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DIFFEOMORPHIC POINT MA TCHING WITH APPLICA TIONS IN MEDICAL IMA GE ANAL YSIS By HONGYU GUO A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2005

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Cop yrigh t 2005 b y Hongyu Guo

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T o m y daugh ter Alicia.

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A CKNO WLEDGMENTS I will b e forev er indebted to m y advisor, Dr. Anand Rangara jan, for his in v aluable guidance and supp ort. I am v ery luc ky to ha v e the opp ortunit y to w ork with him. I am v ery grateful for his appreciation and condence in m y mathematical talen t. Although he is an outstanding researc her in the computer vision and medical imaging eld with man y distingushed con tributions, he is v ery h um ble and do wn to earth. During the y ears of m y study I could kno c k at his do or an y time I needed him and he w ould put do wn his w ork and discuss sparkling ideas with me. W orking with him has b een truly a w onderful and rew arding exp erience. I w ould lik e to express deep est thanks to Dr. Aruna v a Banerjee, Dr. Jorg P eters and Dr. Baba V em uri of the Departmen t of CISE, Dr. Y unmei Chen of the Departmen t of Mathematics and Dr. Haldun Aytug of the Departmen t of Decision and Information Sciences for serving on m y sup ervisory committee. I thank them for their in v aluable criticism and advice. I wish to extend m y gratitude to Dr. Sarang Joshi at the Univ ersit y of North Carolina at Chap el Hill and Dr. Lauren t Y ounes at Johns Hopkins Univ ersit y They are the coauthors of a b o ok c hapter and sev eral pap ers that I ha v e written and this w ork in m y dissertation is greatly b enetted from their previous w ork on dieomorphic landmark matc hing. I w ould lik e to thank Dr. Eric Grimson, the director of MIT AI Lab, for the helpful discussion and v aluable commen ts when he visited the Univ ersit y of Florida. I wish to extend sp ecial thanks to Dr. Alp er for an enligh tening discussion on triangulation and surface reconstruction. iv

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My thanks also go to ev ery one of m y fello w studen ts at the Cen ter for Computer Vision, Graphic sand Medical Imaging at the Univ ersit y of Florida, particularly Bing Jian, San thosh Ko dipak a, Jie Zhang, Adrian P eter, Zhizhou W ang, Eric Sp ellman, F ei W ang, Tim McGra w and Xiaobin W u. I get constan t help from them ev ery da y I am indebted to Dr. John M. Sulliv an at the Univ ersit y of Illinois at UrbanaChampaign for gran ting p ermission to use his graphics image of the Klein b ottle. I w ould lik e to thank m y wife Y anping for her lo v e and supp ort and I thank m y daugh ter Alicia for b eing a w onderful daugh ter. Alicia and I enjo y ed w orking together on man y w eek ends. While she w as fascinated b y the rotating shap es of computer graphics I displa y ed with MA TLAB, I admired her b eautiful and creativ e colorings and dra wings. This researc h is supp orted in part b y the National Science F oundation gran t I IS 0307712. v

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T ABLE OF CONTENTS page A CKNO WLEDGMENTS . . . . . . . . . . . . . . iv LIST OF T ABLES . . . . . . . . . . . . . . . . viii LIST OF FIGURES . . . . . . . . . . . . . . . . ix ABSTRA CT . . . . . . . . . . . . . . . . . . xi CHAPTER1 INTR ODUCTION . . . . . . . . . . . . . . . 1 2 PREVIOUS W ORK . . . . . . . . . . . . . . 17 2.1 Thin Plate Splines (TPS) . . . . . . . . . . . 18 2.2 Bo okstein's Application to 2D Landmark W arping . . . . 19 2.3 More on Splines . . . . . . . . . . . . . . 21 2.4 Repro ducing Kernel Hilb ert Space (RKHS) F orm ulation . . . 23 2.5 The F olding Problem of Splines . . . . . . . . . 24 2.6 Imp osing Restriction on the Jacobian . . . . . . . . 25 2.7 The Flo w Approac h . . . . . . . . . . . . . 25 2.8 Corresp ondence and Softassign . . . . . . . . . . 28 2.9 Distance T ransforms . . . . . . . . . . . . 28 2.10 Implicit Corresp ondence . . . . . . . . . . . 29 2.11 Shap e Con text . . . . . . . . . . . . . . 31 2.12 Activ e Shap e Mo dels . . . . . . . . . . . . 31 2.13 Deterministic Annealing Applied to EM Clustering . . . . 32 2.14 Statistical Shap e Analysis on Dieren tiable Manifolds . . . 34 2.15 Distance Measures from Information Theory . . . . . . 36 3 DIFFEOMORPHIC POINT MA TCHING . . . . . . . . 42 3.1 Existence of a Dieomorphic Mapping in Landmark Matc hing . 42 3.2 Symmetric Matc hing due to Time Rev ersibilit y . . . . . 44 3.3 A Theoretical F ramew ork for Dieomorphic P oin t Matc hing . 46 3.4 A Dieomorphic P oin t Matc hing Algorithm . . . . . . 49 3.5 Applications to 2D Corpus Callosum Shap es . . . . . . 53 3.6 Applications to 3D Shap es . . . . . . . . . . . 55 3.6.1 Exp erimen ts on Syn thetic Data . . . . . . . 59 3.6.2 Exp erimen ts on Real Data . . . . . . . . . 61 vi

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4 TOPOLOGICAL CLUSTERING AND MA TCHING . . . . . 72 4.1 F undamen tals of T op ological Spaces . . . . . . . . 72 4.2 Kohonen Self-Organizing F eature Map (SOFM) . . . . . 76 4.3 T op ological Clustering and Matc hing . . . . . . . . 79 4.3.1 Wh y: the Need for T op ology . . . . . . . . 79 4.3.2 Ho w: Graph T op ology . . . . . . . . . . 80 4.4 Ob jectiv e F unction and the Algorithm . . . . . . . 84 4.5 Prescrib ed T op ology . . . . . . . . . . . . 85 4.5.1 Chain T op ology . . . . . . . . . . . . 85 4.5.2 Ring T op ology . . . . . . . . . . . . 85 4.5.3 S 2 T op ology . . . . . . . . . . . . . 85 4.6 Arbitrary T op ology . . . . . . . . . . . . . 89 5 CONCLUSIONS . . . . . . . . . . . . . . . 98 5.1 Con tributions . . . . . . . . . . . . . . 98 5.2 F uture W ork . . . . . . . . . . . . . . . 99 REFERENCES . . . . . . . . . . . . . . . . . 100 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 106 vii

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LIST OF T ABLES T able page 3{1 Mo died Hausdor distance of the matc hing p oin t sets. . . . . 57 3{2 Geo desic distances and mo died Hausfor distances . . . . . 58 3{3 Matc hing errors on syn thetic data with dieren t noise lev els . . . 60 3{4 Limiting v alue of determined b y the n um b er of clusters . . . 61 3{5 Jensen-Shannon div ergence for v arious pairs of shap es . . . . 70 3{6 Hausdor and mo died Hausdor distance for v arious pairs of shap es 71 viii

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LIST OF FIGURES Figure page 1{1 Grids in R 2 (a) b efore transformation and (b) after transformation . 9 1{2 Klein b ottle immersed in R 3 . . . . . . . . . . . . 9 1{3 (a) The template image and (b) The reference image . . . . 10 1{4 Space deformations . . . . . . . . . . . . . . 10 1{5 Am biguit y of landmark corresp ondence . . . . . . . . 13 1{6 Tw o shap es of a ribb on and the asso ciated landmarks . . . . 15 1{7 Misin terpretation of the shap e . . . . . . . . . . . 16 2{1 Deformation of the thin plate . . . . . . . . . . . 38 2{2 Landmark displacemen ts . . . . . . . . . . . . 38 2{3 Thin-plate Spline in terp olation . . . . . . . . . . . 39 2{4 Dieomorphic in terp olation . . . . . . . . . . . 39 2{5 Tw o distance transformed images of three landmarks . . . . 40 2{6 Lev el sets of t w o distance transformed images . . . . . . 41 3{1 Existence of a dieomorphic mapping . . . . . . . . . 44 3{2 Asymmetry of the matc hing . . . . . . . . . . . 45 3{3 P oin t sets of nine corpus callosum images. . . . . . . . 56 3{4 Clustering of the t w o p oin t sets. . . . . . . . . . . 56 3{5 Dieomorphic mapping of the space. . . . . . . . . . 56 3{6 Matc hing b et w een the t w o p oin t sets. . . . . . . . . . 57 3{7 Ov erla y of the after-images of eigh t p oin t sets with the nin th set. . 57 3{8 Tw o p oin t sets of hipp o campal shap es . . . . . . . . 59 3{9 Matc hing errors on syn thetic data for dieren t n um b er of clusters . 65 3{10 Tw o hipp o campal shap es . . . . . . . . . . . . 66 ix

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3{11 Deterministic annealing in the clustering pro cess . . . . . . 67 3{12 Limiting v alue of determined b y the n um b er of clusters . . . 68 3{13 Clustering of the t w o hipp o campal shap es . . . . . . . 69 4{1 Dieren t top ological spaces . . . . . . . . . . . . 77 4{2 Image con tours of t w o hands . . . . . . . . . . . 80 4{3 Clustering of t w o hands . . . . . . . . . . . . . 81 4{4 Corresp ondence . . . . . . . . . . . . . . . 81 4{5 Finite top ology . . . . . . . . . . . . . . . 82 4{6 T op ological clustering and matc hing of t w o hands . . . . . 86 4{7 Corresp ondence with top ology constrain t . . . . . . . . 86 4{8 T op ological clustering of corpus callosum shap es . . . . . . 87 4{9 Corresp ondence in top ological clustering of corpus callosum shap es . 88 4{10 Sphere top ology . . . . . . . . . . . . . . . 89 4{11 T op ological clustering of hipp o campus with S 2 top ology: the rst set 90 4{12 T op ological clustering of hipp o campus with S 2 top ology: the second set 91 4{13 Graph top ology of hipp o campus shap e through learning . . . . 93 4{14 T op ological clustering of hipp o campus shap e: the rst p oin t set . 94 4{15 T op ological clustering of hipp o campus shap e: the second p oin t set . 95 4{16 Fish shap es . . . . . . . . . . . . . . . . 95 4{17 Graph top ology for the sh shap e with 4 nearest neigh b ors . . . 96 4{18 T op ological clustering and matc hing of the sh shap es . . . . 97 x

