HYBRID METHODS IN PATTERN RECOGNITION
World Scientific
Series in Machine Perception and Artificial Intelligence - Vol. 47
HYBRID METHODS IN PATTERN RECOGNITION Editors
H Bunke University of Bern, Switzerland
A Kandel University of South Florida, USA
li> World Scientific •
New Jersey • London • Singapore • Hong Kong
SERIES IN MACHINE PERCEPTION AND ARTIFICIAL INTELLIGENCE* Editors:
H. Bunke (Univ. Bern, Switzerland) P. S. P. Wang (Northeastern Univ., USA)
Vol. 34: Advances in Handwriting Recognition (Ed. S.-W. Lee) Vol. 35: Vision Interface — Real World Applications of Computer Vision (Eds. M. Cherietand Y.-H. Yang) Vol. 36: Wavelet Theory and Its Application to Pattern Recognition (Y. Y. Tang, L. H. Yang, J. Liu and H. Ma) Vol. 37: Image Processing for the Food Industry (E. R. Davies) Vol. 38: New Approaches to Fuzzy Modeling and Control — Design and Analysis (M. Margaliot and G. Langholz) Vol. 39: Artificial Intelligence Techniques in Breast Cancer Diagnosis and Prognosis (Eds. A. Jain, A. Jain, S. Jain and L. Jain) Vol. 40: Texture Analysis in Machine Vision (Ed. M. K. Pietikaineri) Vol. 41: Neuro-Fuzzy Pattern Recognition (Eds. H. Bunke and A. Kandel) Vol. 42: Invariants for Pattern Recognition and Classification (Ed. M. A. Rodrigues) Vol. 43: Agent Engineering (Eds. Jiming Liu, Ning Zhong, Yuan Y. Tang and Patrick S. P. Wang) Vol. 44: Multispectral Image Processing and Pattern Recognition (Eds. J. Shen, P. S. P. Wang and T. Zhang) Vol. 45: Hidden Markov Models: Applications in Computer Vision (Eds. H. Bunke and T. Caelli) Vol. 46: Syntactic Pattern Recognition for Seismic Oil Exploration (K. Y. Huang) Vol. 47: Hybrid Methods in Pattern Recognition (Eds. H. Bunke and A. Kandel) Vol. 48: Multimodal Interface for Human-Machine Communications (Eds. P. C. Yuen, Y. Y. Tang and P. S. P. Wang) Vol. 49: Neural Networks and Systolic Array Design (Eds. D. Zhang and S. K. Pal)
*For the complete list of titles in this series, please write to the Publisher.
HYBRID METHODS IN PATTERN RECOGNITION
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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HYBRID METHODS IN PATTERN RECOGNITION Series in Machine Perception and Artificial Intelligence — Vol. 47 Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. AH rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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Dedicated to The Honorable Congressman C. W. Bill Young House of Representatives for his vision and continuous support in creating the National Institute for Systems Test and Productivity at the Computer Science and Engineering Department, University of South Florida
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Preface
The discipline of pattern recognition has seen enormous progress since its beginnings more than four decades ago. Over the years various approaches have emerged, based on statistical decision theory, structural matching and parsing, neural networks, fuzzy logic, artificial intelligence, evolutionary computing, and others. Obviously, these approaches are characterized by a high degree of diversity. In order to combine their strengths and avoid their weaknesses, hybrid pattern recognition schemes have been proposed, combining several techniques into a single pattern recognition system. Hybrid methods have been known about for a long time, but they have gained new interest only recently. An example is the area of classifier combination, which has attracted enormous attention over the past few years. The contributions included in this volume cover recent advances in hybrid pattern recognition. In the first chapter by H. Ishibuchi and M. Nii, a novel type of neural network architecture is introduced, which can process fuzzy input data. This type of neural net is quite powerful because it can simultaneously deal with different data formats, such as real or fuzzy numbers and intervals, as well as linguistic variables. The following two chapters deal with hybrid systems that aim at the application of neural networks in the domain of structural pattern recognition. In the second chapter by G. Adorni et al., an extension of the classical backpropagation algorithm that can be applied in the graph domain is proposed. This extension allows us to apply multilayer perceptron neural networks not only to feature vectors, but also to patterns represented by means of graphs. A generalization of self-organizing maps from n-dimensional real space to the domain of graphs is proposed in Chap. 3, by S. Giinter and H. Bunke. In particular, the problem of finding the optimal number of clusters in a graph clustering task is addressed.
