Local stereology

LOCAL STEREOLOGY

ADVANCED SERIES ON STATISTICAL SCIENCE & APPLIED PROBABILITY Editor: Ole E. Barndorff-Nielsen

Published Vol. 1: Random Walks of Infinitely Many Particles by P. Revesz Vol. 4: Principles of Statistical Inference from a Neo-Fisherian Perspective by L Pace and A. Salvan Vol. 5: Local Stereology by Eva B. Vedel Jensen

Forthcoming Vol. 2: Ruin Probability by S. Asmussen Vol. 3: Essentials of Stochastic Finance by A. Shiryaev

Advanced Series on Statistical Science & Applied Probability

LOCAL STEREOLOGY

Eva B. Vedel Jensen University ofAarhus

World Scientific Singapore • New Jersey 'London • Hong Kong

Vol.5

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

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

LOCAL STEREOLOGY Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All 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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ISBN 981-02-2454-0

Printed in Singapore.

Preface The aim of this book is to give a unified exposition of local stereological methods. It is my belief that such a book containing the mathematical and statistical foundations of local stereology is needed since these methods are now in world-wide use in the microscopical study of biological tissue. In many ways this is a very personal book, for a large part based on my own research, and I hope that some of my enthusiasm and joy in working in this area will be acknowledged by the reader. Local stereology has gradually developed during the last fifteen years, as a product of practical needs from the biologists and new advances in the microscopical observation techniques and in the theoretical foundations of local stereology, integral geometry and geometric probability. Three colleagues have had a profound influence on my engagement in local stereology. Roger Miles wrote a fundamental series of papers about stereology and integral geometry in the seventies and the eightties, partly together with Pamela Davy. His paper Some new integral geometric formulae, with stochastic applications, published in Journal of Applied Probability in 1979, became especially important since it contained the tools for constructing local estimators of volume, as we simultaneously realized in 1983. The needs from the users of stereology have been realized through my close collaboration with Hans Jørgen G. Gundersen since 1977. I have during the years tried to cope with his never ending series of questions and their solutions occupy a fair part of the book. The mathematical fundament of local stereology has been built together with my former Ph.D. student Kién Kieu. The book is written for researchers, teachers and graduate students in mathematical statistics and probability. In order to reach a broader audience, the book is not only for specialists in stereology, integral geometry and geometric measure theory. In particular, Chapter 1 is an elementary introduction to stereology. Local stereology involves, however, advanced mathematical tools which is an important part of the book too. Glancing through the book, the reader will realize that this is a book about geometric sampling. Chapter 1 contains an exposition of classical global stereology, including Cavalien's method of volume estimation, estimation via ratios, number estimation using the disector, length and surface area estimation under isotropy, vertical sections. The rest of the book is devoted to local stereology. The mathematical tool for developing local stereological methods is the coarea formula, involving v

vi

PREFACE

generalized Jacobians. This transformation formula is presented in Chapter 2 together with many simple geometric examples and also more complicated examples needed in the following chapters. Chapter 3 is a study of the rotation invariant measure on linear subspaces in Rn. Geometric decompositions of the q—fold product of Lebesgue measure are treated in Chapter 4 and generalized to decompositions of Hausdorff measures in Chapter 5. Local slice formulae and some projection formulae are discussed in Chapter 6. Practical aspects of local methods, including their implementation in the microscopical study of biological tissue and some biological applications are presented in Chapter 7 while Chapter 8 discusses the so-called modelbased approach, a somewhat misleading terminology since the models considered are of a general non-parametric type. I want to thank Ole E. Barndorff-Nielsen for originally suggesting to write this book and for following my work with interest. In the academic year 95/96, I lectured on the subjects of the book at the Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus. I am grateful to the 12 students patiently listening to the lectures, reading the lecture notes, discussing the unclear points, doing the exercises, etc. Especially thanks to my Ph.D. student Jørgen Nielsen and to Kristian Stegenborg Larsen and Niels Væver Petersen. I also want to thank Jørgen Hoffmann-Jørgensen for fruitful discussions about the coarea formula and related matters. Comments on earlier drafts by Morten Bech, Hans Jørgen G. Gundersen and Rick Vitale are also gratefully acknowledged. The illustrations have been made with utmost skill and care by Jakob Goldbach, using the programs Adobe Illustrator 6.0 and STRATA STUDIO Pro VERSION 1.75+. Trine Tandrup and Anette Larsen have provided the photographs of biological tissue. Last but not least I want to thank my husband Kaj and my daughter Jane for patiently listening to all my stories during the writing process and for giving me the opportunity to write this book.

Aarhus January 1997

Eva B. Vedel Jensen

Contents Preface

v

List of notation

xi

1 Introduction to stereology 1.1 Sampling theory 1.2 Stereological estimation of number 1.3 Stereological estimation of volume 1.4 Length and surface area estimation under isotropy 1.5 Local stereology 1.6 Exercises 1.7 Bibliographical notes

1 1 3 11 18 27 29 33

2 The coarea formula 2.1 Hausdorff measures 2.2 The coarea formula 2.3 The special case d = n 2.4 Hausdorff measures on affine subspaces 2.5 Polar decomposition of Lebesgue measure 2.6 Translative decompositions of Hausdorff measures 2.7 A transformation result for surface area measure 2.8 Simplices 2.9 Exercises 2.10 Bibliographical notes

35 35 37 42 44 45 46 49 52 56 61

3 Rotation invariant measures on C™ 3.1 Construction of rotation invariant measures on C™ 3.2 Crofton's formula 3.3 A result on projections 3.4 Pairs of subspaces 3.5 Random subspaces 3.6 Random grids 3.7 Exercises 3.8 Bibliographical notes