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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y DIFFEOMORPHIC POINT MA TCHING WITH APPLICA TIONS IN MEDICAL IMA GE ANAL YSIS By Hongyu Guo August 2005 Chair: Dr. Anand Rangara jan Ma jor Departmen t: Computer and Information Science and Engineering Dieomorphic matc hing of unlab eled p oin t sets is v ery imp ortan t to non-rigid registration and man y other applications but it has nev er b een done b efore. It is a v ery c hallenging problem b ecause w e ha v e to solv e for the unkno wn corresp ondence b et w een the t w o p oin t sets. In this w ork w e prop ose a join t clustering metho d to solv e for a sim ultaneous estimation of the corresp ondence and the dieomorphism in space. The cluster cen ters in eac h p oin t set are alw a ys in corresp ondence b y virtue of ha ving the same index. During clustering, the cluster cen ter coun terparts in eac h p oin t set are link ed b y a dieomorphism and hence are forced to mo v e in lo c k-step with one another. W e devise an ob jectiv e function and design an algorithm to nd the minimizer of the ob jectiv e function. W e apply the algorithm to 2D and 3D shap es in medical imaging. W e further prop ose to use a graph represen tation for the shap e top ology information. Results are giv en for prescrib ed top ologies lik e c hain top ology ring top ology { whic h are v ery common in dealing with 2D con tour shap es { and gen us zero closed surface top ology in 3D. W e also in v estigate the top ology problem in general and the learning of top ology with a nearest neigh b or graph. xi

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CHAPTER 1 INTR ODUCTION This w ork is a rst time eort to study the dieomorphic p oin t matc hing problem with unkno wn corresp ondence. In this c hapter w e in tro duce the need for an eectiv e dieomorphic p oin t matc hing algorithm and the bac kground kno wledge of the shap e matc hing eld. W e giv e some fundamen tal denitions and theorems from dieren tial top ology whic h are essen tial in the dev elopmen t of the follo wing c hapters. W e analyze the nature and dicult y of the corresp ondence problem. Shap e analysis using to ols of mo dern dieren tial geometry and statistical theory has far and wide applications in computer vision, image pro cessing, biology morphometrics, computational anatom y biomedical imaging, and image guided surgery as w ell as arc heology and astronom y Shap es pla y a fundamen tal role in computer vision and image analysis and understanding. In general terms, the shap e of an ob ject, a data set, or an image can b e dened as the total of all information that is in v arian t under certain spatial transformations [ 66 ]. Shap e matc hing and corresp ondence problems arise in v arious application areas suc h as computer vision, pattern recognition, mac hine learning and esp ecially in computational anatom y and biomedical imaging. Shap e matc hing b ecomes an indisp ensable part of man y biomedical applications lik e medical diagnosis, radiological treatmen t, treatmen t ev aluation, surgical planning, image guided surgery and pathology researc h. Shap es ma y ha v e man y dieren t represen tations. They can b e represen ted with the in tensities of pixels of an image, whic h is a function dened in a region of 2D or 3D space, or they can b e represen ted with p oin t sets, curv es or surfaces. In this dissertation, w e fo cus on the p oin t represen tation of shap es. P oin t represen tation of image data is widely used in all areas and there is a h uge amoun t of 1

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2 p oin t image data acquired in v arious mo dalities, including optical, MRI, computed tomograph y and diusion tensor images [ 18 24 ]. The adv an tage of p oin t set represen tation of shap es, as opp osed to curv e and surface represen tations is m ultifarious and that is wh y w e fo cus on p oint shap e matching instead of curv e or surface matc hing [ 64 68 67 ]. The p oin t set represen tation is compact in storage. The computational time when using p oin t set represen tation is dramatically reduced as opp osed to when image in tensities are used. The p oin t set represen tation of shap es is univ ersal and homogeneous. It do es not require the prior kno wledge ab out the top ology of shap es. It has the capabilit y to fuse dieren t t yp es of features in to a global, uniform and homogeneous represen tation. A p oin t set represen tation of shap es is esp ecially useful when feature grouping (in to curv es and the lik e) cannot b e assumed. Statistical analysis on p oin t set shap es is straigh tforw ard, as demonstrated in Co otes et al. [ 18 ] using activ e shap e mo dels. The recen t w ork of Glaunes et al. [ 27 ] is a b old step forw ard to generalize the p oin t set shap es to general Radon measures and distributions in the sense of Sc h w artz generalized functions, in order to mo del shap es represen ted b y a mixture of p oin ts and submanifolds of dieren t dimensions (curv es and surfaces). Although this is viable in theory it b ecomes unpractical when it comes to applications. In their exp erimen ts dealing with a mixture of p oin ts and a curv e, they used the tec hnique of resampling the curv e. That comes bac k to the p oin t set represen tation of the curv e itself. Moreo v er, b y doing so, new free parameters, i.e., the relativ e w eigh t of measure b et w een p oin ts and the curv es, is in tro duced, whic h can b e arbitrary P oin t shap e matc hing is ubiquitous in medical imaging and in particular, there is a real need for a turnk ey non-rigid p oin t shap e matc hing algorithm [ 18 24 19 ]. P oin t shap e matc hing in general is a dicult problem b ecause, as with man y other problems in computer vision, lik e image registration and segmen tation, it

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3 is often ill-p osed. W e try to mak e abstractions out of the practical problems to form ulate precise mathematical mo dels of the problem. P oin t matc hing can b e view ed in the con text, or out of the con text of image registration. In the con text of image registration, the p oin ts are view ed as feature p oin ts in the image. The corresp ondence can b e kno wn or unkno wn. When the cardinalit y of the t w o p oin t sets is the same and when the corresp ondence is kno wn, w e call this the landmark matching pr oblem The n um b er of feature p oin ts ma y also b e unequal. In that case w e are dealing with outliers. The dicult y increases dramatically when the corresp ondence is not kno wn and/or when there are outliers. The p oin t matc hing problem can also exist out of the con text of image registration. In this case, the p oin ts are samples from a shap e and w e ha v e a p oin t represen tation of the shap e. When w e ha v e t w o suc h shap es represen ted b y p oin ts, usually the n um b er of p oin ts in the t w o shap es is dieren t and there is no p oin t-wise corresp ondence. W e w an t to nd the corresp ondence b et w een the t w o shap es. W e call this the p oint shap e matching pr oblem. In the follo wing, w e will rst address the landmark matc hing problem and then the p oin t shap e matc hing problem. W e assume the image domain is d -dimensional Euclidean space R d Usually d = 2 or d = 3. Supp ose w e ha v e t w o images, I 1 : n 1 R and I 2 : n 2 R where n 1 R d and n 2 R d The image registration problems can b e classied in to t w o categories: in tensit y based registration and feature based registration [ 52 ]. In the in tensit y based image registration, w e need to nd a map f : n 1 n 2 suc h that 8 x 2 n 1 ; I 1 ( x ) = I 2 ( f ( x )). This is the ideal registration problem. In feature based registration, w e supp ose w e ha v e t w o corresp onding sets of feature p oin ts, or landmarks, f p i 2 n 1 j i = 1 ; 2 ; :::; n g and f q i 2 n 2 j i = 1 ; 2 ; :::; n g W e need to nd a transformation f : n 1 n 2 suc h that 8 i = 1 ; 2 ; :::; n f ( p i ) = q i In man y applications, w e are required to nd the transformation within some restricted groups, lik e rigid transformations, similarit y transformations, ane

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4 transformations, pro jectiv e transformations, p olynomial transformations, B-spline transformations and \non-rigid" transformations. Dieren t transformation groups ha v e dieren t degrees of freedom, the n um b er of parameters needed to describ e a transformation in the group. This also determines the n um b er of landmark pairs that the transformation can exactly in terp olate. Let us lo ok at some examples. In the t w o dimensional space, where d = 2, a rigid transformation, whic h preserv es Euclidean distance, dened b y 1.1 has 3 degrees of freedom ( '; x 0 ; y 0 ) and cannot in terp olate arbitrary landmark pairs. 264 x 0 y 0 375 = 264 cos sin sin cos 375 264 x y 375 + 264 x 0 y 0 375 (1.1) The landmark pairs to b e matc hed m ust b e sub ject to some constrain ts. That is, they ha v e to ha v e the same Euclidean distance. A similarit y transformation dened b y 1.2 has 4 degrees of freedom ( k ; '; x 0 ; y 0 ) and can map an y 2 p oin ts to an y 2 p oin ts. 264 x 0 y 0 375 = k 264 cos sin sin cos 375 264 x y 375 + 264 x 0 y 0 375 (1.2) An ane transformation dened b y 1.3 has 6 degrees of freedom ( a 11 ; a 12 ; a 21 ; a 22 ; x 0 ; y 0 ) and can map an y 3 non-degenerate p oin ts to an y 3 nondegenerate p oin ts. 264 x 0 y 0 375 = 264 a 11 a 12 a 21 a 22 375 264 x y 375 + 264 x 0 y 0 375 (1.3) A pro jectiv e transformation dened b y 1.4 in term of non-homogeneous co ordinates has 8 degrees of freedom and can map an y 4 non-degenerate p oin ts to an y 4 non-degenerate p oin ts.

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5 264 x 0 y 0 375 = 264 a 11 x + a 12 y + a 13 d 11 x + d 12 y +1 b 11 x + b 12 y + b 13 d 11 x + d 12 y +1 375 (1.4) In three dimensional space, where d = 3, w e write the transformation in a more general form, 266664 x 0 y 0 z 0 377775 = 266664 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 377775 266664 x y z 377775 + 266664 x 0 y 0 z 0 377775 ; (1.5) where A = 266664 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 377775 is a 3 3 matrix, whic h represen ts a linear transformation. A rigid transformation requires A to b e orthogonal and has 6 degrees of freedom. A similarit y transformation requires A to b e a similar matrix and has 7 degrees of freedom. An ane transformation allo ws A to b e the most general form and has 12 degrees of freedom and can map an y 4 non-degenerate p oin ts to an y 4 non-degenerate p oin ts. A projectiv e transformation in 3D as dened b y 1.6 has 15 degrees of freedom and can map an y 5 non-degenerate p oin ts to an y 5 non-degenerate p oin ts. 266664 x 0 y 0 z 0 377775 = 266664 a 11 x + a 12 y + a 13 z + a 14 d 11 x + d 12 y + d 13 z +1 b 11 x + b 12 y + b 13 z + b 14 d 11 x + d 12 y + d 13 z +1 c 11 x + c 12 y + c 13 z + c 14 d 11 x + d 12 y + d 13 z +1 377775 (1.6) The term \non-rigid" transformation is often used in a narro w er sense. Although similarit y ane and pro jectiv e transformations do not preserv e Euclidean distance, they all ha v e nite degrees of freedom. In the literature, \non-rigid"

