edges of image objects. According to Square aperture sampling theorems, optical components always perform the integration of the light intensity which projected onto the photosensitive surface with a fixed size area at fixed interval, the output results are the gray values in an image. As integral time and integral area is fixed, so the outputs depend only on the light intensity distribution on the surface. A pixel gray values output can be expressed as:

f (i , j ) = ∫

j +1 2 j −1 2

∫

i +1 2

i −1 2

g ( x, y )dxdy

(1)

Here, f (i, j ) is pixel gray values, g ( x, y ) is the light intensity distribution of the continuous images. Theoretically, variations of edge gray values should be a gaussian distribution, which is shown in Fig.2, the vertex position of the curve is the precise location of the edge point. Expression of the Gaussian curve is: y=

1 −( x − μ ) 2 exp( ) 2σ 2 2πσ

(2)

μ is a mean value; σ is a standard deviation. Because it is difficult to fit directly, and the purpose here is just to find the vertex position. Therefore perform logarithmic operation on both sides to transform the equation: ln y = −

( x − μ )2 1 + ln 2σ 2 2πσ

(3)

Obviously, the equation is conic style. To simplify the calculation, we can use the values after logarithmic operation to fit a parabola to obtain vertex coordinate. So, we use conic instead of Gaussian curve to improve the efficiency.

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Mm W2

W1

P1

P

Pmax

P2

Fig. 2. Variations of edge gray values

Surpose the conic form is y = Ax2 + Bx + C , the gray value output for each pixel is:

y (n) = ∫

n +1 2

n −1 2

( Ax 2 + Bx + C ) dx

(4)

Let the number of the maximum point of grayscale value difference be 0, and its value be represented as f0. This position can be calculated by classic operator mentioned above. The number of the two points which is nearby the maximum point are represented as -1 and 1, and their values are represented as f-1and f1 , We can get the gray values as follow: −1

1 ⎡1 ⎤ 2 13 ( Ax 2 + Bx + C )dx = ⎢ Ax3 + Bx 2 + Cx ⎥ = A − B + C −3 2 2 ⎣3 ⎦ − 32 12

f −1 = ∫

−1 2

1 A+C 12

(6)

13 A+ B +C 12

(7)

f0 =

f1 =

(5)

Combine the equations from (5) to (7), we can obtain the expression of A, B, C as follows: A=

1 ( f 1 + f −1 − 2 f 0 ) 2

B=

1 ( f 1 − f −1 ) 2

C=

13 1 1 f0 − f −1 − f 12 24 24

(8)

Thus, the abscissa value of the parabola’s vertex is: x=−

f1 − f −1 B = 2 A 2(2 f 0 − f1 − f −1 )

(9)

This solution is the result after taking logarithms in the Gaussian curve and the pixel gray values difference in (9) should be substituted by logarithms, so we get: x=

ln f1 − ln f −1 2(2 ln f 0 − ln f1 − ln f −1 )

(10)

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2.2 The Deficiency of Traditional Gaussian Interpolation Algorithm

For the discrete distribution of digital image’s pixel points, the traditional gaussian interpolation algorithms can only be performed in horizontal, vertical and diagonal direction (± 45 °, ± 135 °). In most cases, edges are in arbitrary direction, so the accuracy of location will be decreased inevitably. Fig.3 shows an object edge of the original image. Fig.4(a) shows the trend curve of gray value change in arbitrary horizontal direction, and Fig.4(b) in gradient direction. As can be seen from the figures, the gray values in normal direction change more drastically than in the other direction, which means that edge location in normal direction will be more accurate.

Fig. 3. Variations of edge gray values

(a)

(b)

Fig. 4. Gray values change chart in horizontal and normal direction

3 Algorithm Optimization 3.1 Interpolation in the Gradient Direction

After obtained the edge location of object using LoG operator, we use Hough transform to get the curve slope of the boundary tangent line, and thus the corresponding normal could be obtained. Suppose the angle of edge normal to horizontal axis is θ, as shown in Fig.5. Take a certain edge point after pixel-precise location as center, rotate the coordinate system to make the normal line be the x-axis, then the edge direction become the y-axis. Do gaussian curve fitting in gradient direction to get the sub-pixel location more accurately.