63 63 72 75 78 82 83 88 93

vii

viii

CONTENTS

4 The classical Blaschke-Petkantschin formula 4.1 A local estimator of planar area 4.2 Decompositions involving lines in Rn 4.3 Proof of the classical Blaschke-Petkantschin formula 4.4 Local estimators of volume 4.5 Local integral geometric formulae for powers of volume 4.6 Exercises 4.7 Bibliographical notes

95 95 97 100 105 112 114 119

5 The generalized Blaschke-Petkantschin formula 5.1 A local estimator of length in R2 5.2 Prerequisites concerning G—factors 5.3 Decomposition of a single Hausdorff measure 5.4 Decomposition of a product of Hausdorff measures 5.5 Local estimators of d—dimensional Hausdorff measure 5.6 An alternative estimator of surface area \1^~1{X) 5.7 Exercises 5.8 Bibliographical notes

121 121 123 126 132 138 142 146 153

6 Local slice formulae 6.1 A local estimator of number in R3 6.2 A local slice formula for d—dimensional Hausdorff measure 6.3 Local slice estimators of d—dimensional Hausdorff measure 6.4 The special case d — n 6.5 The case 0 < d < n 6.6 Some further developments for n = 3 6.7 Exercises 6.8 Bibliographical notes

155 155 157 159 160 164 167 170 173

7 Design and implementation of local stereological experiments 7.1 Optical sectioning 7.2 Implementation of local designs 7.3 Local stereological estimators in use 7.4 Particle aggregates 7.5 Applications of local stereological methods 7.6 Systematic sampling along an axis 7.7 The circular case 7.8 Exercises 7.9 Bibliographical notes

175 175 175 178 181 183 187 192 195 199

CONTENTS

ix

8 The model-based approach 8.1 Point processes in Rn 8.2 Marked point processes in Rn 8.3 A few results from invariant measure theory 8.4 Estimation of the if-function of the reference point process 8.5 Estimation of moments in the mark distribution 8.6 Exercises 8.7 Bibliographical notes

201 201 206 208 208 212 219 221

9 Perspectives and future trends 9.1 Mathematical and statistical aspects 9.2 Affine version of local stereology 9.3 Curvatures and other parameters 9.4 Future trends

223 223 224 226 227

Appendix Invariant measure theory

229

References

233

Subject index

245

This page is intentionally left blank

List of notation Sets Rn

n-dimensional Euclidean space. The i'th coordinate of x G Rn is denoted xi or {x)i ^R1

R O

The origin of Rn

i?+

Positive real numbers

[a, b] (a, b) [a, 6) Bn(x,r) 5n~!

Closed, open and half-closed-open intervals, respectively, between aeRn and b G Rn, e.g. [a, b) = {aa+ (1 - a)b : 0 < a < 1} Open ball in i? n with centre x e Rn and radius r The unit sphere in i? n

5J-1

{u G S™'1 : o;n > 0}

£n

The unit ball in Rn

Lp

p—subspace; p—dimensional linear subspace of Rn

Cp

The set of p—subspaces of Rn

£ntr\

The set of p—subspaces of Rn, containing a fixed r—subspace Lr, say. Note that ££ (0) = ££

Tp

p—slice of thickness 2t, i.e. a set of the form Lp + Bn(0, t)

n

Tp

The set of p-slices of Rn of thickness 2t

TJ7,

The set of p-slices Lp + Bn(0, t) for which L p D L r , where L r is a fixed r—subspace. Note that T 7 ^ = 7^,n

Gg

^—grid, i.e. grid of parallel g-dimensional affine subspaces

QV:

A set of q—grids in Rn, identical up to rotations and translations

Convex set

A set X C Rn with the property [x,y] C X for all x, y G X

xi

xii Starshaped set

LIST OF NOTATION A set X C Rn is said to be star-shaped relative to x G X, if [x, y] C X for all y G X

Constants cdn

The volume (Lebesgue measure) of the unit ball Bn in Rn, = 7r^ n r(l + \n)~x, cf. Exercise 2.7

^

The surface area of the unit sphere Sn'1 Exercise 2.7

c(n,p)

=

in Rn, = 2 ( 7 r ) H \ ± n ) - 1 , cf.

^-i-^-r+i

c(n,0)

=1

5/;.

The /c'th Bernoulli number, A; = 0 , 1 , . . .

Inner product and norm {x,y)

\\x\\

= YsxiViixiy

e

Rn

=(x)x)1/2,xeRn

Concepts relating to linear subspaces span

s p a n { x i , . . . , Xd} is the linear subspace of Rn, spanned by xi,..-,Xd £ ^ n , i-e- a*l vectors of the form a\x\ -\ h a^x^, where ai G R, i — 1 , . . . , d

L1-

The orthogonal complement of the linear subspace L of i ? n , x G L1- if and only if (x, y) = 0 for all y G L

dimL

Dimension of L

L ©M

Orthogonal sum of the linear subspaces L and M of Rn, i.e. L ø M i s the sum of L and M , and L and M are orthogonal

Le M

Orthogonal difference of L and M , i.e. L e M = L n ( L n M)" 1

7T^

Orthogonal projection onto L. Recall that if a\,..., orthogonal basis of L, then

TTUJ

ITLX

= J2 n'tf ad

Orthogonal projection onto span{a;},u; G Rn

a^ is an

LIST OF NOTATION

xiii

Matrices In

n x n identity matrix

det(A)

Determinant of the matrix A

T

A

Transposed matrix of A

SO(n)

The group of rotations in Rn, special orthogonal transformations

SO(n, Lr)

The subgroup of SO{n) consisting of rotations, keeping the r—subspace Lr fixed

Set operations X +y

= X + {y}, X C Rn, y G Rn

X

={-x

X\Y

Set difference, = {z e Rn : z e X, z > # 0 )