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6 transformation usually refers to a transformation of innite degrees of freedom, whic h can p oten tially map an y nite n um b er of p oin ts to the same n um b er of p oin ts. So w e immediately see a big dierence b et w een nite degree of freedom transformations and non-rigid transformations. Giv en a xed n um b er of landmark pairs to b e in terp olated, the former is easily o v er constrained; the latter is alw a ys under constrained. This is one of the reasons that the non-rigid p oin t matc hing problem is m uc h more dicult. T o nd a unique non-rigid transformation, w e need further constrain ts. W e call this regularization. Tw o desirable prop erties of non-rigid transformations are smo othness and top ology preserving. Let n 1 R d and n 2 R d A transformation f : n 1 n 2 is said to b e smo oth if all partial deriv ativ es of f up to certain orders, exist and are con tin uous. If transformation f : n 1 n 2 preserv es the top ology then n 1 and Img ( f ) = f p 2 2 n 2 j9 p 1 2 n 1 ; p 2 = f ( p 1 ) g ha v e the same top ology A transformation that preserv es top ology means w e will not ha v e tears in space or in the image. A transformation that preserv es top ology is called a home omorphism and its formal denition is: Denition 1. L et n 1 and n 2 b e two top olo gic al sp ac es. A map f : n 1 n 2 is a home omorphism if f is a bije ction; f is c ontinuous; the inverse f 1 is c ontinuous. A smo oth transformation f : n 1 n 2 ma y not preserv e the top ology Namely a smo oth map ma y not b e a homeomorphism. It is easy to see this b ecause a smo oth map ev en ma y not b e a bijection. On the other hand, a homeomorphism ma y not b e smo oth b ecause in the denition, w e only require con tin uit y in b oth f and its in v erse but w e do not require dieren tiabilit y It is strongly desirable that

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7 the transformation is b oth smo oth and top ology preserving. What w e w an t is a die omorphism whic h is dened as follo ws. Denition 2. L et M 1 and M 2 b e dier entiable manifolds. A map f : M 1 M 2 is a die omorphism if f is a bije ction; f is dier entiable; the inverse f 1 is dier entiable. Because the concept of dieomorphism is essen tial to this w ork, w e w ould ha v e a little more discussion here in order to clarify some common misconceptions ab out this concept. It is ob vious that a dieomorphism is b oth a smo oth map and a homeomorphism from the denitions. Ho w ev er, what is not so ob vious is that b y requiring a dieomorphism, w e are asking for more than something that is b oth smo oth and homeomorphism. Let us lo ok at some of follo wing coun terexamples and w e will learn what factors ma y con tribute to mak e the transformation fail to b e a dieomorphism. First, a smo oth bijection is not necessarily a dieomorphism. Let M 1 R b e [0 ; 2 ) and M 2 = S 1 the unit circle. f : M 1 M 2 dened b y f ( ) = (cos( ) ; sin ( )) is smo oth and bijectiv e, but not a dieomorphism b ecause the in v erse f 1 is not con tin uous, and hence not dieren tiable. In fact, f is not a homeomorphism. Second, a smo oth homeomorphism is not necessarily a dieomorphism. Consider f : R R with f ( x ) = x 3 It is smo oth and it is a homeomorphism but the in v erse f 1 is not dieren tiable at x = 0. A related concept is lo c al die omorphism and it is dened as: Denition 3. A dier entiable map f : M 1 M 2 is a lo c al die omorphism if for e ach x 2 M 1 ther e exists a neighb orho o d U of x such that f j U : U f ( U ) is a die omorphism.

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8 F urthermore, it is useful that w e list without pro of some kno wn facts in dieren tial top ology [ 9 8 50 10 31 ] ab out dieomorphism and lo cal dieomorphism. Theorem 1. A map f : M 1 M 2 is a lo c al die omorphism if and only if its tangent map is an isomorphism. With some simple facts in linear algebra, the ab o v e theorem can b e rewritten as the follo wing. Theorem 2. A map f : M 1 M 2 is a lo c al die omorphism if and only if its Jac obian of the tangent map is nowher e e qual to zer o. Theorem 3. A map f : M 1 M 2 is a die omorphism if and only if it is a bije ction and a lo c al die omorphism. With the help of these theorems w e can visualize more situations when a smo oth map fails to b e dieomorphism. One situation is when the tangen t map fails to b e an isomorphism ev erywhere. Namely the Jacobian is zero at some p oin ts. In that case, the smo oth map ev en fails to b e a lo cal dieomorphism. The other situation is that the smo oth map is a lo cal dieomorphism but not a global dieomorphism. The follo wing example demonstrates this. Consider f : R 2 R 2 with f ( x; y ) = ( e x cos y ; e x sin y ). The Jacobian is J = @ f x @ x @ f x @ y @ f y @ x @ f y @ y = e x cos y e x sin y e x sin y e x cos y = e 2 x 6 = 0 8 ( x; y ) 2 R 2 : So f is a lo cal dieomorphism. Ho w ev er, f is not a dieomorphism as can b e seen in Figure. 1{1 Notice that b ecause the function is p erio dic in y in the co domain, the image of the function has innitely man y sheets o v erlaid. Another related example is the self in tersection of the immersion of the Klein b ottle in R 3 as sho wn in Figure. 1{2 No w let us lo ok at an example of another smo oth transformation, namely the thin-plate spline (TPS) [ 70 ].

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9 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -6 -4 -2 0 2 4 6 200 150 100 50 0 50 100 150 200 150 100 50 0 50 100 150 (a) (b) Figure 1{1: Grids in R 2 (a) b efore transformation and (b) after transformation Figure 1{2: Klein b ottle immersed in R 3 Image courtesy of John Sulliv an.(h ttp://torus.math.uiuc.edu/jms/Images/klein.h tml)

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10 (a) (b) Figure 1{3: (a) The template image and (b) The reference image 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 (a) (b) Figure 1{4: Space deformations (a) obtained with thin-plate spline in terp olation. The folding of space is illustrated b y the deformation of the grid lines. (b) The desired space transformation: a dieomorphism, whic h eliminates the space folding problem.

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11 Figure 1{3 a sho ws a template image. Figure 1{3 b is the reference image obtained b y w arping the template. Some landmarks are selected and sho wn in the images. Figure 1{4 a demonstrates the transformation of space b y sho wing the deformation of the rectangular grid. W e can see the folding of space. This is the dra wbac k of the thin-plate spline in terp olation. Due to the folding of space, features in the template ma y b e smeared in the o v erlapping regions. And furthermore, the transformation is not in v ertible. A dieomorphic transformation is strongly desirable, whic h preserv es the features, the top ology and whic h is smo oth as sho wn in Figure 1{4 b. W e still face the unkno wn corresp ondence problem, whic h is a dicult problem. There are t w o scenarios in whic h w e encoun ter the corresp ondence problem. The rst scenario is when w e do landmark based image registration. If the landmarks are selected b y hand, w e do kno w the corresp ondence. Ho w ev er, the pro cess of hand pic king landmarks is painstaking and it requires exp ert kno wledge ab out the image and the sub ject area and it ma y in v olv e h uman error. The complete automation of landmark selection is still not ac hiev ed at presen t but there is go o d progress to w ards the goal. If the landmarks are automatically selected, then the corresp ondence is unkno wn and the corresp ondence needs to b e automatically obtained. The second scenario is when w e ha v e a p oin t set represen tation of the shap es. Eac h shap e is represen ted b y a large p oin t set (p oin t cloud) and w e ha v e no kno wledge ab out the corresp ondence b et w een the t w o p oin t sets. What mak es the unkno wn corresp ondence problem more dicult is that it is ill-p osed. The corresp ondence b et w een t w o p oin t sets is a v ery in tuitiv e concept whic h ev ery one seems to understand. Ho w ev er, as is w ell kno wn within the medical image analysis comm unit y it is v ery dicult to dene corresp ondence precisely This creates a problem for v alidation since an irreducibly sub jectiv e factor seems to b e presen t in deciding what is a go o d corresp ondence.

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12 W e con tin ue with a short discussion aimed at reac hing a b etter understanding of p oin t corresp ondence. Supp ose w e ha v e t w o p oin t sets S 1 = f p i 2 n 1 j i = 1 ; 2 ; :::; n g and S 2 = f q i 2 n 2 j i = 1 ; 2 ; :::; n g where n 1 R d and n 2 R d usually with d = 2, or d = 3. When S 1 and S 2 are p oin t sets of equal cardinalit y with p oin ts randomly distributed in space, what is the corresp ondence b et w een S 1 and S 2 ? There are n bijections, or p erm utations b et w een the t w o sets. W e ha v e no a priori kno wledge allo wing us to judge that one corresp ondence is b etter than the other. Finite p oin t sets ha v e no extra structure o v er and b ey ond their discrete structure. So when w e talk ab out the corresp ondence b et w een p oin t sets, w e alw a ys imply that the p oin t set is the represen tation of some underlying shap e, whic h is a top ological space. And, as is fairly standard in the literature, w e can b ypass discussing p oin t corresp ondence b y fo cusing on the corresp ondence of the underlying top ological space, whic h is a homeomorphism, and most desirably a dieomorphism. Ev en so, the homeomorphism b et w een the t w o top ological spaces is not unique. Here w e mak e our rst assumption: the optimal corresp ondence b et w een the t w o p oin t sets is the one that induces the least deformation of space. Here w e understand the notion of space deformation in tuitiv ely but w e will carefully dene it later. It is easy to realize that this assumption is not alw a ys true b ecause w e are only giv en the p oin t set and w e ha v e no explicit kno wledge of the underlying shap e. This is easily illustrated with a simple example. In Figure 1{5 a and 1{5 b w e ha v e t w o p oin t sets. It is not clear at all what the corresp ondence is b et w een the t w o p oin t sets. Figure 1{5 c and 1{5 d put the t w o p oin t sets on top of t w o underlying images, the images of t w o w omen. Figure 1{5 c is the dra wing My Wife and My Mother-in-law published in 1915 b y the carto onist W. E. Hill. Figure 1{5 d is the image of He ad of a Woman in Pr ole (cropp ed) b y Jean-Pierre Da vid. If y ou in terpret Figure 1{5 c as the image of a y oung girl, namely the p oin t C as the

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13 50 100 150 200 50 100 150 200 250 A B C D E F G H 50 100 150 200 50 100 150 200 250 A B C D E F G H I J K L M (a) (b) A B C D E F G H A B C D E F G H I J K L M (c) (d) Figure 1{5: Am biguit y of landmark corresp ondence

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14 c hin and H as the ear, then w e ma y ha v e the corresp ondence b et w een the t w o p oin t sets from Figure 1{5 a to 1{5 b: A A, B B, C C, D D, E E, F F, G G and H H. F rom another p ersp ectiv e, if w e view the image as that of an old w oman, namely p oin t C as the tip of the nose, E the c hin and H the left ey e, w e ma y ha v e the corresp ondence: A A, B B, C I, D J, E C, F K, G L and H M. Let us lo ok at another example. Figure 1{6 a and Figure 1{6 b are t w o images of a blac k strip with the latter non-rigidly deformed. W e select four landmarks A1 through A4 for shap e A and four landmarks B1 through B4 for shap e B. Figure 1{6 c sho ws the landmarks for shap e A without the underlying ribb on shap e. Figure 1{6 d sho ws the landmarks for shap e B without the underlying ribb on shap e. Giv en the fact that w e can see the shap es, w e kno w the corresp ondences are A 1 B 1, A 2 B 2, A 3 B 3, A 4 B 4. Figure 1{7 a sho ws this landmark corresp ondence and Figure 1{7 b sho ws the deformation of space for this landmark corresp ondence. Ho w ev er, if w e only ha v e the t w o sets of p oin ts as sho wn in Figure 1{6 c and Figure 1{6 d, without the kno wledge of the underlying shap es, the ab o v e corresp ondence is not the one that w e will nd since it do es not giv e the least deformation of space. In fact, another corresp ondence, A 1 B 1, A 2 B 3, A 3 B 2, A 4 B 4 sho wn in Figure 1{7 d giv es a smaller deformation solution as sho wn in Figure 1{7 e. This is in fact a misin terpretation of the shap e, with Figure 1{7 c as the original shap e and Figure 1{7 f the misin terpreted shap e, due to the lac k of information regarding the underlying shap es. Consequen tly b y making the ab o v e assumption of least space deformation, w e are really assuming that the p oin ts are dense enough suc h that the underlying shap e is w ell represen ted b y the p oin ts. Adding more p oin ts can help resolv e this am biguit y and hence k eep the ab o v e assumption|that the correct corresp ondence giv es the smallest deformation of space|appro ximately v alid.