Vll
Vlll
Preface
In Chap. 4, A. Bargiela and W. Pedrycz introduce a general framework for clustering through identification of information granules. It is argued t h a t the clusters, or granules, produced by this method are particularly suitable for hybrid systems. T h e next two chapters describe combinations of neural networks and hidden Markov models. First, in Chap. 5, G. Rigoll reviews a number of possible combination schemes. Most of t h e m originated in the context of speech and handwriting recognition; however, they are applicable to a much wider spectrum of applications. In Chap. 6, by T. Artieres et al., a system for on-line recognition of handwritten words and sentences is investigated. T h e main building blocks of this system are a hidden Markov model and a neural net. T h e following three chapters address the emerging field of multiple classifier systems. First, in Chap. 7, T. K. Ho provides a critical survey of the field. She identifies the lessons learned from previous work, points out the remaining problems, and suggests ways to advance the state-of-the-art. Then, in Chap. 8, F . Roli and G. Giacinto describe procedures for the systematic generation of multiple classifiers and their combination. Finally, in Chap. 9, A. Verikas et al. propose an approach to the integration of multiple neural networks into an ensemble. B o t h the generation of the individual nets and the combination of their o u t p u t s is described. In the final three chapters of the book applications of hybrid methods are presented. In Chap. 10, A. Klose and R. Kruse describe a system for the interpretation of remotely sensed images. This system integrates methods from the fields of neural nets, fuzzy logic, and evolutionary computation. In Chap. 11, D.-W. J u n g and R.-H. P a r k address the problem of fingerprint identification. T h e authors use a combination of various methods t o achieve robust recognition at a high speed. Last b u t not least, M. Junker et al. describe a system for automatic text categorization. Their system integrates symbolic rule-based learning with subsymbolic learning using support vector machines. Although it is not possible to cover all current activities in hybrid pattern recognition in one book, we believe t h a t the papers included in this volume are a valuable and representative sample of up-to-date work in this emerging and important branch of p a t t e r n recognition. We hope t h a t the contributions are valuable and will be useful to many of our colleagues working in the field.
Preface
IX
The editors are grateful to all the authors for their cooperation and the timely submission of their manuscripts. Finally, we would like to thank Scott Dick and Adam Schenker of the Computer Science and Engineering Department at the University of South Florida for their assistance and support.
Horst Bunke, Bern, Switzerland Abraham Kandel, Tampa, Florida August 2001
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Contents
Preface H. Bunke and A. Kandel Neuro-Fuzzy Systems Chapter 1 Fuzzification of Neural Networks for Classification Problems H. Ishibuchi and M. Nii Neural Networks for Structural Pattern Recognition Chapter 2 Adaptive Graphic Pattern Recognition: Foundations and Perspectives G. Adorni, S. Cagnoni and M. Gori Chapter 3
Adaptive Self-Organizing Map in the Graph Domain S. Giinter and H. Bunke
Clustering for Hybrid Systems Chapter 4 From Numbers to Information Granules: A Study in Unsupervised Learning and Feature Analysis A. Bargiela and W. Pedrycz Combining Neural Networks and Hidden Markov Models Chapter 5 Combination of Hidden Markov Models and Neural Networks for Hybrid Statistical Pattern Recognition G. Rigoll Chapter 6
From Character to Sentences: A Hybrid Neuro-Markovian System for On-Line Handwriting Recognition T. Artieres, P. Gallinari, H. Li, S. Marukatat and B. Dorizzi
vii
1
33
61
75
113
145
xii
Contents
Multiple Classifier Systems Chapter 7 Multiple Classifier Combination: Lessons and Next Steps T. K. Ho Chapter 8 Chapter 9
171
Design of Multiple Classifier Systems F. Roll and G. Giacinto
199
Fusing Neural Networks Through Fuzzy Integration A. Verikas, A. Lipnickas, M. Bacauskiene and K. Malmqvist
227
Applications of Hybrid Systems Chapter 10 Hybrid Data Mining Methods in Image Processing A. Klose and R. Kruse
253
Chapter 11 Robust Fingerprint Identification Based on Hybrid Pattern Recognition Methods D.-W. Jung and R.-H. Park