I = 1 , . . . , :V. cf. Exercise 1.2. The set of particles hit by the plane is therefore a size-biased sample. Under the assumption that L^^Zi) > 0 for all i, we can construct the Horvitz™ Thompson estimator of N,

ies In practice, L(irwX) can easily be measured directly, cf. Figure 1,1» while L(7rwZi) cannot be determined from a single plane section. Under the assumption that all particles ha¥e the same» known and simple shape like spherical shape, the estimator can be simplified, cf. Wicksell (1925) for the first reference. Craz-Orive (1980a)

6

1. INTRODUCTION TO STEREOLOGY

suggested to determine the projection lengths using local serial sectioning» thereby avoiding specific shape assumptions. A more convenient procedure was however suggested by Sterio (1984). Di'sector sampling design. A breakthrough in the number estimation came in 1984. when Sterio (1984) suggested to sample particles by means of two planes a short, but known distance h apart, a so-called disector (di=two). The idea is to sample all particles hit by one of the planes» the reference plane, but not by the other» the look-up plame, cf. Figure 1.3.

Figure 1.3. A disector consists of a reference plane and a look-up plane, a distance h apart.

The reference plane is denoted by i#2 and is distributed as before, the look-up plane is L-2 — JtUJ.

The disector sample is S = {i € V : Zi H L2 ± 0. Z, P (L-2 - M - 0}. Under weak assumptions about particle shape, the sampling probabilities can be calculated» as shown in the proposition below. As we will see» the sampling probability does not depend on the particle considered. In essence, we sample from the population of particle caps instead of whole particles» so all particles are 'made into* the same height.

1.2 STEREOLOGICAL ESTIMATION OF NUMBER

7

Proposition 1.3. Suppose that for all i G V, the projection ir^Zi is a line-segment of length at least h, i.e. KujZi = {uu : a,i < u < bi}, where a^bi G R,bi — ai > h,i = 1 , . . . , N. Then the sampling probabilities of the disector design are the same for all objects, viz. p(i)

= p(i eS) = h/LfaX^ie

V.

Proof. Recall that L 2 has the same distribution as L2(o) + Uu. For any particle Z we have Z n (L2(o) + Uu) ^ 0 & Uu G iruZ. Therefore, we get P(i eS) =

P{UUJ

= P(Uu

G irUJZl, {U - h)u 0 TTLUZ1) G ITuZi^TTuZi

= LfaZiXfruZi +

+

hu))

hu^/L^^X)

( }

= h/L(ir„X),

where we at (*) have used the assumption of the proposition.

□

The Horvitz-Thompson estimator of the particle number N therefore becomes

In order to determine \S\, it must be possible for all i G V and for all u G R to identify the set {Zi D (L2(o) + uu), Zi D (L2(o) + 0 - /i)^)} generated by a disector with position u. This requires either prior knowledge about the particles or extra information collected from e.g. moving optical planes. In the latter case, the assumption of Proposition 1.3 concerning a minimal 'height' of the particles is not necessary, cf. Chapter 7. Systematic disector sampling design. Usually, it is not enough to collect data from a single disector. Instead, a systematic set of disectors can be used. The reference planes of the disectors constitute a series of equidistant, parallel planes with distance A>h between neighbour planes. The series is {L2(0)+ (U + iA)w}°°=-00,

8

1. INTRODUCTION TO STEREOLOGY

cf. Figure 1,4, where I' is now a uniform random variable in the interval [0,A), say. If we let Siu) = {ieV:Zin

(L 2(0) + 1^) ^ ø.Z, n (L 2(0) + (u - h)u) = 0}

be the set of particles sampled by a disector with position m, thee the sample generated by the systematic set of disectors is

Figure 1.4. A systematic set of disectors.

In the proposition below» the sampling probabilities are derived. Proposition 1.4. Suppose that for all i € V, the projection ir^Zi is an interval of length, at least h, cf. Proposition 1.3. Then the sampling probabilities of the systematic disector design are />(/) = P(ie

S) = h/Xi

GT.

Proof. Because of the assumption of the proposition» we have for all u S(u + ji A) fl S(u +

JL> A)

= 0. for all ji # J2-

1.2 STEREOLOGICAL ESTIMATION OF NUMBER

9

(A particle is sampled in at most one disector.) If 1{-} denotes indicator function, we therefore get oo

P(ieS)=

£

P(isS(U

+ jA))

j=-oo

= f; Ji{ieS{u + jA)}% ;=-oo

0

^'+1>A

oo

=i E

/

J=-oo

i{ies(«)}d«

j A

oo

= i

M { i G 5(u)}/}'^_0 of bounded, space-filling, non-overlapping and congruent subsets of lf\ In particular, the /.Vs satisfy

Dhr\Dh =

^h±j2.

Since the D,\s are congruent we can find UJ € R:i such that Dj = for all j ,

DQ

+ Uj = {u +

UJ

: u G D0}.

12

1. INTRODUCTION TO STEREOLOGY

Definition 1.5. A uniform spatial point grid is a set of random points in R3 distributed as Go(£/) = {£/ + ^ : j = 0 , l , . . . } , where U is uniform random in Do-

□

Below, we write Go as short for GQ(U), when convenient. The definition of a uniform spatial point grid is illustrated by the 2-dimensional analogue shown in Figure 1.7.

Figure 1.7. The uniform planar point grid. The random variable U is uniform random in the region DQ.

In practice, V(Z) is estimated, using the spatial point grid, by

V(Z) =

V(D0)\ZnG0\.