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15 A1 A2 A3 A4 B1 B2 B3 B4 (a) (b) 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 A1 A2 A3 A4 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 B1 B2 B3 B4 (c) (d) Figure 1{6: Tw o shap es of a ribb on and the asso ciated landmarks. (a) Shap e A, (b) Shap e B, (c) Landmarks of shap e A without the underlying shap e, (d) Landmarks of shap e B without the underlying shap e

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16 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 A1 B1 A4 B4 A2 B2 B3 A3 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 B1 B2 B3 B4 (a) (b) (c) 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 A1 B1 A4 B4 A2 B2 B3 A3 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 B1 B2 B3 B4 (d) (e) (f ) Figure 1{7: (a) The \correct" corresp ondence. (b) The space deformation according to the \correct" corresp ondence is not the smallest. (c) The \correctly" deformed shap e. (d) Another p ossible but \incorrect" corresp ondence. (e) The space deformation with this \incorrect" corresp ondence is the smallest. (f ) The misin terpretation of the shap e due to the \incorrect" corresp ondence. The rest of the dissertation is organized as follo ws. In Chapter 2 w e briery review the previous related w ork in the past ten y ears. In Chapter 3 w e describ e our theory of dieomorphic p oin t matc hing, dev elop an algorithm to solv e the problem and apply the algorithm to 2D and 3D medical imaging applications. Chapter 4 is an extension to the w ork in Chapter 3 and in it w e describ e a metho d of top ological clustering and matc hing. In Chapter 5 w e summarize our con tributions and p oin t out the directions of future w ork.

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CHAPTER 2 PREVIOUS W ORK In this c hapter, w e giv e a brief accoun t of the previous researc h that is related to our w ork. Most of the w ork w as dev elop ed in the past ten y ears but some w ork ma y b e traced bac k to the 1970s. W e start with thin-plate splines, whic h pla y an imp ortan t role in shap e matc hing, and all other splines w ork with the same principle. The Repro ducing Kernel Hilb ert Space form ulation giv es us a great insigh t on the nature of the en tire class of spline w arping problems. W e then discuss the folding problem with splines and v arious approac hes to o v ercome this, including imp osing constrain ts on the Jacobian and the ro w approac hes. W e also discuss the corresp ondence problem. In general there are t w o classes of approac hes to tac kle the corresp ondence problem. One is to confron t it directly while the other is to try to circum v en t it. With the second class of approac hes, p eople ha v e used implicit corresp ondence b y using a shap e distance that is the function of the t w o p oin t sets, instead of dep ending on the p oin t-to-p oin t corresp ondence, and there are also some metho ds that transform the p oin t matc hing problem to image matc hing problem through distance transform. With the rst class of approac hes, the direct approac h, there are metho ds that treat the corresp ondence and matc hing separately lik e softassign and there are metho ds that handles b oth at the same time lik e the join t clustering and matc hing (JCM) algorithms. Shap e con text metho d mak es use of extra lo cal con text information to help b etter resolv e the am biguous p oin t corresp ondence problem. Activ e shap e mo dels pla y as a bridge b et w een the landmark based metho ds and image in tensit y based metho ds, and also a bridge b et w een rigid matc hing and non-rigid matc hing. W e sp end a section discussing the deterministic annealing metho d for EM clustering, whic h is an 17

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18 eectiv e metho d to a v oid lo cal minima and w e adopt suc h an annealing approac h in our algorithm. Also closely related is the w ork of statistical shap e analysis on dieren tiable manifold and some distance measures from Fisher information theory 2.1 Thin Plate Splines (TPS) A spline is a long strip of w o o d or metal that is xed at a n um b er of p oin ts. This has long b een used as a drafting to ol to dra w smo oth curv es that are required to pass certain p oin ts (called \duc ks," or \dogs," or \rats"). These w o o den strips are not only used in drafting, but also used in constructions. The W righ t brothers used one to shap e their wings. In the old da ys splines w ere used in shipbuilding. They are also used for b ending the w o o d for m usical instrumen ts lik e pianos, violins, violas, etc. In the mo dern da ys, splines are used to mo del the b o dy of automobiles. In all those uses, it is actually the ph ysical realization of some smo oth curv es. In ph ysics, the shap e of the w o o d strip has to tak e the form of suc h a curv e that the b ending energy of the strip is minimized. Sheo en b erg [ 63 62 ] is credited to b e the rst to study spline functions started with one dimensional problems, the cubic splines. The thin plate spline is the natural generalization of the cubic spline. In the drafting practice, the w o o den strip is long but v ery thin and narro w, so that it is abstracted as a one dimensional curv e. In the thin plate case, w e ha v e a plate whic h is v ery thin, so the problem is t w o dimensional. In b oth cases, the same ph ysical b ending energy of the strip or the thin plate in minimized. The problem of thin plate spline in terp olation is stated as follo ws. Giv en f p i 2 R 2 j i = 1 ; 2 ; :::; n g and f z i 2 R j i = 1 ; 2 ; :::; n g w e w an t to nd a function f : R 2 R 2 suc h that it in terp olates the p oin ts with f ( p i ) = z i i = 1 ; 2 ; :::n and minimizes the thin plate b ending energy E ( f ) = Z Z R 2 ( @ 2 f @ x 2 ) 2 + 2( @ 2 f @ x@ y ) 2 + ( @ 2 f @ y 2 ) 2 dxdy : (2.1)

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19 If w e in terpret the f z i 2 R j i = 1 ; 2 ; :::; n g as the displacemen ts of the thin plate in the z direction at p oin ts f p i 2 R 2 j i = 1 ; 2 ; :::; n g in the x y plane, w e can easily see the ph ysical in tuition of this in terp olation problem, as sho wn in Figure. 2{1 Sometimes it is desirable that w e sacrice the exact in terp olation. W e only seek an appro ximation of the in terp olation as a trade o of less deformation of the thin plate. So the thin plate smo othing problem is dened as giv en f p i 2 R 2 j i = 1 ; 2 ; :::; n g and f z i 2 R j i = 1 ; 2 ; :::; n g w e w an t to nd a function f : R 2 R 2 suc h that it minimizes the cost function 1 n n X i =1 ( z i f ( p i )) 2 + E ( f ) : (2.2) 2.2 Bo okstein's Application to 2D Landmark W arping Bo okstein [ 4 5 ] applied thin plate splines to the landmark in terp olation problem. F or simplicit y w e discuss the problem in 2D space. Ev erything in the 2D form ulation easily applies to 3D except w e ha v e a dieren t k ernel for 3D. The goal is to nd a smo oth transformation f : n 1 n 2 that in terp olates n pairs of landmarks f p i 2 n 1 j i = 1 ; 2 ; :::; n g and f q i 2 n 2 j i = 1 ; 2 ; :::; n g It is ob vious that suc h a smo oth transformation is not unique. W e will giv e it further constrain ts. One c hoice is to require the to transformation minimize some functional, in this case E = 2 X h =1 Z Z R 2 ( @ 2 f h @ x 2 ) 2 + 2( @ 2 f h @ x@ y ) 2 + ( @ 2 f h @ y 2 ) 2 dxdy ; (2.3) where f 1 and f 2 are the x and y comp onen ts of the mapping, f ( x; y ) = ( f 1 ( x; y ) ; f 2 ( x; y )) : (2.4) If w e in terpret eac h of f 1 and f 2 as the b ending in the z direction of a metal sheet, or thin plate, extending in the x y plane, the energy in ( 2.3 ) is the analogy of the

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20 thin plate b ending energy The problem can b e solv ed using a standard Green's function metho d. The Green's function for this problem is the function satisfying the equation 2 U ( x ; x 0 ) = ( @ 2 @ x 2 + @ 2 @ y 2 )( @ 2 @ x 2 + @ 2 @ y 2 ) U = ( x ; x 0 ) (2.5) where ( x ; x 0 ) is the Dirac delta function and x = ( x; y ) and x 0 = ( x 0 ; y 0 ) 2 R 2 This can b e solv ed with the solution U ( r ) = r 2 l og r 2 ; (2.6) where r is the distance p x 2 + y 2 Because an ane transformation has a zero con tribution to the b ending energy the transformation allo ws a free ane transformation. Dene the matrices K = 266666664 0 U ( r 12 ) ::: U ( r 1 n ) U ( r 21 ) 0 ::: U ( r 2 n ) ::: ::: ::: ::: U ( r n 1 ) U ( r n 2 ) ::: 0 377777775 ; whic h is n n ; (2.7) P = 266666664 1 x 1 y 1 1 x 2 y 2 ::: ::: ::: 1 x n y n 377777775 ; whic h is 3 n ; (2.8) and L = 264 K P P T O 375 ; whic h is ( n + 3) ( n + 3) ; (2.9) where the sym b ol T is the matrix transp ose op erator and O is a 3 3 matrix of zeros.