275
Chapter 12
Text Categorization Using Learned Document Features M. Junker, A. Abecker and A. Dengel
301
CHAPTER 1 FUZZIFICATION OF N E U R A L N E T W O R K S FOR CLASSIFICATION PROBLEMS
Hisao Ishibuchi Department of Industrial Engineering, Osaka Prefecture University 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan E-mail:
[email protected] Manabu Nii Department of Computer Engineering, Himeji Institute of Technology 2167 Shosha, Himeji, Hyogo 671-2201, Japan E-mail: nii@comp. eng. himeji-tech. ac.jp
This chapter explains the handling of linguistic knowledge and fuzzy inputs in multi-layer feedforward neural networks for pattern classification problems. First we show how fuzzy input vectors can be classified by trained neural networks. The input-output relation of each unit is extended to the case of fuzzy inputs using fuzzy arithmetic. That is, fuzzy outputs from neural networks are denned by fuzzy arithmetic. The classification of each fuzzy input vector is performed by a decision rule using the corresponding fuzzy output vector. Next we show how neural networks can be trained from fuzzy training patterns. Our fuzzy training pattern is a pair of a fuzzy input vector and a non-fuzzy class label. We define a cost function to be minimized in the learning process as a distance between a fuzzy output vector and a non-fuzzy target vector. A learning algorithm is derived from the cost function in the same manner as the well-known back-propagation algorithm. Then we show how linguistic rules can be extracted from trained neural networks. Our linguistic rule has linguistic antecedent conditions, a non-fuzzy consequent class, and a certainty grade. We also show how linguistic rules can be utilized in the learning process. That is, linguistic rules are used as training data. Our learning scheme can simultaneously utilize linguistic rules and numerical data in the same framework. Finally we describe the architecture, learning, and application areas of interval-arithmetic-based
1
H. Ishibuchi & M. Nii
2
neural networks, which can be viewed as a basic form of our fuzzified neural networks. 1. Introduction Multilayer feedforward neural networks can be fuzzified by extending their inputs, connection weights and/or targets to fuzzy numbers (Buckley and Hayashi 1994). Various learning algorithms have been proposed for adjusting connection weights of fuzzified neural networks (for example, Hayashi et al. 1993, Krishnamraju et al. 1994, Ishibuchi et al. 1995a, 1995b, Feuring 1996, Teodorescu and Arotaritei 1997, Dunyak and Wunsch 1997, 1999). Fuzzified neural networks have many promising application areas such as fuzzy regression analysis (Dunyak and Wunsch 2000, Ishibuchi and Nii 2001), decision making (Ishibuchi and Nii 2000, Kuo et al. 2001), forecasting (Kuo and Xue 1999), fuzzy rule extraction (Ishibuchi and Nii 1996, Ishibuchi et al. 1997), and learning from fuzzy rules (Ishibuchi et al. 1993, 1994). The approximation ability of fuzzified neural networks was studied by Buckley and Hayashi (1999) and Buckley and Feuring (2000). Perceptron neural networks were fuzzified in Chen and Chang (2000). In this chapter, we illustrate how fuzzified neural networks can be applied to pattern classification problems. We use multilayer feedforward neural networks with fuzzy inputs, non-fuzzy connection weights, and nonfuzzy targets for handling uncertain patterns and linguistic rules such as "If X\ is small and x 0,
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Fuzzification
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25
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Fig. 4. Compiling the encoding network from the recursive network and the given data structure by function ipr.
the state variable from the state of the children q^xxv, Q21xv, q^xxv and the label uv attached to the node. Unlike Fig. 4, in m a n y real world problems the knowledge in a recursive network R yields topological constraints t h a t often make it possible t o cut the number of trainable parameters significantly. Let us consider a directed ordered graph. For any node v one can identify a set, eventually empty, of ordered children ch[u]. Let ajch[v] be the state associated with set ch[w] and 0 be vector of learning parameters. T h e state xv and the o u t p u t yv of each node v follows the equations Xv = /(»ch[v], UV: V, 0 ) , Vv = 9(Xv, uv, v,
9).