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Fig. 5. Interpolation in edge normal direction

(a) Input image

(b) Output image

Fig. 6. The interpolation if image gray values

The algorithm is possible to achieve a high precision theoretically. However, there is a critical problem that must be resolved: because the discrete distribution of digital image pixels, the original pixel point f-1 and f1 in the new rotated coordinate system '

'

'

'

( f −1 and f1 ) may not be integers. The gray values of f −1 and f1 after coordinate rotation must be determined. To solve this problem, reintroduce gray value interpolation method, as shown in Fig. 6. Suppose the existing pixel gray values as f(u1,v1) f u2 v2 … f un vn , the new pixel gray values in the new coordinate system can be expressed as:

、 （ ， ）， ， （ ， )

n

f (u 0 , v0 ) = ∑ f(u i ,vi )h(u i − u0 , vi − v0 )

(11)

i =1

h(·,·) is interpolation kernel function,

f(ui ,vi )(i = 1,2,0..., n )

is weight coefficient.

3.2 The Chosen of Interpolation Kernel Function

The precision and calculation of interpolation algorithm depend on the interpolation kernel function, and the design of kernel function is the core of algorithm. The paper presents a weighing Lagrange interpolation algorithm to keep a balance between precision and complexity. From mathematical analysis we found that if any function f(x) has n+1 derivative at point x0, then the function can be expanded as a Taylor series in that neighborhood:

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f ( x ) = f ( x 0 ) + f' ( x 0 )( x − x 0 ) +

f (n ) (x 0 ) 1 (x − x 0 ) + R n (x ) f " ( x 0 )( x − x 0 ) 2 + " + n! 2!

f (n +1) (ξ )

(12)

n +1

In the formula, R n ( x ) = (n + 1)! (x − x 0 ) , which is Lagrange remainder. In General, second order Taylor series is sufficient to approach the original function and this paper try to use Taylor series to the interpolation function. As shown in Fig.7, a single pixel is surrounded by 4 pixels and we can choose any 3 points 3

( C 4 ) to do Lagrange interpolation. Finally, we take the average as the output gray value for the pixel. (0,0)

(0,1) (x,y)

(1,0)

(1,1)

Fig. 7. Interpolation of pixels

Suppose the gray value for points (0, 0), (0, 1) (1, 0), (1, 1) is y0, y1, y2 and y3 respectively. Do second order Lagrange interpolation for point (0, 0), (0, 1) and (1, 0) to get f0(x, y): f 0( x, y) = y 0

( x − x 0 )( x − x 2 ) ( x − x 0 )( x − x 1 ) ( x − x1 )( x − x 2 ) + y1 + y2 ( x 0 − x1 )( x 0 − x 2 ) ( x1 − x 0 )( x1 − x 2 ) ( x 2 − x1 )( x 2 − x 0 )

(13)

Similarly, Lagrange interpolation for points (0, 0), (0, 1) and (1, 1), we get f1(x,y): f 1(x, y) = y0

(x − x1 )(x − x 3 ) ( x − x 0 )(x − x 3 ) ( x − x 0 )(x − x1 ) + y1 + y3 ( x 0 − x1 )(x 0 − x 3 ) ( x1 − x 0 )(x1 − x 3 ) ( x 3 − x1 )(x 3 − x 0 )

(14)

Lagrange interpolation for points (0, 1), (1, 0) and (1, 1), we get f2(x, y): f 2( x , y) = y1

( x − x 2 )( x − x 3 ) ( x − x 1 )( x − x 3 ) ( x − x 2 )( x − x 1 ) + y2 + y3 ( x 1 − x 2 )( x 1 − x 3 ) ( x 2 − x 1 )( x 2 − x 3 ) ( x 3 − x 1 )( x 3 − x 2 )

(15)

Lagrange interpolation for points (0, 0), (1, 0) and (1, 0), we get f3(x, y): f 3( x , y) = y 0