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21 Let V = ( v 1 ; :::; v n ) b e an y n -v ector and write Y = ( V j 0 0 0) T Dene the v ector W = ( w 1 ; :::; w n ) and the co ecien ts ( a 1 ; a x ; a y ) b y the equation L 1 Y = ( W j a 1 a x a y ) T : (2.10) Finally w e ha v e the solution f ( x; y ) = a 1 + a x x + a y y + n X i =1 w i U ( j P i ( x; y ) j ) : (2.11) 2.3 More on Splines Bo okstein's seminal w ork applying thin-plate splines to the landmark problems in 2D [ 4 5 ] is w ell kno wn in the computer vision and imaging comm unit y What is less kno wn is the theoretical foundations for the thin-plate spline laid b y Duc hon [ 21 22 23 ] and Meinguet [ 48 ] in the 1970s. This solution is correct and simple. Ho w ev er it is not n umerically stable b ecause it in v olv es the in v erse of the large k ernel matrices. A n umerically stable solution is giv en b y W ah ba [ 70 ] using the QR decomp osition. W ah ba has an accoun t for a more generalized form ulation for d -dimensional space and the energy in v olving m th order partial deriv ativ es. Here w e only discuss the sp ecial thin plate spline with d = 2 and m = 2. Later in our w ork in 3D situations, 3D thin plate splines are used where d = 3 but it is straigh tforw ard to generalize from 2D to 3D, while the k ernel in 3D is dieren t than that for 2D. Let f ; g 2 W k ; 2 (n), where W k ; 2 (n) is the Sob olev space. The inner pro duct of f and g is dened as A thin-plate smo othing problem in 2D is to nd f 2 W k ; 2 (n) to minimize the functional 1 n n X i =1 jj q i f ( p i ) jj 2 + E ( f ) : (2.12)

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22 The n ull space is spanned b y 1 ( t ) = 1 ; 2 ( t ) = x; 3 ( t ) = y ; (2.13) whic h is an ane space. Duc hon [ 23 ] pro v ed that if f p i 2 n 1 j i = 1 ; 2 ; :::; n g are suc h that least squares regression on 1 ; 2 ; 3 is unique, then 2.12 has a unique minimizer ^ f with the represen tation ^ f ( t ) = 3 X =1 d ( t ) + n X i =1 c i K ( t; p i ) ; (2.14) where K is a Green's function for the double iterated Laplacian. K has the form K = K ( j s t j ) ; (2.15) and K ( ) = 2 log j j : (2.16) By using in tegration b y part, w e obtain that c; d are the minimizers of 1 n jj y T d K c jj 2 + c 0 K c (2.17) sub ject to T 0 c = 0, where K is a n n matrix with ij th en tries K ( p i ; p j ) : Let us do a QR decomp osition on T T = ( Q 1 : Q 2 ) 0B@ R 0 1CA (2.18) where ( Q 1 ; Q 2 ) is orthogonal and R is lo w er triangular. Q 1 is n 3 and Q 2 is n ( n 3). Since T 0 c = 0, c m ust b e in the column space of Q 2 with c = Q 2 r for some n 3 v ector r Substituting in 2.17 and the energy is in the form of

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23 1 n jj Q 02 y Q 02 K Q 2 r jj 2 + 1 n jj Q 01 y R d Q 01 K Q 2 r jj 2 + r 0 Q 02 K Q 2 r : (2.19) The solutions are d = R 1 ( y K Q 2 r ) c = Q 2 r ; (2.20) with r = ( Q 02 K Q 2 + nI ) 1 Q 02 y (2.21) where I is the iden tit y matrix. 2.4 Repro ducing Kernel Hilb ert Space (RKHS) F orm ulation It is p ossible to come to the same result for the minimizer using repro ducing k ernels and from this p oin t of view it is easier to understand the existence and uniqueness of the solution [ 70 7 ]. Let jj f jj 2 = E = R n ( f 2 xx + 2 f 2 xy + f 2 y y ) dxdy where jj f jj is the norm of f in W k ; 2 (n). Since W k ; 2 (n) is a Hilb ert space, from Riesz represen tation theorem, for an y x 2 n, the ev aluation linear functional x : W k ; 2 (n) R ; x ( f ) = f ( x ) (2.22) has a represen ter u x 2 W k ; 2 (n) suc h that f ( x ) = < u x ; f > : (2.23) No w the problems is transformed to a new problem: nd a function f 2 W k ; 2 (n) with minimal norm jj f jj sub ject to constrain ts

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24 < u X i ; f > = v i ; i = 1 ; 2 ; :::; n : (2.24) F or X a ; X b 2 n, u ( X a ; X b ) = u X a ( X b ) is the k ernel of the repro ducing k ernel Hilb ert space. Let T b e the linear subspace spanned b y u x i ; i = 1 ; 2 ; :::; n An y function f 2 W k ; 2 (n) can b e decomp osed in to f = f T + f ? where f T 2 T and f ? is in the orthogonal complemen t of T and hence < u X i ; f ? > = 0. W e kno w if f T satises ( 2.24 ), then f also satises ( 2.24 ) only with jj f jj > jj f T jj if f ? 6 = 0. So w e only need to searc h for the solution in T The general solution th us can b e written as f ( X ) = a 0 + a 1 x + a 2 y + n X i =1 w i u ( X i ; X ) ; where functions of the form a 0 + a 1 x + a 2 y span the n ull space. With this form, E can b e rewritten as n X i =1 ;j =1 w i U ij w j = W U W + : (2.25) 2.5 The F olding Problem of Splines This is all v ery nice except as men tioned ab o v e there is no mec hanism to guaran tee a bijection in order to b e homeomorphic or dieomorphic. In tuitiv ely this problem is kno wn as the folding of space. Figure 2{2 sho ws the displacemen t of landmarks. Figure 2{3 is the thin-plate spline in terp olation. W e can see the folding of space. This is the dra wbac k of thinplate spline in terp olation. Due to the folding of space, features in the template ma y b e smeared in the o v erlapping regions. And furthermore, the transformation is not in v ertible. A dieomorphic transformation is strongly desirable, since it preserv es the features, the top ology and is smo oth as sho wn in Figure 2{4

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25 2.6 Imp osing Restriction on the Jacobian One straigh tforw ard approac h to nd a dieomorphism for practical use is to mak e some remedy on the thin-plate spline. W e can restrict our searc h space to the set of dieomorphisms and the ideal one should minimize the thin-plate energy W e mak e the observ ation that if the Jacobian of the transformation f c hanges sign at a p oin t, then there is folding. W e put the constrain t requiring the Jacobian is alw a ys p ositiv e. There is some literature on this approac h but most of these approac hes do es not guaran tee that the transformation is smo oth ev erywhere [ 38 11 ]. 2.7 The Flo w Approac h Another approac h is to utilize a ro w eld[ 28 39 49 ]. W e in tro duce one parameter, the time t in to the dieomorphism. Let t : n n b e the dieomorphism from n to n at time t A p oin t x is mapp ed to the p oin t t ( x ). Sometimes w e also denote this as ( x; t ). It is easy to v erify that for all the v alues of t t form a one parameter dieomorphism group. If x is xed, then ( x; t ) traces a smo oth tra jectory in n. The in terp olation problem b ecomes: nd the one parameter dieomorphic group ( ; t ) : n n suc h that giv en p i 2 n and q i 2 n 8 i = 1 ; 2 ; :::; n ( x; 0) = x and ( p i ; 1) = q i W e in tro duce the v elo cit y eld v ( x; t ) and construct a dynamical system using the transp ort equation @ ( x; t ) @ t = v ( ( x; t ) ; t ) : (2.26) The in tegral form of the relation b et w een ( x; t ) and v ( x; t ) is ( x; 1) = x + Z 1 0 v ( ( x; t ) ; t ) dt : (2.27) Ob viously suc h a ( x; t ) is not unique and there are innitely man y suc h solutions. With the analogy to the thin-plate spline, it is natural that w e require the desirable dieomorphism results in minimal space deformation. Namely w e require the deformation energy

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26 Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt (2.28) to b e minimized, where L is a giv en linear dieren tial op erator. The follo wing theorem [ 39 ] states the existence of suc h a v elo cit y eld and sho ws a w a y to solv e for it. Theorem 4. L et p i 2 n and q i 2 n 8 i = 1 ; 2 ; :::; n The solution to the ener gy minimization pr oblem ^ v ( ) = arg min Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt (2.29) subje ct to ( p i ; 1) = q i ; 8 i = 1 ; 2 ; :::; n (2.30) wher e ( x; 1) = x + Z 1 0 v ( ( x; t ) ; t ) dt (2.31) exists and denes a die omorphism ( ; 1) : n n The optimum velo city eld ^ v and the die omorphism ^ ar e given by ^ v ( x; t ) = n X i =1 K ( ( x i t ) ; x ) n X j =1 ( K ( ( t )) 1 ) ij ^ ( x j ; t ) (2.32) wher e K ( ( t )) = 0BBBBBBBBBB@ K ( ( p 1 ; t ) ; ( p 1 ; t )) K ( ( p 1 ; t ) ; ( p n ; t )) K ( ( p n ; t ) ; ( p 1 ; t )) K ( ( p n ; t ) ; ( p n ; t )) 1CCCCCCCCCCA (2.33)

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27 with ( K (( ( t )) ij denoting the ij 3 3 blo ck entry ( K ( ( t )) ij = K ( ( p i ; t ) ; ( p j ; t )) and ^ ( p n ; ) = arg min ( p n ; ) Z 1 0 X ij ( p i ; t ) T ( K ( ( t )) 1 ) ij ( p j ; t ) dt (2.34) subje ct to ( p i ; 1) = q i ; i = 1 ; 2 ; :::; N with the optimal die omorphism given by ^ ( x; 1) = x + Z 1 0 ^ v ( ^ ( x; t ) ; t ) dt : (2.35) The pro of [ 39 ] is omitted here. With this theorem, w e can con v ert the original optimization problem lo oking for the v ector eld ^ v ( x; t ) to a problem of nite dimensional optimal con trol with end p oin t conditions. This problem is called the exact matching pr oblem b ecause w e required the images of the giv en p oin ts p i ; i = 1 ; 2 ; :::; n are exactly another set of giv en p oin ts q i ; i = 1 ; 2 ; :::; n The exact matc hing problem is symmetric with resp ect to t w o sets of landmarks or t w o p oin t shap es. Sw apping the t w o p oin t sets f p i 2 n 1 j i = 1 ; 2 ; :::; n g and f q i 2 n 2 j i = 1 ; 2 ; :::; n g results in the new optimal dieomorphism to b e the in v erse of the old dieomorphism. The exact matc hing problem can b e generalized to the inexact matching pr oblem. In the inexact matc hing problem, w e do not require the p oin ts exactly matc h. Instead, w e seek a compromise b et w een the closeness of the matc hing p oin ts and the deformation of space. W e minimize Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt + n X i =1 jj q i ( p i ; 1) jj 2 ; (2.36) whic h can b e similarly solv ed. As seen from the form ulation of the problem, there ma y b e innitely man y dieomorphisms that in terp olate the t w o sets of landmarks. A usual w a y to nd a particular desirable dieomorphism is to require the dieomorphism to minimize a