As shown in the proposition below, this estimator can be regarded as a HorvitzThompson estimator. Proposition 1.6. Let G0 be a uniform spatial point grid and let Z = U ^ Z » be any decomposition of Z, such that Z{ n Z2■ = 0,z / j . Suppose that the decomposition is chosen so fine that for all u G DQ \ZinGo(u)\e{0,lhi

= l,.-.,N.

(1.1)

1.3 STEREOLOGICAL ESTIMATION OF VOLUME

13

Let V = { 1 , . . . , N} and associate to object i the characteristic y(i) = V(Zi), i G V. Let S be the random sample given by S =

{ieV:ZinGo^Qi}.

Then, the estimator V(Z) is the Horvitz-Thompson estimator associated with this sampling design. □ Note that one possible decomposition satisfying (1.1) is the finite number of non­ empty sets among { Z n D j } ^ . Proof of Proposition 1.6. First note that V(Z) can be expressed as the population total

V(Z) = £V(Z;) = 5>(z) = yp. lev

lev

The next step is to determine the sampling probabilities. We find P(i eS) = P(Zt n G 0 jt 0) {1

=]E\ZtnG0\ OO

= E^2i{u

+

UjeZi}

j=0 OO

= Y,P(u + ujeZi) 3=0

J

-°Do 1

°° f J

= =

Dj

VW) 11{u e Zl}du V(Zi)/V(D0).

The Horvitz-Thompson estimator of V(Z) therefore becomes

v(z) = Y/y-j\ = v(Do)\s\.

14

1. INTRODUCTION TO STEREOLOGY

The last step is to show that | 5 | = \Z n G0\. We get N

2= 1

N

i=l

= \s\.

n The spatial point grid design cannot be used in all cases of practical interest. Very often, the 'smallness' of the structure of interest Z suggests a cascade of different sampling levels at different magnifications, whereby the object phase of one level becomes the reference phase of the next level, cf. Cruz-Orive and Weibel (1981), X = Y0DY1D---DYN

= Z.

The parameter of interest V(Z) is then estimated via the equation

The first quantity V(YQ)= V(X) is estimated using the spatial point grid design, or simply by fluid displacement, while the ratios are estimated using different types of sampling designs depending on the magnification. At intermediate levels, light microscopy may be used corresponding to a magnification of the order of 102 to 103, while at higher levels, electron microscopy may be necessary giving magnifications from 103 to 105. Bounded point grid design. We will now discuss the bounded point grid design which is used in cases where high magnifications are needed. Let us assume that the object is to estimate the ratio

VV = V(Z)/V(Y), where Y = Y/v-i is an appropriately chosen reference space containing Z — Y/v. Both Y and Z are assumed to be open and bounded subsets of Æ3. The set X is cut into small blocks and a section of negligible extent compared to X is made through each one of a random sample of blocks. Only sections hitting Y are analysed. Equivalently, we consider small sections hitting Y as the one shown in Figure 1.8. The sections are equipped with a finite set of points.

1.3 STEREOLOCilCAL ESTIMATION OF VOLUME

15

Figure 1.8. The bounded point grid design. The .set Y is here a cell population while the set 2 is the darker regions inside the cells.

Formally» we will define a uniform bounded point grid hitting Y, using a set of n points m T\) =

{itoii-.-iUu,,}.

In the applications, we ha¥e in mind, the points will lie in the same plane but this is not necessary for the theoretical considerations that follow. Definition 1.7. A uniform bounded point grid hitting Y is a set of random points in Ä 3 with the same distribution as

C\)(L:) = r -f r 0 = { r + uoi

r + no,,}.

where U is uniform random in the hitting set {'/ € RA : Y n (u ~t- 7o) ^ 0}«

□

As before, we write (/;. instead of GVT: when convenient, Let us find the density of U. Since Y

" ..

ies

Note that we always have p(i) > 0. The estimator TV is bounded. To see this, let

Vt = {ieV:

11*11 < t},

1.6 EXERCISES

29

dmax = max||zi|| and am[n — arcsin(t/d m a x ). Then

N = \snvt\+ Yl

P^1

iesyPt = - | S | c The local design described above is an example of a design of the type p(i) = c(i)y(i),i eV ies

v

'

The quantity c(i) depends on distance measurements which can be measured in an actual application. More details about practical aspects will appear in Chapter 7.

1.6 Exercises Exercise 1.1. Prove Proposition 1.2. Show also that the variance can be rewritten as \r (- \

ST1-P(i) lev

HK J

/-x2 ,

v^ ievjev,i^j

P{hJ)~ P{i)p(j) r\ ( -\ HK JHKJJ

Exercise 1.2. Derive the sampling probabilities p(i) = P(i eS) = L(iruZi)/L(iruX),i

G V,

associated with the one plane sampling design. Exercise 1.3. This exercise concerns the variance of the estimator ^ L ^ X ) | 5 | h of N based on the disector sampling design. 1. Let Lij = L(7TwZi fl irwZj\[(TTwZi + huj) U (TTWZJ + hu)]),

30

1. INTRODUCTION TO STEREOLOGY ij

= 1,.. .,7V. Show that

Var(tf) = : É ^ £ > , ■ - * ' . Hint. It is a good idea to start by finding the second-order sampling probabilities.

2. Show that Var(7V)>^^7V-iV2. 3. Let us suppose that the particles are positioned such that the minimum of the variance is attained, i.e. - N2.

Var(iV) = ^ p - N Show under this assumption that | 5 | ~ b(l,p)

w tn

i

parameter p =

Nh/L(7ri0X).