( x − x 2 )( x − x 3 ) ( x − x 0 )(x − x 3 ) ( x − x 2 )(x − x 0 ) + y2 + y3 ( x 0 − x 2 )( x 0 − x 3 ) ( x 2 − x 0 )(x 2 − x 3 ) ( x 3 − x 0 )(x 3 − x 2 )

(16)

Finally, Calculated the pixel gray value f(x, y) for point (x,y): f ( x, y) =

1 (f 0 ( x, y) + f1 ( x, y) + f 2 (x, y) + f 3 ( x, y)) 4

(17)

After obtaining the gray values for the interpolation points in the gradient direction under the new coordinates system, we can relocate the edge points for sub-pixel precision in the gradient direction based on the Gaussian interpolation algorithm.

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4 Experiment Results In order to verify the algorithm, we adopted the standard Gauge which has a high quality of straight edges for sub-pixel location experiment. Fig.8(a) illustrate a standard Gauge with 20mm of working length (the long side is the working face) and Fig.8(b) shows the gray value data of gradual partial edge.

(a)

(b)

Fig. 8. Original image of Gauge and gray values of partial edge Table 1. The list of edge point coordinates by different edge extraction methods

No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Pixel precision (437,444) (437,445) (437,446) (438,447) (438,448) (438,449) (439,450) (439,451) (439,452) (440,453) (440,454) (440,455) (441,456) (441,457) (441,458) (442,459) (442,460) (442,461) (442,462) (443,463) (443,464) (443,465)

Traditional Interpolation (436.7507, 444.1063 ) (437.1611,444.7419 ) (437.3867,446.5000) (437.7011,446.7521) (438.2676,448.4210) (438.4088,448.5000) (438.5354,449.8514) (439.1241, 451.3852) (439.3931,452.0611) (439.5411, 453.4563) (440.1897,454.4351) (440.3845,454.7162) (440.6585,455.8461) (441.0398, 456.6891) (441.2947,458.0000) (441.6210,458.6502) (441.7600,460.4556) (442.2098, 461.4322) (442.3906,461.6704) (442.8143,463.4500) (443.0912,464.6721) (443.3627,465.2967)

Improved algorithm (436.9439,444.0181) (437.1932,444.9377) (437.3832,445.8765) (437.9123,447.0282) (438.2705,447.9128) (438.4052,448.8693) (438.7990,450.0648) (439.1645,450.9470) (439.3851,451.8758) (439.7986,453.0649) (440.2177,453.9298) (440.3815,454.8770) (440.8256,456.0562) (441.1015,456.9673) (441.2945,457.9050) (441.3920,459.1961) (441.9337,460.0214) (442.2306,460.9253) (442.3854,461.8757) (442.9165,463.0269) (443.1451,463.9532) (443.3624,464.8831)

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Execute the LoG edge extraction (σ=1, thresh=70) and we can get the edge binary image. Do Hough transform for the left side edge we get the line equation as y=3.0994x-909.3564. To precisely locate the edge point, rotate the coordinate system about the LoG zero crossing point to make the edge be y-axis and the normal direction become x-axis. Conduct sub-pixel location according to the algorithm mentioned above. Notice that the operation is only carried out for the zero crossing point along the normal direction. Table 1 lists the edge point coordinates for pixel precision, sub-pixel precision by traditional Gaussian Interpolation and the sub-pixel precision by improved algorithm. Fig.9 shows the 2D curve drawn by positions of each sub-pixel points. We can find that the curve made by pixel level precision shapes like a zigzag, traditional gaussian interpolation method remedy such errors to some extent. However, because it was not carried on the edge’s normal, so it inevitably has some errors compared with the real edge. The improved algorithm executes second sampling in edge’s normal and uses 470 pixel accuracy edge traditional sub-pixel edge improved sub-pixel edge

465

y coordinate

460

455

450

445

440 436

437

438

439

440 x coordinate

441

442

443

444

Fig. 9. Comparison of edge point curve made from different algorithms

Fig. 10. Ring gauge of Ф 20

Fig. 11. Comparison of edge point curve

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second-order Lagrange interpolation algorithm to restore the gray values in the gradient direction. Consequently, this method can improve the accuracy of edge location to some extent. As shown in the figure, line connected by sub-pixel points obtained by executing improved algorithm, get even closer to the real straight line. Fig.10 is a ring gauge of Ф20. Similarly, We obtained the edge point coordinate by pixel precision, sub-pixel precision by tradition Gaussian interpolation, sub-pixel precision by improved algorithm. Fig.11 is the comparison of the edge curve derived from different methods. Fig.12 shows the error data curve between the improved arc sub-pixel location method and pixel-level location method. 