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28 certain ob jectiv e function. Camion and Y ounes [ 7 ] prop osed a dieren t ob jectiv e function in the form of E ( v ; q ) = Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt + n X i =1 jj dq i ( t ) dt v ( q i ( t ) ; t ) jj 2 dt (2.37) o v er all time dep enden t v elo cities v ( x; t ) on n and o v er all tra jectories q 1 ( t ) ; :::; q n ( t ). This can b e in terpreted as a geo desic distance b et w een t w o p oin ts on the conguration manifold. This is generalized to exact matc hing b y Marsland et al. [ 46 ]. 2.8 Corresp ondence and Softassign So far all w ork has assumed landmark matc hing with a kno wn corresp ondence. When the corresp ondence is unkno wn, the problem is dramatically complicated. One approac h is the softassign [ 54 43 ] metho d, whic h solv es for the corresp ondence as a p erm utation problem using linear programming in the space of doubly sto c hastic matrices [ 65 3 ]. The problem of nding the corresp ondence b et w een t w o sets of p oin ts can b e form ulated as nding the p erm utation matrix. This approac h is exp ensiv e in computational time. Ch ui et al. [ 12 17 14 13 15 ] adopted the join t clustering sc heme, in whic h the corresp ondence and space deformation is estimated sim ultaneously Ho w ev er, in all the w ork, splines are used and a p oten tial dra wbac k is that a dieomorphic mapping in space is not guaran teed. 2.9 Distance T ransforms An un usual w a y of getting around the p oin t corresp ondence problem is to use distance transforms to con v ert the p oin t matc hing problem to an image matc hing problem. The distance transform w as rst in tro duced b y Rosenfeld and Pfaltz [ 60 ], and it has a wide range of applications in image pro cessing, rob otics, pattern recognition and pattern matc hing. P aragios et al. [ 53 ] giv e one example of using distance transforms to establish lo cal corresp ondences for compact represen tations

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29 of anatomical structures. Distance transform applies to binary images, as w ell as p oin t sets, whic h can b e though t as a sp ecial case of a binary image. Supp ose w e ha v e a domain n, and a p oin t set S n. F or eac h p oin t x 2 n, w e assign to it a non-negativ e real n um b er, whic h is the shortest distance from x to all the p oin ts in S This w a y w e obtain a scalar eld in domain n. W e treat this scalar eld as a gra y scale image, w e call it the distance transformed image of the p oin t set. If w e ha v e t w o p oin t sets, w e can rst p erform the distance transform on the t w o sets resp ectiv ely and later register the t w o distance transformed images. Ho w ev er, there are problems with this approac h. If the matc hing of the t w o distance transformed images is in tensit y based, then the original p oin ts ma y not b e matc hed exactly Ev en in the inexact matc hing case, the optimization for the distance transformed images is o v er the en tire image region and that ma y not b e optimal for the original p oin t set. If the matc hing of the t w o distance transformed images is lev el set based, the lev el sets in the t w o images ma y not b e top ologically equiv alen t, as sho wn in Figure 2{5 and Figure 2{6 In Figure 2{5 there are t w o distance transformed images, with three p oin ts eac h in the image. Figure 2{6 sho ws the lev el sets of the t w o distance transformed images. Hence this metho d cannot guaran tee to obtain a dieomorphism. Due to the indirect approac h of transforming p oin t sets in to distance transforms, this metho d has not seen wide applicabilit y for p oin t sets. 2.10 Implicit Corresp ondence It is p ossible to dene distance measures b et w een t w o shap es, with the shap es view ed as p oin t sets, without the kno wledge of corresp ondence. Hausdor distance b et w een t w o p oin t sets is a w ell-kno wn distance measure for suc h purp oses. Huttenlo c her et al. [ 37 ] dev elop metho ds comparing shap es using Hausdor distance. Glaunes et al. [ 27 ] use another metho d to circum v en t the corresp ondence problem b y in tro ducing a distance measure of the t w o shap es using an external

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30 Hilb ert space structure. If the t w o shap es are p oin t sets, the distance measure b et w een t w o p oin t sets used in Glaunes et al. [ 27 ] is 1 2 R j v1 v j 2I (2.38) where j j 2I is the norm squared in some Hilb ert space I and j j 2I = N 1 X i =1 N 2 X j =1 c i c j k I ( x i ; y j ) (2.39) where c i ; c j are constan t co ecien t. x i are the p oin ts in the rst sets and y j are the p oin ts in the second sets. k I is some k ernel. In their exp erimen ts, they used radial basis function k ernels k I ( x; y ) = f I ( j x y j 2 2 I ) ; (2.40) with f I ( u ) = e u and f I ( u ) = 1 1+ u The idea of distance measure without explicit corresp ondence actually go es bac k to Grimson et al. [ 29 ] and Lu and Mjolsness [ 45 ] in 1994. The distance measure used is X i X j e jT l i m j j 2 2 2 : (2.41) As p oin ted out in Rangara jan et al. [ 55 ] this is equiv alen t to using E ( M ; T ) = X ij M ij D ij ( T ) + X ij M ij log M ij ; (2.42) where M ij is the explicit corresp ondence and T is the space transformation and D ij is the distance measure b et w een p oin t pairs that are in corresp ondence. Guo et al. suggested a join t clustering algorithm for solving 2D dieomorphic p oin t matc hing problems, with unkno wn corresp ondence, using explicit corresp ondences [ 33 32 ]. The dieren t form ulations with explicit corresp ondence and implicit corresp ondence

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31 are closely related to eac h other, through a Legendre transform, as p oin ted out b y Mjolsness and Garrett [ 51 ]. 2.11 Shap e Con text In the approac h of shap e con text [ 2 ], the corresp ondence is expressed explicitly and the corresp ondence problems is tac kled directly Moreo v er, the corresp ondence is solv ed separately from the space transformation with the help of shap e c ontext. Shap e con text is the lo cal shap e information at eac h p oin t. F or eac h p oin t q and all other p oin ts p i w e can dra w a v ector from q to p i that is p i q The lo cal shap e con text information is all stored in this set of n 1 v ectors. This information is ric h and in practice the distribution of these n 1 v ectors giv es us more robust, compact and discriminativ e descriptor. F or eac h p oin t p i w e compute a histogram h i of these n 1 v ectors h i ( k ) = # f q 6 = p i j ( q p i ) 2 bin( k ) g : (2.43) The histogram is dened to b e the shap e con text of p oin t p i Bins that are uniform in log-p olar space are used to mak e the nearb y con text p oin ts more imp ortan t the the far a w a y con text p oin ts. The cost for the corresp ondence b et w een t w o shap es is the sum of the cost of corresp onding p oin t pairs H ( ) = X i C ( p i ; q ( i ) ) ; (2.44) whic h is a function of the p erm utation and the cost with eac h individual pair of p oin ts is dened as C ij = C ( p i ; q j ) = 1 2 K X k =1 [ h i ( k ) h j ( k )] 2 h i ( k ) + h j ( k ) : (2.45) 2.12 Activ e Shap e Mo dels Co otes et al. prop ose a metho d, whic h they call Activ e Shap e Mo dels [ 18 ] to lo cate ob jects in images, with the help of mo del shap es and the training of

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32 these mo del shap es. This approac h pla ys as a bridge b et w een the landmark based metho ds and image in tensit y based metho ds, and also a bridge b et w een rigid matc hing and non-rigid matc hing. They use landmark p oin ts on the mo dels but not on the test images. The landmarks are hand pic k ed and hand lab eled. The automatic corresp ondence problem is circum v en ted b y the h uman in v olv emen t of the landmark pic king and lab eling pro cess. The deformation mo del of the template is similarit y whic h is v ery close to rigid, allo wing translation, rotation plus a scaling. What mak es it applicable to non-rigid deformation is that they learn the statistics of the mo del shap es through a set of training samples and nd the mean and v ariance in higher dimensional space and apply the PCA analysis with the v ariances. When it applies to lo cating the mo del in the image, they use the snak e mo del, whic h is non-rigid in nature, with the help and restriction of the statistics of the mo del shap es. 2.13 Deterministic Annealing Applied to EM Clustering Deterministic annealing is an eectiv e tec hnique used in the clustering problems. The clustering problem is a non-con v ex optimization problem. The traditional clustering tec hniques use descen t based algorithms and they tend to get trapp ed in a lo cal minim um. Rose et al. [ 59 ] prop osed an annealing approac h using analogies to statistical ph ysics. The clustering problem is to partition a set of data p oin ts, f x i 2 R d j i = 1 ; 2 ; :::; n g in to K clusters C 1 ; C 2 ; :::; C K with the cen ters r 1 ; r 2 ; :::; r K resp ectiv ely In the fuzzy clustering literature, w e call the probabilit y of eac h p oin t p i b elonging to eac h cluster C j the fuzzy mem b ership. Hard clustering is a marginal sp ecial case, where eac h p oin t is deterministically asso ciated with a single cluster. Let E j ( x i ) denote the energy asso ciated with assigning a data p oin t x i to cluster C j

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33 The a v erage total energy is < E > = X i X j P ( x i 2 C j ) E j ( x i ) : (2.46) Since w e do not mak e an y assumption ab out the data distribution, w e apply the principle of maxim um en trop y It is w ell kno wn from statistical ph ysics, the asso ciation probabilities P ( x i 2 C j ) that maximize the en trop y under constrain t ( 2.46 ) are Gibbs canonical distributions, P ( x i 2 C j ) = 1 Z i e E j ( x i ) ; (2.47) where Z i is the partition function Z i = X k e E k ( x i ) : (2.48) The parameter is the Lagrange m ultiplier determined b y the giv en v alue of < E > in ( 2.46 ). In the analogy in statistical ph ysics, is in v ersely prop ortional to the temp erature T W e ha v e assumed that w e ha v e a xed set of clusters. W e w an t to extend this to include optimization o v er the n um b er of clusters as w ell. The optimal solution then is the one that minimizes the free energy F = 1 ln Z : (2.49) The set of cluster cen ters are the ones that satisfy @ F @ r j = 0 ; 8 j : (2.50) And the solution is r j = P i x i P ( x i 2 C j ) P i P ( x i 2 C j ) : (2.51) This pro cedure determines a set of cluster cen ters f r j 2 R d j j = 1 ; 2 ; :::; n g for eac h xed Generally c hanging K the imp osed n um b er of clusters, will mo dify the p ositions of the set of cluster cen ters. Ho w ev er, there exists some n c suc h that