Exercise 1.4. In Figure 1.5, a planar section with N — 15 particles is shown. The Horvitz-Thompson estimator is in this example of the form TV = 9 ^ 2 / ( 2 ) , ies where S is the systematic sample of windows and y(i) is the number of particles counted in window i, i.e. the number of particles first seen in window i. Find the distribution of N and show that EN = 15. Exercise 1.5. This exercise concerns the 1-dimensional analogue of the spatial point grid design. We consider an open and bounded subset Z of the real line R. The object is to estimate the length (1-dimensional Lebesgue measure) of Z, using a uniform point grid with distance A between neighbour points, say. This point grid is distributed as {U + i A } ^ z _ 0 0 , where U is uniform random in the interval [0,A). The estimator of L(Z) becomes 00

L(Z) = A ^2 !{U + j=-oo

1. Show directly that EL(Z)

= L(Z).

jAeZ}.

1.6 EXERCISES

31

2. Suppose that Z is a line-segment of length L(Z) — kA + 6, where A; is a nonnegative integer and 0 < 6 < A. Show that the distribution of L(Z) in this case is given by

P(L(Z) = A(fc + l)) = A P(L(Z) = Afc) = 1 - -^. Also show that the variance of the estimator is VarL(Z) = (A - 6)6. Notice that the variance does not depend on the length of the interval. Is this reasonable? 3. Show for an open and bounded set ZC R that oo

VarL(Z) = A ^

L(Z)2

L(Z n (Z + jA)) -

j=-oo

Exercise 1.6. Prove Proposition 1.9, using the standard strong law of large numbers and the central limit theorem. Exercise 1.7. Buffons's needle problem. Let Z be a line-segment in the plane R2 of length I. Let be a random set of parallel lines with distance A between neighbour lines. Here, Li(fi) is the line through the origin O with unit normal vector £1 chosen as a uniform point on the unit semicircle, and U is independent of £2 and uniform in an interval of length A. The distribution of Q, is the same as that of (cos $, sin ), where 3> is uniform in the interval [0,7r). 1. Show under the assumption I < A that, conditionally on Q, P(Z is hit by a line|fi = u) =

L

^

Z

\

Hint. Recall that Z n (Li(u) + (u + jA)uu) / 0 ^> (u + jA)cj e TTujZ. 2. Show that L(TTQZ)/1 has the same distribution as |cos$|, where $ is uniform random in [0,7r). Hint. Without loss of generality Z can be assumed to be parallel to the x—axis. 3. Show that the unconditional hitting probability becomes 2/ P(Z is hit by a line) = —-. 7rA

32

1. INTRODUCTION TO STEREOLOGY

Exercise 1.8. Let Z be a line-segment in R3 and let ft G 5+ be an isotropic direction. Show that the random variable L(TT^Z)/L(Z) is uniform in the interval [0,1]. Hint. Without loss of generality, it can be assumed that Z is parallel to the z—axis. In that case L(7r n Z)/L(Z) = cos/(fi), where /(ft) G [0, f ] is the angle between ft and the z-axis. transformation result (1.3), that /(ft) has density

Show, using the

Exercise 1.9. Let Z be a line-segment in Æ3 and let G 2 = G2(f/,ft) = {L2(ft) + ([/ + j A ) f t : j = 0 , ± l , ± 2 , . . . } be a uniform and isotropic plane grid, as defined in Definition 1.10. Finally, let L(Z) = 2A\ZnG2\. 1. Show that the conditional mean value and variance of L(Z) are E(L(Z)\Q = uj) = 2L(7rUJZ) Var(L(Z)|ft = co) = 4(A - «„)«„, where ^ is defined by L^^Z) = k^A + 6U, where A^ is a non-negative integer and 0 < du < A. Hint. Exercise 1.5. 2. Show that the unconditional mean value and variance are E(L(Z)) = L{Z) Var(L(Z)) - \h{Zf

+ 4£((A -

Sn)Sa).

Hint. It is useful to use the identity Var(L(Z)) = Var£(L(Z)|Q) + £Var(L(Z)|fi). 3. Show that \\mN^{L{Z))=l-L{Z)\ The variance does not tend to 0, although the number of planes hitting Z is increasing to infinity. The reason for this is of course that under the limiting procedure only one direction is represented.

1.7 BIBLIOGRAPHICAL NOTES

33

1.7 Bibliographical notes Stereological methods can be applied in biology, medicine, metallography, min­ eralogy and many other fields. For this reason, the same stereological method has often been invented independently in different fields. It was first in 1962 that the International Society for Stereology was founded and a forum was created for mutual exchange among scientists with interests in stereology. The society is responsible for organizing international congresses (every fourth year) and publishes Journal of Microscopy (together with The Royal Microscopical Society) and Acta Stereologica. Since 1994, Advances in Applied Probability has included the subsection Stochas­ tic Geometry and Statistical Applications (SGSA) where papers on more theoretical aspects of stereology have been published. Workshops on stochastic geometry, stere­ ology and image analysis have been held every second year since 1981. Abstracts from the workshop in 1995 can be found in Jensen (1996). A solid theoretical framework for stereology has been laid down by Roger Miles and Pamela Davy in a series of papers from the seventies and onwards, cf. Davy & Miles (1977), Miles & Davy (1976, 1977) and Miles (1978a, b). See also Coleman (1979). These papers have had a profound impact on the further development of stereological methods. Another series of interesting papers from the eighties has been written by Luis Cruz-Orive and coauthors, cf. Cruz-Orive & Myking (1981), Cruz-Orive & Weibel (1981) and Cruz-Orive (1982, 1983a). This series of papers is more close to biological applications. Analyses of stereological data are included. Reviews of stereology can be found in Weil (1983), Jensen et al. (1985), CruzOrive (1987b), Stoyan (1990) and Baddeley (1993). The monographs Weibel (1979, 1980) and the more recent papers by Gundersen et al. (1988a, b) present stereology to biologists and other users. Baddeley (1993) has recently emphasized that modern stereology can be regarded as spatial sampling theory. In this chapter, I have introduced stereology from this viewpoint and described a stereological method as a geometric sampling design and an associated Horvitz-Thompson estimator. As also noted by Baddeley (1993), this viewpoint does not lead to refined methods of estimating the variance of the estimators. The main problem is that the second-order sampling probabilities may well be zero. As can be seen from the exercises at the end of this chapter, the variance depends in a complicated way on the spatial arrangement of the structure in question. The disector design discussed in Section 1.2 is a sampling design which uses local 3-dimensional information and represents a major breakthrough. It does not only solve the problem of estimating particle number but provide a statistical satisfactory solution to the classical sphere size problem, stated in Wicksell (1925), cf. e.g. Jensen & Gundersen (1987b). A nice review of earlier solutions to the sphere size problem