ẃᑺᮍ৥ᅮԡ䇃Ꮒ





Error values 





                      







Pixel point পḋ⚍ᑣ߫

Fig. 12. Error data curve

Because standard gauge boasts high-quality straight lines and curve edges, we can figure out the sub-pixel location precision by checking the shape errors of the object edges from the image. The shape errors of gauge are shown in Table 2 and we can find that the new algorithm improved the precision of sub-pixel location obviously. Table 2. Shape Errors of gauge image

Algorithm

Pixel-level Location

Straightness Radian

0.849 0.868

Traditional Location 0.318 0.403

Sub-pixel Improved Location 0.223 0.238

Sub-pixel

5 Summary Edge detection and sub-pixel precision location for edge points are the basis for image measurement. Classical Gaussian interpolation location can achieve high speed but low precision. As for arbitrary oriented edges, gaussian interpolation on the edge gradient can improve the location precision. To restore the gray values on the gradient direction, perform second order Lagrange interpolation and then the gaussian interpolation of sub-pixel relocation. Experiment proves that improved sub-pixel interpolation algorithm get much better precision than traditional algorithm. Acknowledgment. Supported by Chinese Universities Scientific Fund.

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References 1. Steger, C., Ulrich, M., Wiedemann, C.: Machine Vision Algorithms and Applications, pp. 1–2. Tsinghua University Press, Beijing (2008) 2. Qu, Y.D.: A fast subpixel edge detection method using Sobel-Zernike moment operator. Image and Vision Computing 23, 11–17 (2005) 3. Tabatabai, A.J., Mitchell, O.R.: Edge location to sub-pixel values in digital imagery. IEEE Trans. Pattern Anal. Machine Intell. PAMI-6(2), 188–201 (1984) 4. van Assen, H.C., Egmont-Petersen, M., Reiber, J.H.C.: Accurate object localization in gray level images using the center of gravity measure:accuracy versus precison. IEEE Transactions on Image Processing 11(12), 1379–1384 (2002) 5. Malamas, E.N., Petrakis, E.G.M., Zervakis, M., et al.: A survey on industrial vision systems, applications and tools. Image and Vision Computing 21, 171–188 (2003) 6. Li, Y., Pang, J.-x.: Sub-pixel edge detection based on spline interpolation of D2 and LoG operator. Journal of Huazhong University of Science and Technology 28(3), 77–79 (2000)

The Dynamic Honeypot Design and Implementation Based on Honeyd Xuewu Liu1, Lingyi Peng2, and Chaoliang Li3 1

Hunan University of Commerce Beijin College 2 Hunan First Normal University 3 School of Computer Hunan University of Commerce, Changsha, China {12870595,494680234,522396825}@qq.com

Abstract. Along with the rapid development of Internet technology, Network security has become a very serious problem. At present the main security technologies include firewall technology, intrusion detection technology, access control technology, data encryption technology and so on. These safety technologies are based on the passive defence, so they are always in a passive position when thay are face to the up-to-date attack means. So,we put forward a kind of active defense network security technology -Honeypot technology and research detailedly the dynamic honeypot design and implementation based on Honeypot. Keywords: Network security, Honeyd, Dynamic honeypot, Virtual honeypot.