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34 for all K > n c one gets only n c distinct cluster cen ters while the remaining K n c cluster cen ters are rep etitions from this set. Th us at eac h giv en w e get at most n c clusters. Here w e assume K n c at a giv en and w e only consider them without rep etitions. The free energy F and are Legendre transform images of eac h other. Fixing one of them determines the other. F or = 0, eac h data p oin t is uniformly asso ciated with all clusters and all the cen ters ha v e the same lo cation, the cen troid of the data. Clearly for = 0 w e ha v e a single minim um, whic h is the global minim um, for F and the en tire data set in in terpreted as one cluster. A t higher the free energy ma y ha v e man y lo cal minima, and the concept of annealing emerges here can b e view ed as trac king the global minim um while gradually increasing Moreo v er, at = 0 there is only one cluster ( n c = 1), but at some p ositiv e w e shall ha v e n c > 1. In other w ords, this cluster will split in to smaller clusters, and will th us undergo a phase tr ansition The rst phase transition o ccurs at a critical v alue for c = 1 2 max ; (2.52) where max is the largest eigen v alue of C xx the co v ariance matrix of the data set. Giv en = 1 =T and for Gaussian mixture clustering T = 2 2 T it is quite understandable that the critical v alue c for the prescrib ed T is c = p max 2.14 Statistical Shap e Analysis on Dieren tiable Manifolds Da vid Kendall rst in tro duced the idea of represen ting shap es in complex projectiv e spaces. The idea is dev elop ed to represen t shap es on general dieren tiable manifolds, or shap e spaces [ 66 ]. A shap e is constructed from a sequence of landmarks p 1 ; p 2 ; :::p n with p i 2 R 2 Namely eac h landmark is a p oin t in 2-dimensional Euclidean space. The t w o sequences of landmarks in R 2 are considered of the same shap e if they dier only b y a similarit y transformation in R 2 A shap e then is

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35 dened as an equiv alen t class of these landmark sequences and the shap e space is carv ed out of the quotien t space dened b y the equiv alen t relation. One w a y to distill the shap e information out of the landmark sequence is to remo v e the lo cation, scale and orien tation. T o remo v e the lo cation, w e can dene r i = x i x ; (2.53) where x = 1 n n X j =1 x j (2.54) is the mean or cen troid of the landmarks. That is, to remo v e the lo cation, w e mak e the landmarks mean zero. T o remo v e the scale w e can mak e the v ariance of the landmarks as one. So w e dene i = x i x q P nj =1 jj x j x jj 2 : (2.55) W e refer the v ector as the pr e-shap e of the landmarks. The pre-shap e space is the in tersection of the ( n 2)-dimensional subspace F 2 n 2 = f ( x 1 ; :::; x n ) 2 R 2 n j n X j =1 x j = 0 g (2.56) with the unit sphere S 2 n 1 = f ( x 1 ; :::; x n ) 2 R 2 n j n X j =1 jj x j jj 2 = 1 g : (2.57) The in tersection S 2 n 3 = F 2 n 2 \ S 2 n 1 (2.58) is a (2 n 3)-dimensional sphere within the am bien t Euclidean space R 2 n It is more dicult to remo v e the orien tation of the shap e. T o do this, w e dene the orbit of a pre-shap e 2 S 2 n 3 is the circle O ( ) = f ( ) j 0 < 2 g S 2 n 3 : (2.59)

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36 Tw o pre-shap es are of the same shap e if they are on the same orbit. A shap e is dened as the equiv alen t class of the pre-shap es. If 1 and 2 are t w o preshap es, then the great circle distance b et w een 1 and 2 is giv en b y d ( 1 ; 2 ) = cos 1 ( < 1 ; 2 > ) : (2.60) The induced metric on n2 is then dened as d [ O ( 1 ) ; O ( 2 )] = inf f d ([ 1 ( 1 ) ; 2 ( 2 )] j 0 1 ; 2 < 2 g : (2.61) This is called Pr o cruste an distanc e. It can b e pro v ed that if 1 and 2 are t w o represen tativ es of t w o shap es, the pro crustean distance b et w een the t w o shap es can b e expressed as d ( 1 ; 2 ) = cos 1 ( j n X k =1 1 k 2 k j ) : (2.62) The standard statistics lik e means and v ariances can b e p erformed on the shap e manifold. One dra wbac k of this approac h is that the shap e is treated as the se quenc e of landmarks instead of sets. That means if w e t w o iden tical sets of landmarks and only lab el them dieren tly this theory treats them as t w o distinct shap e. It do es not consider to mak e an equiv alen t class from the p erm utation of landmark p oin ts. 2.15 Distance Measures from Information Theory Information geometry is an emerging discipline that studies the probabilit y and information b y w a y of dieren tial geometry In information geometry every probabilit y distribution is a p oin t in some space. A family of distributions corresp onds to p oin ts on a dieren tiable manifold. Endres et al. [ 25 ] prop osed a metric for t w o distributions. Giv en t w o probabilit y distributions P and Q and R = 1 2 ( P + Q ), w e can dene a distance D P Q

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37 b et w een the t w o distributions b y D 2 P Q = 2 H ( R ) H ( P ) H ( Q ) = D ( P jj R ) + D ( Q jj R ) = N X i =1 ( p i log 2 p i p i + q i + q i log 2 p i p i + q i ) : (2.63) This metric can also b e in terpreted as the square ro ot of an en trop y appro ximation to the logarithm of an evidence ratio when testing if t w o samples ha v e b een dra wn from the same underlying distribution. 1 2 D 2 P Q is named Jensen-Shannon div ergence, whic h is dened as D ( P ; Q ) = D ( P jj R ) + (1 ) D ( Q jj R ) (2.64) R = P + (1 ) Q and therefore 1 2 D 2 P Q = D 1 2 ( P ; Q ) : (2.65) With probabilit y distribution P ( x j ) where is a set of parameters 1 ; :::; n the Fisher information is dened as [ 26 ] G ij ( ) = E @ 2 log p ( x j ) @ i @ j = Z p ( x j ) @ 2 log p ( x j ) @ i @ j dx : (2.66) C. R. Rao [ 57 ] suggested this is a metric. In fact, it is the only suitable metric in parametric statistics and it is called Fisher-Rao metric.

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38 5 10 15 20 5 10 15 20 0.2 0 0.2 0.4 Figure 2{1: Deformation of the thin plate 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 90 100 Figure 2{2: Landmark displacemen ts

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39 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 160 Figure 2{3: Thin-plate Spline in terp olation 20 0 20 40 60 80 100 120 140 20 0 20 40 60 80 100 120 140 160 Figure 2{4: Dieomorphic in terp olation

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40 0 100 200 300 400 500 600 100 200 300 400 500 600 (a) 0 100 200 300 400 500 600 100 200 300 400 500 600 (b) Figure 2{5: Tw o distance transformed images of three landmarks

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41 0 100 200 300 400 500 600 100 200 300 400 500 600 (a) 0 100 200 300 400 500 600 100 200 300 400 500 600 (b) Figure 2{6: Lev el sets of t w o distance transformed images

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CHAPTER 3 DIFFEOMORPHIC POINT MA TCHING In this c hapter w e in v estigate the dieomorphic p oin t matc hing theory and apply the theory to shap es in medical imaging. In Section 3.1 w e pro v e a theorem ab out the existence of a dieomorphic mapping matc hing the landmarks. In Section 3.2 w e pro v e another theorem ab out the symmetric nature in the case of exact landmark matc hing. In Section 3.3 w e form ulate a theory of dieomorphic p oin t matc hing with unkno wn corresp ondence and devise an ob jectiv e function. In Section 3.4 w e design an algorithm to solv e the problem using join t clustering and deterministic annealing. In Section 3.5 w e apply the algorithm to 2D corpus callosum shap es. In Section 3.6 p erform matc hing on 3D hipp o campus shap es. Both corpus callosum and hipp o campus are parts in the h uman brain and the matc hing of these shap e ha v e great signicance in medical treatmen t and medical researc h. 3.1 Existence of a Dieomorphic Mapping in Landmark Matc hing W e ha v e discussed in Chapter 1 that in 2D a similarit y transformation can map exactly 2 giv en p oin ts to 2 giv en p oin ts. A ane transformation can map exactly 3 giv en p oin ts to 3 giv en p oin ts. A pro jectiv e transformation can map exactly 4 giv en p oin ts to 4 giv en p oin ts. No w giv en n arbitrary distinct p oin ts f p i 2 R 2 j i = 1 ; 2 ; :::n g and another set of n arbitrary distinct p oin ts f q i 2 R 2 j i = 1 ; 2 ; :::n g w e w an t to nd a dieomorphism f : R 2 R 2 suc h that f ( p i ) = q i It is natural to ask the question, do es suc h a dieomorphic mapping alw a ys exist? Our in tuition is it exist and there are innitely man y suc h dieomorphisms. This is stated as our rst theorem and the pro of follo ws. 42

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43 Theorem 5. A die omorphic tr ansformation that interp olates arbitr ary numb er of n p airs of landmarks always exists. Pr o of. W e sho w the existence b y construction. W e construct a simple, although most lik ely undesirable in most of the applications, dieomorphism. The in tuitiv e idea is to \dig canals" connecting the landmark pairs. W e rst c ho ose the rst pair of landmarks p 1 and q 1 Assume no other landmarks lie on the line connecting p 1 and q 1 Establish a co ordinate system suc h that p 1 and q 1 are on the x axis. Let the signed distance from p 1 to q 1 b e d Construct the transformation f 1 : n 1 n 2 suc h that f 1 ( x; y ) = ( x 0 ; y 0 ), x 0 = x + de v 2 (3.1) y 0 = y where v = tan ( 2 y ), for an y arbitrarily small It is easy to sho w that f 1 is a dieomorphism and it maps p 1 to q 1 and k eeps all other landmarks q 2 ,..., q n xed. This is v ery m uc h lik e the ro w of viscous ruid in a tub e. Similarly w e can construct dieomorphism f i that maps p i to q i and k eeps all other landmarks xed, for i = 1 ; 2 ; :::; n The comp osition of this series of dieomorphisms f = f n :::f 2 f 1 (3.2) is also a dieomorphism and ob viously f maps p i to q i for i = 1 ; 2 ; :::; n If some landmark q k lies on the line of p i and q i w e can nd suc h a direction suc h that w e dra w a line l k through q k and there are no other landmarks on the line. Then w e mak e a dieomorphism h transp orting q k to a nearb y p oin t q 0 k along the line without mo ving an y other landmarks, using the same canal of viscous ruid tec hnique. Then w e mak e dieomorphism f i as describ ed b efore. After that, w e mo v e landmark q 0 k bac k to the old p osition with the in v erse of h 1 So w e use F i = h 1 f i h in place of f i

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44 Figure 3{1: Existence of a dieomorphic mapping 3.2 Symmetric Matc hing due to Time Rev ersibilit y Asymmetry exist in man y image and shap e matc hing situations. Supp ose w e ha v e a p oin t set f p i 2 R 2 j i = 1 ; 2 ; :::n g and another p oin t set f q i 2 R 2 j i = 1 ; 2 ; :::n g w e nd the dieomorphic mapping f : R 2 R 2 whic h minimize the energy functional E ( f ) sub ject to the constrain ts f ( p i ) = q i No w w e dene a rev erse problem, namely to nd a mapping g : R 2 R 2 whic h minimize the same energy functional E ( f ) sub ject to the constrain ts f ( p i ) = q i as sho wn in Figure 3{2 In general, dep ending on the ob jectiv e function to minimize, g 6 = f 1 This is called asymmetry of the matc hing. In some cases w e do ha v e g = f 1 The matc hing is called symmetric then and this is a nice prop ert y to ha v e. The follo wing theorem states that for the exact dieomorphic landmark matc hing case, the matc hing is symmetric due to the time rev ersibilit y of the ro w.