34

1. INTRODUCTION TO STEREOLOGY

has been written by Cruz-Orive (1983b), see also Jensen (1984) and Coleman (1989). Another very useful design closely related to the disector design is the fractionator design, cf. Gundersen (1986). The variance of the estimator of volume based on a spatial point grid design, cf. Section 1.3, can be approximated, using the theory of regionalized variables, cf. Matheron (1965, 71). Recently, these methods have been reconsidered and further developed, cf. Souchet (1995), Kiéu (1997) and Kiéu et al. (1997). See also Chapter 7. Practical considerations concerning the efficiency of stereological procedures can be found in Gundersen & Østerby (1981). An interesting biological example of volume estimation using magnetic resonance imaging is described in Roberts et al. (1993). The invention of vertical designs by Baddeley (1983, 1984) has had a major importance in practice. The sin-weighted lines illustrated in Figure 1.14 can be replaced by a cycloid test system, cf. Baddeley et al. (1986). Vertical designs have also been developed in local stereology. A dual model-based approach has earlier been studied, cf. Mathieu et al. (1983) and Cruz-Orive et al. (1985), using methods from directional statistics. A vertical design for estimating curve length has been developed by Gokhale (1990). See also Gokhale (1992, 1993) and Batra et al. (1995).

Chapter 2

The coarea formula The coarea formula is a transformation formula for integrals with respect to Hausdorff measures. In this chapter we will present this formula and derive a number of applications which are important for the development of local stereological methods.

2.1 Hausdorff measures Let ujd = 7T2^r(l + \d)~l be the d—dimensional Lebesgue measure of the ball in R with radius 1. In particular, LOQ — 1, u\ — 2, U2 = TT and ^3 = |7r. Let A C Rn. For any e > 0 we then let d

\dn(A,e) = i n f { £ w d ( ^ ^ )

d

: A C U ^ d i a m A , < e,Vj},

(2.1)

3

where diamAj = sup{||x — x'\\ : x £ Aj,xf € Aj}. The number of sets in a covering {Aj} of A may be finite or infinite. The sum in the definition of A^(A, e) may accordingly be finite or infinite. Note that 6i<e2^\dn(A,61)>\dn(A,e2).

(2.2)

This is easy to see because the set of coverings appearing in (2.1) is decreasing when e decreases. We now define the d—dimensional Hausdorff measure A^ in Rn by \dn{A)=\\m\dn(A,e). ejO

Note that because of (2.2), we also have A ^ 4 ) = supA^4,e). €>0

35

36

2. THE COAREA FORMULA

As mentioned earlier, the Hausdorff measures have been introduced to measure d—dimensional volume in Rn. In particular, A° is counting measure, cf. Exercise 2.1, and AJJ = An, the Lebesgue measure in Rn, cf. Hoffmann-Jørgensen (1995b, p. 6-7). Note that

ua{

-) = 2s

3

diam

^ i ' for d = 1

J

and, in case A is a spatial curve, the sum of the diameters of the covering sets is converging, when e tends to zero, to what is usually called the length of the curve, cf. Figure 2.1 and Hoffmann-Jørgensen (1995b, p. 6-7). In Exercise 2.2, it is shown that \ln([a,b)) =

\\a-bla,beRn.

Figure 2.1. The sum of the diameters of the circles converges to the length of the curve when e goes to zero.

Furthermore, as we will show later in this chapter, if A is a subset of a d-dimensional affine subspace of Rn, then X^(A) is the ^-dimensional Lebesgue measure defined on this affine subspace of the set A, cf. Figure 2.2. The Hausdorff measures also satisfy the following relation for a G Ä\{0}, a e Rn and A C Rn \dn(aA + a) = cf. Exercise 2.3.

\a\d\dn(A),

' 2.2 ;,THB "CÖÄRIA 'TCRftRJLA

37

Figure 2.2. If A is a subset of a ^-dimensional affine subspace, then Xrl,iA) is the J-dimensional Lebesgue measure defined on this affine subspace of the set A.

2.2 The coarea formula The coarea formula IEYOIYCS transformations between differentiable manifolds. Intuitively» X C Rn is a d—dimensional manifold if X locally looks like R*. Formally, X is said to be a d—dimensional manifold, if there exists for all x £ X a neighbourhood U of x and a homeomorphism ^ : f." — R
Figure 2.3. A ^-dimensional differentiable manifold in Rn.