1 Preface Along with the the rapid development of Internet technology, Network information safety has to be face to a serious threat. The current network security technologies mainly use the passive defense methods, but these methods are very tough to deal with complex and changeable attacks from hacker. Since passive defense modes are difficult to deal with the complex and changeable attacks, we must solve the problem of defensive measure which is from the passive into active.This is also our research new topic. In this context , We put forward a kind of active defense network security technology—Honeypot. The Honeypot system elaborate network resources for hackers, which is a strict monitoring network deception system.The system aims at attracting hacker attacks through offerring real or analog networks and services ,,collecting the information and analyzing its attack behavior and process during the hacker attacks.In this way,we can hold the hackers’ motivations and goals, repair security holes The system attacked before , which can avoid the attacks occurred.

2 Honeyd Analysis and Research The Honeyd is designed by the Niels Prowvos from the Michigan university.It’s a application-oriented honeypot with low interactive. The Honeyd’s software frame S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 93–98, 2011. © Springer-Verlag Berlin Heidelberg 2011

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includes configuration database, central bag dispensers, agreement processor, personality engines and an optional routing component several parts. The structure is shown in figure 1:

Fig. 1. Logical frame of honeyd

After Honeyd receives packets, The central bag splitter will check IP packet length and confirm bag checksum. Honeyd main response three Internet protocols which are ICMP, TCP and UDP , Other protocols are discarded after credited into log .Before Packets are processed, The central bag splitter will search the honeypot configuration corresponding with the packet destination address. If they can't find the corresponding configuration, the System uses a default configuration..After the configuration is given, Packet and corresponding configuration will be assigned to specific protocol processor.

3 The Dynamic Honeypot Design Based on Honeyd 3.1 The Design of Dynamic Honeypot Environment Obtain The main purpose obtaining the environment around is to learn about the Internet environment. It’s the necessary conditions to solve the honeypot system configuration , That is, to solve allocation problems must know surrounding network environment first. 1) Active detection technology In order to get the network operating system and server types, We can use tools Nmap in detecting the entire network, After that ,we can get the feedback of target system that help us to determine its operating systems and services provided by it. But if active detection by excessive used can also cause faults, Namely excessive active detection will consume extra bandwidth,,which may cause the system shutdown. 2) Passive fingerprint identification technology Passive fingerprint identification technology is based on the principles which each operating system IP protocol has its own characteristic, maintains a fingerprinting

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database, Records data packets characteristic of different kinds of operating system . After it catches data packets in the network ,it will compare with the record of the database ,thereby it can judge the operating system categories. 3) Design of the active detection combining with the passive fingerprint According to the above active detections and passive fingerprint designs, we will be able to determine approximately the kind of operating system and obtain the basic situation of the environment. Its design is shown in figure 2:

Fig. 2. Design of the active detection combining with the passive fingerprint

3.2 The Architecture Design of the Dynamic Honeypot System Dynamic honeypot technology is first proposed as a kind of design method by the honeynet organization. For the challenge existing in the honeypot configuration and maintenance, we have to analyze it with dynamic honeypot technology. The system mainly uses active detection technology, passive fingerprint identification technology and Honeyd technology. The overall structure design which is made for the above content is in the following figure 3:

Fig. 3. The dynamic honeypot overall design based on Honeyd

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4 The Dynamic Honeypot Realization Based on Honeyd 4.1 Set Virtual Machine We can use installing Linux operating system hosts to do the honeypot host, Linux is main operating system in the machine honeypot mainframe, And let the system with bridge function, Another installation Vmware virtual machine, Used to support multiple guest operating system. Also installed the virtual machine to support guest operating system. In this honeypot system, We adopt Settings gateway for 2 Bridges mode, Through the use of two-layer gateway, Honeypot system with real system in a network environment, So in tracking and understand the external network attack, Can understand the internal network security problems. In addition, we through the source code to realize system support bridge mode. 4.2 Establish Honeyd Configuration Files The system through the Honeyd to simulate the virtual honeypot. Through creating Honeyd.config files to configure the template. The system created a default host templates, Used to store those not in other templates defined in packets, In addition, also created a Windows operating system template and a XP operating system template and router template. 4.3 Data Control Honeyd usually has two layers of data control, Respectively is a firewall data control and router data control, Firewall data control mainly through a firewall to control the honeypot out connection, Firewall adopt "wide into severe out" strategy, In fire prevention wall of honeypot machine from outside sends the number of connections set a rational threshold, Generally allow outside sends number of connections Settings for 5 to 10 more appropriate, Won't cause invaders doubt, Also avoid honeypot system become the invaders against other systems and tools. Routing control by routers completed, Basically is to use routers to go out the access control function of the packet filtering, Lest honeypot is used to attack other parts of a network. Mainly used to prevent Dos attack, IP deception or some other deceptive attack. In this system, we adopt gateway to replace, The advantage of using gateway is: Gateway no network address, Control operation will more latent, Hackers perceive is not easy. We adopt Honeynet development of rc. Firewal scripts to the configurations and realization, And using IPTables to restrict. IPTables is Linux self-contained open source firewall, According to the need to Forsake a bag, In a given period allowed only a certain number of new connection, Possibly through discard all packages to completely isolate the honeypot system. Every time the connection initialization out the connection, Firewall count, When the total limit is reached then, IPTables will block any Honeypot launched from any connection. Then IPTable reset itself, Allow each time period allowed out connection number. In this script installed per hour allow TCP and UDP, ICMP or other arbitrary IP packet out number of connections, When an intruder outside sends a packet to specified value, Automatically cut off all foreign connections, To reduce the network risk.

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4.4 Data Capture Data capture is the key of the honeypot system, We need to use data capture information to determine the invaders behavior and motivation, In order to determine the invaders gain access after had done, We need to capture data can provide invaders keystroke records and attack effect. 1) Realize data capture by snort Snort is a lightweight intrusion detection system, It has three working mode: Sniffer, packet recorder, network intrusion detection system. We mainly use Snort intrusion detection model. Above configuration files is Snort collected data output to local called Snortdb Mysql database, The user name is Snortoper, Verification code is Password. At the same time will be recorded in Tcpdump format packets Snort.log file. 2) Realize data capture by sebek Sebek is a based on the kernel's data capture tools, It can be used to capture the honeypot concealed all activities. Sebek caught in the packet encryption has great advantage, Because no matter what kind of encrypted data to the destination host to have after action, Will be decrypted to call system calls. Hackers who get packets, Use its own agreement will the packet on the Internet, Thus obtained by the Sebek Server. Sebek Client are through some hidden technology makes the invaders feel oneself be monitored, Convenient for us to capture the real data. Sebek Client capture the data package into UDP packets, Through the nic driver sent to the Internet, Avoid being invaders may install the sniffer to detect. Sebek consists of two parts: The client and the server. The client from the honeypot capture data and the output to network lets server-side collection. The server have two ways to collect data: The first kind is directly from the network activity packet capture, The second from Tcpdump format preservation packets files. When data collected can upload the relational database, Also can instantly display keystroke records. 4.5 Log Record log record is mainly to the honeypot host capture data recorded, Its main function is to collect and record hackers behavior, For the future analysis hackers the tools used, strategies and their attack purposes or take lawsuit hackers crime to provide evidence. In order to ensure that capture hacker attacks data security, We design a log oportunidades programme to the backup data in the system. The honeypot host is running with Linux ep-red Hat 9.0 operating system of real host,the Syslog of Linux Red Hat 9.0 function is powerful, Syslog can send recording system kernel and tools generated information. We can configure their Syslog. Conf files, To realize the virtual honeypot collected log message transferred to log server. By modifying Syslog. Conf, Realized the local log information transfer to remote log server for backup. Finally, we began to capture the hacker information for analysis, Thus learning hackers means and

、

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methods, In view of its attack means to take corresponding defensive measures, To this, the Honeyd based on dynamic honeypot is realized basically.

5 Summary Dynamic honeypot although is based on virtual honeypot, But it is a low interaction of honeypot, With the interaction between the attacker is very limited, Capture data also is very limited, Improve the honeypot interactivity can gain more attack information, To study the attacker to attack has very great help. We can interact with high virtual honeypot honeypot combined, To capture more attacks, information. Dynamic virtual honeypot on network security role is mainly indirectly, Namely recognition threat, divert attacks flow. Thus the honeypot with other security technology, Such as firewall technology and intrusion detection system combining but also the future of a very important developing direction. Along with the development of honeypot technology, Some problems will be solved step by step, Some new techniques and applications will further development, Make honeypot technology better for network security provided protection.