(4)
This is a straightforward extension of classic causal models in system theory. T h e hypothesis of dealing with directed acyclic graph t u r n s out to be useful for carrying out a forward computation, and the hypothesis of considering ordered sets of children is used in order to define the position of the parameters in functions / and g. Alternatively, one can keep essentially the same computational scheme for directed positional acyclic graphs in which the children of each node are associated with an integer. T h e difference with respect to directed ordered acyclic graphs is t h a t they consider only ordered sets of children, and do not include the case in which the children of a given node are not given in a sequential ordering. For instance, in Fig. 3, the difference of the two p a t t e r n s is kept in the graphical representation in t h e case of directed positional acyclic graphs, b u t is lost in the case of a representation based on directed acyclic graphs. Given
46
G. Adorni, S. Cagnoni & M. Gori
the recursive network R and any DOAG u, we can construct an encoded representation of u on the basis of the independence constraints expressed by R, t h a t is ur = ipr(R,u). T h e scheme adopted for compiling ur is depicted in Figure 4, while a detailed description of the mathematical process involved is given in Ref. 1. From the encoding network depicted in Fig. 4 we can see a pictorial representation of the computation taking place in the recursive neural network. Each n i l pointer, represented by a small box, is associated with a frontier state xv, which is in fact an initial state used to terminate the recursive equation. T h e graph plays its own role in the computation either because of the information attached to its nodes or for its topology. Any formal description of the computation of the input graph requires sorting the nodes, so as to define for which nodes the state must be computed first. As already pointed out for the computation of the activation of the neurons of a feedforward neural network, the computation can be based on any topological sorting. One can use a d a t a flow computation model where the state of a given node can only be computed once all the states of its children are known. To some extent, the computation of the o u t p u t yv can be regarded as a transduction of the input graph u to an o u t p u t y with the same skeleton 6 as u. These IO-isomorph transductions are the direct generalisation of the classic concept of transduction of lists. When processing graphs, the concept of IO-isomorph transductions can also be extended to the case in which the skeleton of the graph is also modified. Because of the kind of problems considered in this chapter, however, this case will not be treated. T h e classification of DOAGs is in fact the most important IOisomorph transduction for applications to p a t t e r n recognition. T h e o u t p u t of the classification process corresponds with ys, t h a t is the o u t p u t value of the variables attached to the supersource in the encoding network. Basically, when the focus is on classification, we disregard all the o u t p u t s yv of the IO-isomorph transduction apart from the final values ys of the forward computation. T h e information attached to the recursive network, however, needs to be integrated with a specific choice of functions / and g which must be suitable for learning the parameters 0. T h e connectionist assumption for functions / and g t u r n s out to be adequate especially to fulfill computational e
T h e skeleton of a graph is the structure of the data regardless of the information attached to the nodes.
Adaptive Graphic Pattern Recognition: Foundations
and Perspectives
47
complexity requirements. The extension to the case of DO AGs is straightforward. Let o be the maximum outdegree of the given directed graph. The dependence of node v on its children chjt;] can be expressed by pointer matrices Av(k) € M n,n , k = 1 , . . . ,o. Likewise, the information attached to the nodes can be propagated by weight matrix Bv £ M.n'm. Hence, the first-order connectionist assumption yields
xv = a I ^2 Av(k) • q^Xy + Bv • uv I .
(5)
The output can be computed by means of yv = a(Cv • xv + Dv • uv), where Cv € W'n and Dv e K p , m . Hence the learning parameters can be grouped for each node of the graph in 6V = {[Av(l),...
, Av(o)}, Bv, Cv, Dv} .
The most attracting feature of connectionist assumptions for / and g is that they provide universal approximation capabilities by means of a graphical structure with units of a few different types (e.g. the sigmoid). The strong consequence of this graphical representation for / and g is that, for any input graph, an encoding neural network can be created which is itself a graph with neurons as nodes. Hence, the connectionist assumption makes it possible to go one step further than beyond the general independence constraints expressed by means of the concept of recursive network. The encoding neural network un associated with equation 5 is constructed by replacing each node of the encoding network ur with the chosen connectionist map; that is 4>n = -ipn ° Ipr • Un -» 4>n{ur) = i)n(lpr(R,u))
.