38

2. THE COAREA FORMULA

The manifold X is said to be a differentiable manifold if, intuitively speaking, X is smooth. Formally, we require that if (U,(p) and (V,^) are two coordinate neighbourhoods of, say» x and y, respectively» such that U n V ^ 0, thee the two mappings (po^r1 and tp o ip""1 are diffeomorphisms, cf. e.g. Boothby (1975). The unit sphere Sn^1 is an (n — t)—dimensional differentiable manifold. If A" is a (/-dimensional differentiable manifold» then there exists for each x £ X a »/-dimensional linear suhspace of /?", denoted Tan|X,x] and called the tangent space of A* at .r £ A". Let H\p) be a coordinate neighbourhood of x. Then, the tangent space can be determined as Tan [A*. ./•; — s]>an{ai,..., atj} where «,• £ It" are given by

/— 1 (L and (^ _ 1 )j is the j ' t h coordinate function of s~[.j ----- 1 //. For a '/--dimensional differentiable manifold A. we denote A.'.'AA the length. surface area or volume of X, depending on whether d --- L // — 1 or //. Below, we present the version of the coarea formula appropriate tor our purposes. We use the notation dxd as a short notation for X^(dx). Furthermore, vectors are represented as column vectors and Är is the transposed matrix of A. fn Figure 2.4, the situation in Theorem 2.1 is illustrated.

Figure 2,4. In Theorem 2.1» the mapping / is defined on an open subset D of Rn, Here» we have for simplicity chosen I) ■■■- />'".

2.2 THE COAREA FORMULA

39

Theorem 2.1. Let D C Rn be an open set and let y C i?fc be a differentiable manifold of dimension p. Let / : Z) —» y be a differentiable mapping. Furthermore, let I C D be a differentiable manifold of dimension d, where d > p. Then, there exists a function Jf(-;X) : X —> i?+ U {0}, called the Jacobian, such that for any non-negative measurable function # defined on X we have y g{x)Jf(x- X)dxd = J X

Let Df(x)

g(x)dxd-?dyF.

j

Y

Xnf-i(y)

be the k x n matrix of partial derivatives

and let keiDf(x) = {y e Rn : Df(x)(y) Jf(x]X) > 0 iff the dimension of

= O} be the kernel of Df(x).

Tan[X, x] n (kerD/(x) n Tan[X, x])"1

Then, (2.3)

is equal to p. If the subspace (2.3) has dimension p, then the Jacobian can be calculated as Jf(x;X)

y/det{Df(x;X)Df{x;X)T},

=

where Df(x; X) is the following p x k matrix r Tn

I1] Df(x;X)=

•

\Df(x)T

\eT and e i , . . . , ep is an orthonormal basis of the subspace (2.3). In particular, if g{x) = h(f(x)) J h(f(x))Jf(x; x

we get the following transformation result

X)dxd = y h(y)Xdn-P(X n Z " 1 ^ ) ) ^ y

Example 2.2. Let £>

=

□

JR

2

X = (0,TT) x (0,2TT) C L>

y = 5 2 c R3

(2.4)

40

2. THE COAREA FORMULA

and consider the mapping / : R2 - S2 (#, (/)) —> (sin 9 cos 0, sin 9 sin 0, cos 9) Here, n = 2, fc = 3, d = 2 and p = 2. The matrix af partial derivatives can be written as [cos 9 cos 0 — sin 9 sin ø 1 Df(0,4>)= cosØsinø sin 9 cos <j) .

[ -sin(9

0

J

The rank of Df(0,4>) is 2 and keiDf(9, ) = {O}. Therefore, Tan[X, (0, 0)] n (ker£>/((0, ø)) n Tan[X, (0, 0)]) X = Æ2n

({OjnR2)1-

= R2. The orthonormal basis becomes

ei=

G)' e 2 = (0

and Z?/((0,0); X) = I2Df{0, f.

The Jacobian can now be calculated as Jf((9, ); X) = yJdet{Df(0,4>)TDf(e,<j>)}

=

V d 6 t { [j sin%]>

= sinØ. Using the coarea formula with a function of the form g{9,4>) = h(f(0, )), we get the following transformation result, cf. (2.4), / / /i(sm 0 cos , sin 0 sin 0, cos 0) sin ØdødØ = oo s2

h(y)dy2.

In particular, we get the transformation result (1.3) of Chapter 1 by restricting attention to functions vanishing on S2\S+. □ Example 2.3. Let D = X = R2\{0} Y = S1 CR2

C R2

2.2 THE COAREA FORMULA

41

and consider the mapping / : R2\{0} x->x/\\x\\.

S1

Note that the mapping / has a simple geometric interpretation as the mapping associating to a point x G R2\{0} the direction of the ray starting at O and passing through x, cf. Figure 2.5.

Figure 2.5. The mapping / from Example 2.3. The matrix of partial derivatives of / is

Df(x) = ^ \

^

IWJ

1

~~ iN 7 r s p a n ^ J ~' where ix

s x±

denotes the orthogonal projection onto spanjx} . Accordingly, kevDf(x) — span{x},

and Tan[X, x] n (keiDf(x) n Tan[X, x]r = R2 n (span{a:} O Ä 2 )- 1 =span{x} has dimension 1 for all x e R2\{0}. described in Theorem 2.1.

We can therefore calculate the Jacobian as

42

2. THE COAREA FORMULA Let e = ||x|| _ 1 (-X2,xi) T be a unit vector spanning (keiDf(x)) Df{x;R2\{0})

= eTDf(xf

= (Df(x)(e)f

. It follows that

= \\x\\~leT

and Jf(x;R2\{0})

= W - y d e t { e T e } = »iir1.

We then have the following transformation result, cf. Theorem 2.1, j g(x)\\x\\-ldx2

= f

2

1

R

S

f

g{x)dx1duj\

f-i(u)

At the left-hand side, we have replaced R2\{0} with R2. Note that f~l(u) is the ray starting in O with direction u.

D

2.3 The special case d = n In this section, we present three corollaries of Theorem 2.1 which are concerned with the case where the dimension of the set X appearing in Theorem 2.1 is n. Corollary 2.4. Let the situation be as in Theorem 2.1. Assume in addition that d = n, p = k and that the rank of Df(x) is k. Then, the Jacobian can be calculated as

Jf(x;X) =

^det{Df(x)Df(xf}.