References 1. Fu, X., Yu, W., Cheng, D., et al.: On Recognizing Virtual Honeypots and Countermeasures. In: The 2nd IEEE Interational Symposium on Dependable, Autonomic and Secure Computing, vol. 9, pp. 220–230 (2006) 2. Leita, C., Mermoud, K., Daeier, M.: SeriptGen:all automated script generation tool for honeyd. In: 21st Annual Computer Security Applications Conference, vol. 9, pp. 125–135 (2008) 3. Zhang, F., Zhou, S., Qin, Z., et al.: Honeypot:a supplemented active defense system for network security. In: Proceedings of the Fourth International Conference, PDCAT 2009, vol. 8, pp. 231–235 (2003) 4. Kreibich, C., Crowcroft, J.: Honeycomb-Creating intrusion Detection Signatures Using Honeypots (EB/PDF), http://www.sigcomm.org/I-IotNets-II/papershaoneycomb.pal.2010 5. Domscif, M., Holz, T., Mathes, J., Weisemoller, I.: Measuring Security Threats with Honeypot Technology, 21–29 (2009) 6. Kwong, L., Yah: Virtual honeynet srevisited.SMC Information Assurance Workshop. In: Proceedings from the Sixth Annual IEEE, vol. 9, pp. 230–240 (2010)

Research of SIP DoS Defense Mechanism Based on Queue Theory Fuxiang Gao, Qiao Liu, and Hongdan Zhan College of Information Science and Engineering, Northeastern University, Shenyang 110819, China [email protected], {dongqin4060432,hongdanzhan}@163.com

Abstract. The SIP is becoming the core of multimedia communication network through the IP, based on which the 3G network has been operated in our country. The easy implementation, high destruction and hard tracking against attacking source which are the characters of DoS attack cause the great security threat to SIP. Focusing on the research on defense mechanisms of DoS attacks in SIP protocol, this paper proposes a model based on queue theory to approximately analyze the DoS attack, and then presents a DoS attack defense method. Simulation was taken to analyze the performance of the defense mechanism. Keywords: SIP, DoS attack, queue model, defense mechanism.

1 Introduction SIP was proposed in 1999 by IETF as a signaling protocol that based on IP network environment. And it has been widely used in NGN at present. SIP is vulnerable to DoS attack because it runs on IP network environment and has openness. For example, attackers can create false messages which have false source addresses and via field. And the SIP proxy server which was attacked is set as request initiator, then these messages are sent to large numbers of SIP users. Hence the spoofed users will send many DoS attack messages to the attacked server. Therefore, research on DoS detection defense system of SIP has become a hot point at present and also has been a problem that urgently needed to be solved in NGN deployment. According to the definition of VolPSA, DoS attack problems of SIP system can be divided into five categories: request flooding, malformed requests and messages, QoS abuse, spoofed messages, call hijacking [1]. This paper mainly researches on the flooding problems of SIP, and combining with the M/M/1/K mathematical model it proposes a SIP DoS attack defensive scheme that based on the queue theory. At last, this paper simulates and analyzes the performance of the scheme.

2 Analytical Model of SIP DoS Attack 2.1 Related Assumptions We assume that the arrival time of SIP request messages obeys exponential distribution and the average arrival rate is λ / s. We also assume that the consuming S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 99–104, 2011. © Springer-Verlag Berlin Heidelberg 2011

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time that the server processes the unit message is Q seconds, the size of messages obeys exponential distribution whose average is L, the average service rate of SIP system is µ and the average service time is S, thus: S=1/μ=LQ

(1)