The construction of the encoding neural network un from the encoding neural network ur is depicted in Fig. 5. For the particular case of stationary models, in which the parameters 0V are independent of the node v. Encoding neural networks turns out to be weighed graphs, that is there is always a real variable attached to the edges (weight). Note that the architectural choice expressed by equation 5 can be easily extended so as to express functions / and g by general feedforward neural architectures. Of course, the composition of directed acyclic graphs (data) with the local node computation based on feedforward neural networks, which are directed acyclic
G. Adorni, S. Cagnoni & M. Gori
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graphs, yields in general encoding neural networks which are still acyclic graphs. As a result, the supervised of learning a given set of DOAGs results in the supervision of the corresponding encoding neural networks. Because of the non-stationarity hypothesis, the parameters are independent of the node and, therefore, the learning of the weights 8 can be framed as an optimization problem. We can thus use the Backpropagation algorithm for training. Since the Backpropagation of the error takes place on neural networks which encode the structure of the given examples, the corresponding algorithm for the gradient computation, in this case, is referred to as Backpropagation through structure.1'31 This algorithm uses the classical forward
Adaptive Graphic Pattern Recognition: Foundations
and Perspectives
49
and backward steps, the only difference being that the parameters of the different encoded neural networks must be shared. 4.3. Cycles, Non-stationarity,
and
Beyond
Our proposed computation scheme proposed for directed acyclic graphs is a straightforward extension of the case of static data. The hypothesis that the children of each node are ordered is fundamental, and allows us to attach the appropriate set of weights to each child. The assumption that the graph is acyclic yields acyclic encoded neural networks for which the Backpropagation algorithm holds. Finally, the non-stationarity hypothesis makes it possible to attach the same set of weights to each node. The hypothesis of dealing with ordered graphs can be relaxed in different ways. A straightforward solution is to share the pointer matrices Ak among the children. In so doing, a unique matrix A is used for all the children, which overcomes the problem of defining the position in the computation of functions / and g. Alternatively, given chbj] one can consider the set of its permutations 'P(ch[ti]) and calculate functions / and g by an appropriate sharing of the weights.32 The second solution is more general than the first in terms of computational capabilities, but turns out to be effective only when the outdegree of the graphs is quite small. Otherwise, the cardinality oiV(ch[v}) explodes. The construction of the encoding neural network gives rise to feedforward neural networks in the case of acyclic graphs. As shown in Fig. 6, for general graphs and directed graphs with cycles, the same construction of the encoding neural networks produces a recurrent neural network. As a result, the computation of each graph cannot be performed by simple forward step. The feedback loops in the neural network can produce complex dynamics, which do not necessarily correspond with convergence to an equilibrium point. It is worth mentioning that cyclic and undirected pattern representations can be extracted in a more natural way than directed ordered graphs. However, the drawback to this approach is that the corresponding learning process is significantly more expensive. In general, given a planar graph, one can construct a corresponding DOAG provided that an anchor node is also specified. Unfortunately, in pattern recognition one cannot always rely on the availability of such an anchor; there are cases in which the corresponding graphical extraction is likely not to be very robust.
50
G. Adorni, S. Cagnoni & M. Gori
Fig. 6. The encoding of cyclic graphs yields cyclic encoding networks, which in turn, gives rise to recurrent neural network architectures.
The proposed models represent a natural extension of the processing of sequences by causal dynamical systems. In pattern recognition, the hypothesis of causality could be profitably removed, since there is no need to carry out an on-line computation at node level. Having homogeneous computations at node level may not be adequate for many pattern recognition problems. This has been already pointed out in Ref. 33, where a simple solution has been adopted to account for nonstationarity. The graphs are partitioned into different sets depending on the number of nodes, and are processed separately. A more general and computational scheme has been devised in Ref. 34, where a linguistic description of non-stationarity is given which is used to compile the encoding neural networks. 5. Graphical Pattern Recognition The Ref. Ref. The and,
term adaptive graphical pattern recognition was first introduced in 33, but early experiments using this approach were carried out in 35. Graphs are either in the data or in the computational model. adopted connectionist models inherit the structure of the data graph moreover, they have their own graphical structure that expresses the
Adaptive Graphic Pattern Recognition: Foundations
and Perspectives
51
dependencies on the single variables. Basically, graphical pattern recognition methods integrate domain structure into decision-theoretic models. The structure can be introduced at two different levels. First, we can introduce a bias on the map (e.g. receptive fields). In so doing, the pattern of connectivity in the neural network is driven by the prior knowledge in the application domain. Second, each pattern can be represented by a corresponding graph. As put forward in the previous section, the hypothesis of directed ordered graphs can be profitably exploited to generalize the forward and backward computation of classical feedforward networks. The proposed approach can be pursued in most interesting pattern recognition problems. In this chapter we focus attention on supervised learning schemes, but related extensions have recently been conceived for unsupervised learning. 5.1.