Proof. Since d > p we have under the assumptions of the corollary that n > k. Furthermore, if fi denotes the z'th coordinate function of / , i — 1 , . . . , k, we get since Tan[X, x] — Rn Tan[X, x] n (keiDf(x)

n Tan[X, x})1-

= (kerD/(x)) ± = span{D/i(x),...,

Dfk(x)}.

Since Df(x) has full rank k, the vectors Dfi(x),..., Dfk(x) are linearly independent and the subspace spanned by these vectors has dimension p = k. We can therefore use the procedure of Theorem 2.1 to calculate the Jacobian. Let e i , . . . , e& be an orthonormal basis of span{D/i ( £ ) , . . . , £ / / , ( > ) } .

2.3 A SPECIAL CASE

43

We then have r T-]

\l I Df(x) = A\ • ,

where A is a A; x fc invertible matrix. Therefore, £>/(*; X ) D / 0 r ; X ) T r T ~i

r T i

1

p1

r1 =

[ei,...,ejfe]i4T>l

"

Lei J

•

[ei,...,efc]

14 J

It follows that

Jf{x;X) =

^ået{ATA}

=

jdet{AAT}

[eil = [detM

•

[ei,...,e fe ]j4 T }] 1 / 2

lek J

=

y/det{Df(x)Df(xf}. D

Corollary 2.5. Let the situation be as in Theorem 2.1. Assume in addition that d = n, p = n and that the rank of Df(x) is n. Then, Jf(x; X) = V / det{D/(x) T D/(x)}. Proof. Since p < k we have under the assumptions of the corollary that n < k. Since Df(x) is of rank n, the columns of Df{x) are n linearly independent vectors in Rk. Accordingly, keiDf(x)

= {O} =» (kerD/^))" 1 = i T

44

2. THE COAREA FORMULA

Therefore, if In is the n x n identity matrix,

= Df(x)T

Df(x;X) = InDf(xf and the result follows immediately.

□

Note that Example 2.2 is of the type described in Corollary 2.5 above. Corollary 2.6. Let the situation be as in Theorem 2.1. Assume in addition that d = p = k = n and that Df{x) has rank k = n. Then,

Jf(x]X)

= \det{Df(x)}\.

□ Corollary 2.6 gives the usual way of calculating the Jacobian.

2.4 Hausdorff measures on affine subspaces As mentioned in Section 2.1, if A is a subset of a d—dimensional affine subspace of Rn, then X^(A) is the d—dimensional Lebesgue measure defined on this affine subspace of the set A. In the proposition below, this result is proved, using the coarea formula, see also Figure 2.2. Proposition 2.7. Let Fj be a d—dimensional affine subspace of Rn Fd = span{ai,...,ad} + a, where a\,..., a^ is a set of d orthonormal vectors in Rn. A' C Zi^ be the unique set such that

Let A C F^ and let

d A =

{^XiUi

+ a : (xi,...,xd)

€ A'}.

2= 1

Then, A^(A) = \d(Af), where Arf is ordinary Lebesgue measure in Rd. Proof. Let / be the 1-1 mapping

f-.&^Fd d

(xi,...,xd)

- > ^Xidi 2=1

+ a.

2.5 POLAR DECOMPOSITION OF LEBESGUE MEASURE. Note that A — f(Af).

45

The matrix of partial derivatives of / is DfUi

rf/) = [ a i , . . . , a j .

In particular, Df(X{... .. xj) does not depend on ( x i , . . . , rr^). We can now use the coarea formula with D = X = Rd and Y = Fj. According to Corollary 2,5, •//((■n

xd); Rd) = \/dex{Df(jL^...<xd)TDf{x1,...,xd)}

= 1.

Since / is 1—1, the version of the coarea formula given in (2.4) now becomes

I h(f{x))dxd = I h(y)dyd. By using h(y) = l{y e A}, we find Xd(Af) = Xi(A).

D

2.5 Polar decomposition of Lebesgne measure This section concerns the generalisation to Rn of Example 2.3. This result will be important in the coming chapters» especially in Chapter 4. For later use, we formulate the result in Proposition 2.8 below. The mapping / of the proposition is illustrated in Figure 2.6.

Figure 2.6.- The mapping / from Proposition 2.8. Proposition 2.8. Let f : Rn\{0}

-

.r^x/Wxl

S"" 1

46

2. THE COAREA FORMULA

Then, for any non-negative function g on Rn, f g{x)\\x\f{n~1)dxn

=

f

f

g{x)dx1dujn-\

Proof. We want to find the Jacobian of the mapping / . Note that although d — n we cannot use Corollary 2.4 or 2.5. The matrix of partial derivatives of / is given by

Df(x)(y) = \\x\f\y

{

-

-^x)

=M ^ ^ ^ y

G R\

cf. Exercise 2.8, and therefore, kerD/(x) = span{x}. Let e i , . . . , en-i be an orthonormal basis of span{x} . Then, r

Df(x;Rn\{0}) =

T

1

■

r

D/(a:)T = W - 1

T ~\

•

T T len-l J len-l J = [Df(x)(e1),...,Df(x)(en-1)]T

r ei l

= INI _ 1

• T

len-l J and r

Jf(x; Rn\{0}) = Wxr^-^deti

T

i

■ [elt..., e^]})1'2 T len-l J

=

M^^. D

2.6 Translative decompositions of Hausdorff measures It is well-known that ordinary Lebesgue measure in Rn is a product measure of n copies of Lebesgue measure in R. Furthermore, if Lq is a