For convenient analysis, we assume that the service time and the interarrival time still obey exponential distribution during the system under DoS attacking [2]. We also assume that the whole system has only a core processing unit and the size of the cache is K, namely there are a maximum of K request messages wait to be processed, the rest of the arrival messages will be discarded [3]. So we can analyze the whole system using the M/M/1/K model. 2.2 The Establishment of the Analytical Model For the M/M/1/K system, we write ρ as ρ= λ / μ, thus we can obtain the rate (p0) when the queue’s length of the system is 0:

⎛ k ⎞ p0 = ⎜ ∑ ρ i ⎟ ⎝ i =1 ⎠

−1

(2)

The rate of messages being discarded when the system statistical balance is: П=pk=ρkp0

(3)

The average waiting time of messages when there are no messages are lost is: ρ0 1 ⎧ 1 P (1 − k ρ k −1 + ( k − 1) ρ k ) ( ρ ≠ 1) 2 ⎪1 − Π − (1 − ρ ) μ ⎪ W =⎨ ⎪ 1 ρ 1 ( k − 1) k ⎪1 − Π 0 μ 2 ⎩

(4)

The average delay of the messages equals the waiting time of the messages in the queue plus the average service time, thus the average response time of the system is: R = W ( λ , L, K ) + S ( L )

(5)

We can see that the response time of system is related with three parameters: the arrival rate of the request messages λ, the average size of the messages L and the size of the system cache K. Normally, the value of K is sure, so the response time of system depends on λ and L.

3 The Design of SIP DoS Attack Defensive Scheme According to the M/M/1/K model above, the new coming message will be discarded when the number of the received messages larger than k. That means that other legal users will not be able to accept normal services if the server is suffering a DoS attack and causing its own buffer space exhausted.

Research of SIP DoS Defense Mechanism Based on Queue Theory

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For example, if the illegal INVITE message blocks up the header of the queue, the legal INVITE message behind it will not be able to obtain the service so that the call can not be established according to the normal process, which will lead to sessions is failed because of the timeout at last. Even if the illegal INVITE message does not block up the header of the queue, a large number of illegal messages block at the head of the queue, that may cause the new coming session response message cannot be accepted or be discarded, the call of legal users will also be timeout because of not accepting any response [4], what is shown in Fig.1. 180 Running The normal INVITE message

The illegal INVITE message

The normal INVITE message

200 OK Programming

Incoming message Programming

The illegal INVITE message

Fig. 1. The blocked queue of INVTE message. The former is the situation of blocking the header of the queue and the later is blocking the queue. Others

Others

SIP message

INVITE

INVITE Programming

Others Programming

Programming

INVITE Programming

Fig. 2. The Sketch maps of single queue and priority queue. The former is the situation of the signal queue and the later is the priority queue.

In order to reduce the influence that the INVITE flooding has on the proxy server, we consider the scheme in which the priority queue is introduced. SIP server generally uses FIFO queue when it processes the message queue. When there is a DoS attack, it will produce the problem above. If we adopt the priority queue, the INVITE messages will be assigned with low priority and be put into low priority queue, others that are not INVITE messages will get high priority and be put into high priority queue. The two queues respectively follow FIFO principle, but only when the higher priority queue is empty, they will process the messages of lower priority, what is shown in Fig.2. Thus, we can assign the original cache sources of server to two queues, and the way of assigning the sources according to the actual conditions. For convenient analysis, this paper assign the cache sources to two queues averagely, namely two M/M/1/K/ queues [5]. The priority queue adopts the non-preemptive priority rules, a message which is accepting services is allowed to finish this service without interference, even if there is a higher priority message arrives. Every priority message has an independent queue. When the server can be used, the first message of the non-preemptive priority queue that has the highest priority will be the first to accept the service. It will give the corresponding average delay equations for every priority category. For the M/M/1/ (K/2) system which has multiple priority levels, firstly, we define various parameters as followed: qwi is the average length of queue whose priority is i; Wi is the average waiting time whose priority is i; p0i is the rate when the length of the message

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F. Gao, Q. Liu, and H. Zhan

queue is 0 whose priority is i; ρi= λi / μi is the utilization rate of the message whose priority is i on the system; TR is the average residual service time. Assuming that the whole utilization rate of the system is lesser than 1, namely: ρ1+ρ2+…+ρn