Classification
Recursive neural networks seem to be very appropriate for either classification or regression. Basically, the structured input representation is converted to a static representation (the neural activations in the hidden layers), which is subsequently encoded into the required class. This approach shares the advantages and disadvantages of related. MLP-based approaches for static data. In particular, the approach is well-suited for complex discrimination problems. The effectiveness of recursive neural networks for pattern classification has been shown in Ref. 36 by massive experimentation on logo recognition. In particular, it has been shown that the network performance is improved by properly filtering the logo image before extracting the data structure. The patterns were represented using trees extracted by an opportune modification of the contour-tree algorithm. That modification plays a fundamental role in the creation of data structures that enhance the structure of the pattern. The experimental results show that, though in theory the contour tree rotation invariance no longer holds, as a matter of fact, there is a very slight dependence of the performance on the rotation angle. These experimental results indicate that adaptive graphical pattern recognition is appropriate when we need to recognize patterns in the presence of noise, and under rotation and scale invariance. These very promising results suggest that the proposed method bridges nicely decision-theoretic approaches based on numerical features and syntactic and structural approaches.
52
G. Adorni, S. Cagnoni & M. Gori
Network growing a n d pruning can b e successfully used for improving the learning process. It is worth mentioning t h a t recursive neural networks can profitably be used for classification of highly structured inputs, like image documents representations by XY-trees. Unfortunately, in this particular kind of application the major limitation t u r n s out to be t h a t the number of classes is fixed in advance, a limitation which is inherited from multilayer networks. Neural networks in structured domains can be used in verification problems, where one wants to establish whether a given p a t t e r n belongs to a given class. Unlike p a t t e r n classification, one does not know in advance the kind of inputs t o be processed. It has been pointed out t h a t sigmoidal multilayered neural networks are not appropriate for this task. 3 7 Consequently, our recursive neural networks are also not appropriate for verification tasks. However, as for multilayer networks, the adoption of radial basis function units suffices to remove this limitation. 5 . 2 . Image
Retrieval
T h e neural networks introduced in this chapter and their related extensions are good candidates for many interesting image retrieval tasks. In particular, the proposed models introduce a new notion of similarity, which is constructed on the basis of the user feedback. In most approaches proposed in the literature, queries involve either global or local features, and disregard the p a t t e r n structure. T h e proposed approach makes it possible to retrieve p a t t e r n s on the basis of a strong involvement of the p a t t e r n structure, since t h e graph topology plays a crucial role in t h e computation. On the other hand, since the nodes contain a vector of real-valued features, the proposed approach can also be able to exploit the sub-symbolic nature of the p a t t e r n s . Figure 7 shows a possible graphical representation of the images of a given database. T h e database has been created using an a t t r i b u t e plex g r a m m a r as described in Ref. 38. Unlike p a t t e r n classification, in which the learning scheme is a straightforward extension to backpropagation for static d a t a , learning the notion of similarity requires the definition of an appropriate target function. For each pair of images, the user provides feedback on how relevant the retrieved image is to the query. Consequently, the learning process consists of adapting the weights so as to incorporate the user feedback. Given any two pairs of images the user is asked whether they look similar and is expected to
Adaptive Graphic Pattern Recognition: Foundations
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Graph Extraction
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provide a simple Boolean answer. In the case the images are not similar (see e.g. Fig. 8) their corresponding points in the hidden layer of the recursive neural network must be moved far apart, whereas in the case the images are similar, the corresponding points must be moved close to each other (see e.g. Fig. 9). Let Ui and u-2 be the graphical representations of two images for which the user is evaluating the similarity. u\ and u