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45 g f Figure 3{2: Asymmetry of the matc hing Theorem 6. If ( x k ; 1) = y k and ( x; t ) and v ( x; t ) minimize the ener gy E = Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt ; then the inverse mapping maps the landmarks b ackwar d 1 ( y k ; 1) = x k and 1 ( x; t ) and v ( x; t ) also minimize the ener gy E Pr o of. First, from the kno wn prop ert y of the dieomorphism group of suc h a dynamical system, ( x; t 1 + t 2 ) = ( ( x; t 1 ) ; t 2 ), it is easy to sho w that 1 ( x; t ) = ( x; t ). This is b ecause ( :; t ) ( :; t )( x ) = ( :; t ) ( :; t )( x ) = ( ( x; t ) ; t ) = ( x; t + ( t )) = ( x; 0) = x And ( x; t )and v ( x; t ) also satisfy the transp ort equation @ ( x; t ) @ t = v ( ( x; t ) ; t ). Supp ose ( x; t ) and v ( x; t ) minimize the energy E = R 1 0 R n jj Lv ( x; t ) jj 2 dxdt but 1 ( x; t ) = ( x; t ) and v ( x; t ) do not minimize the energy E = R 1 0 R n jj Lv ( x; t ) jj 2 dxdt Let the minimizer b e ( x; t ) and u ( x; t ) suc h that 8 k ( y k ) = x k and R 1 0 R n jj Lu ( x; t ) jj 2 dxdt < R 1 0 R n jj Lv ( x; t ) jj 2 dxdt Then, w e can construct 1 ( x; t ) = ( x; t ) suc h that 1 ( x; t ) and u ( x; t ) satisfy the transp ort equation and 1 ( x k ; 1) = y k Ho w ev er R 1 0 R n jj Lu ( x; t ) jj 2 dxdt < R 1 0 R n jj Lv ( x; t ) jj 2 dxdt con tradicts the assumption that v ( x; t ) is the minimizer of the energy E

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46 3.3 A Theoretical F ramew ork for Dieomorphic P oin t Matc hing There are dieren t w a ys to solv e for the unkno wn p oin t corresp ondences [ 56 14 17 16 ]. Essen tially within the framew ork of explicit p oin t corresp ondences|as opp osed to the distance function framew ork of implicit corresp ondence|w e ha v e a c hoice b et w een i) solving for an optimal p erm utation and ii) letting corresp onding \lab eled" p oin ts disco v er their optimal lo cations. W e opt for the latter in this w ork b ecause of its simplicit y The clustering in fact serv es t w o purp oses. First, it is the metho d to nd the unkno wn corresp ondence. W e initialize the t w o sets of cluster cen ters around the cen troids of their data p oin ts, resp ectiv ely The cluster cen ters are lab eled with iden tical lab els in the t w o sets denoting corresp ondence. The cluster cen ters ev olv e during the iterations of an incremen tal EM algorithm and they are link ed b y a dieomorphism and are forced to mo v e in lo c k-step with one another. Second, clustering is the mo deling of the real data sets, with noise and/or outliers b ecause with t w o shap es represen ted b y p oin t samples, w e cannot assume a p oin t-wise corresp ondence. The corresp ondence is only b et w een the t w o shap es and clustering is a useful w a y to mo del the shap es. W e use a Gaussian mixture mo del to describ e the clustering of the p oin t sets. F or more details on this approac h along with justications for the use of this mo del, please see [ 15 17 ]. The Gaussian mixture probabilit y densit y is p ( x j r ; T ) = 1 N N X k =1 1 (2 2 T ) d= 2 exp( 1 2 2 T jj x r k jj 2 ) (3.3) with x b eing a p oin t in R d r as the collectiv e notation of a set of cluster cen ters and 2 T as the v ariance of eac h Gaussian distribution. The reason for the notation of subscript T will b e discussed in Section 3.4 in the con text of annealing ( T will b e the temp erature. T and T are related b y T = 2 2 T .) Here w e just understand 2 T as the pr escrib e d v ariance in the Gaussian mixture mo del as opp osed to the actual measured v ariance 2 from data tting.

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47 The clustering pro cess is the estimation of the parameters r that leads to the maxim um log-lik eliho o d of the observ ed sample log p ( x j r ; T ) = N 1 X i =1 log N X k =1 exp( 1 2 2 jj x i r k jj 2 ) : (3.4) The solution can b e found b y applying the EM algorithm. As p oin ted out b y Hatha w a y [ 34 ], in the mixture mo del con text, the EM algorithm maximizing ( 3.4 ) can b e view ed as an alternativ e maximization of the follo wing ob jectiv e F ( M ; r ) = 1 2 2 T N 1 X i =1 N X k =1 M x ik jj x i r k jj 2 (3.5) N 1 X i =1 N X k =1 M x ik log M x ik : This is equiv alen t to minimizing E ( M ; r ) = F ( M ; r ) (3.6) = 1 2 2 T N 1 X i =1 N X k =1 M x ik jj x i r k jj 2 + N 1 X i =1 N X k =1 M x ik log M x ik with simplex constrain ts on M The clustering of the other p oin t set is iden tical. F or the join t clustering and dieomorphism estimation, w e put together the clustering energy of the t w o p oin t sets and the dieomorphic deformation energy induced in space giving us an

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48 ob jectiv e function E ( M x ; M y ; r ; s ; v ; ) (3.7) = 1 2 2 T N 1 X i =1 N X k =1 M x ik jj x i r k jj 2 + N 1 X i =1 N X k =1 M x ik log M x ik + 1 2 2 T N 2 X j =1 N X k =1 M y j k jj y j s k jj 2 + N 2 X j =1 N X k =1 M y j k log M y j k + 1 2 2 T N X k =1 jj s k ( r k ; 1) jj 2 + Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt: In the ab o v e ob jectiv e function, the cluster mem b ership matrices satisfy M x ik 2 [0 ; 1], 8 ik M y j k 2 [0 ; 1], 8 j k and P Nk =1 M 1 ik = 1, P Nk =1 M 2 j k = 1. The matrix en try M x ik is the mem b ership of data p oin t x i in cluster k whose cen ter is at lo cation r k The matrix en try M y j k is the mem b ership of data p oin t y j in cluster k whose cen ter is at p osition s k The dieomorphic deformation energy in n is induced b y the landmark displacemen ts from r to s where x 2 n and ( x; t ) is a one parameter dieomorphism: n n. Since the original p oin t sets dier in p oin t coun t and are unlab eled, w e cannot immediately use the dieomorphism ob jectiv e functions as in Joshi and Miller [ 39 ] or Camion and Y ounes [ 7 ] resp ectiv ely Instead, the t w o p oin t sets are clustered and the landmark dieomorphism ob jectiv e is used b et w een t w o sets of cluster cen ters r and s whose indices are alw a ys in corresp ondence. The dieomorphism ( x; t ) is generated b y the ro w v ( x; t ). ( x; t ) and v ( x; t ) together satisfy the transp ort equation @ ( x;t ) @ t = v ( ( x; t ) ; t ) and the initial condition 8 x ( x; 0) = x holds. This is in the inexact matc hing form and the displacemen t term P Nk =1 jj s k ( r k ; 1) jj 2 pla ys an imp ortan t role here as the bridge b et w een the t w o systems. This is also the reason wh y w e prefer the deformation energy in this form b ecause the coupling of the t w o sets of clusters app ear naturally through the inexact matc hing term and w e don't ha v e to in tro duce external coupling terms as

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49 in Guo et al. [ 33 ]. Another adv an tage of this approac h is that in this dynamic system describ ed b y the dieomorphic group ( x; t ), the landmarks trace a tra jectory exactly on the ro w lines dictated b y the eld v ( x; t ). Also, the feedbac k coupling is no longer needed as in the previous approac h b ecause with this deformation energy describ ed ab o v e, if ( x; t ) is the minimizer of this energy then 1 ( x; t ) is the bac kw ard mapping whic h also minimizes the same energy The T in ( 3.7 ) is a xed parameter. It is not a v ariable during the minimization. It is an attribute of the p oin t set and can b e a priori estimated for the p oin t shap es. The reason that w e ha v e co ecien t 1 2 2 T in fron t of the term P Nk =1 jj s k ( r k ; 1) jj 2 instead of another free parameter is that T is a natural unit of measuremen t for distan t discrepancies in clustering and it do es not mak e sense to mak e this co ecien t to o big or to o small. Since T is a constan t, w e m ultiply the ob jectiv e function b y a constan t 2 2 T and w e get the nal form of the ob jectiv e function as E ( M x ; M y ; r ; s ; v ; ) (3.8) = N 1 X i =1 N X k =1 M x ik jj x i r k jj 2 + 2 2 T N 1 X i =1 N X k =1 M x ik log M x ik + X N X k =1 M y j k jj y j s k jj 2 + 2 2 T N 2 X j =1 N X k =1 M y j k log M y j k + N X k =1 jj s k ( r k ; 1) jj 2 + 2 2 T Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt: 3.4 A Dieomorphic P oin t Matc hing Algorithm Our join t clustering and dieomorphism estimation algorithm has t w o comp onen ts: i) clustering and ii) dieomorphism estimation. F or the clustering part, w e use the deterministic annealing approac h. The clustering problem is a non-con v ex optimization problem. The traditional clustering

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50 tec hniques use descen t based algorithms and they tend to get trapp ed in a lo cal minim um. Rose et al. [ 59 ] prop osed an annealing approac h using analogies to statistical ph ysics. The clustering cost function is seen as the free energy of a Gibbs canonical distribution. The minimization of clustering cost function is seen as the sim ulation of a ph ysical annealing pro cess in whic h free energy is minimized. Let T b e the temp erature of the system and in the clustering system T = 2 2 T Let = 1 =T b e the recipro cal temp erature. Initially let = 0. W e ha v e a single minim um for the energy in ( 3.6 ), whic h is the global minim um and all the cluster cen ters are lo cated at the same p oin t, whic h is the cen ter of mass of all the data p oin ts and eac h data p oin t is uniformly asso ciated with all clusters. In the n umerical implemen tation, w e initialize all the cluster cen ters on a sphere of a v ery small radius and there are no data p oin t within the sphere. When the temp erature is lo w ered gradually at a certain critical v alue of temp erature, the clusters will split in to smaller clusters and a phase transition o ccurs. A t lo w er T the free energy ma y ha v e man y lo cal minima but the annealing pro cess is able to trac k the global minim um. F or the dieomorphism estimation, w e expand the ro w eld in term of the k ernel K of the L op erator v ( x; t ) = N X k =1 k ( t ) K ( x; k ( t )) (3.9) where k ( t ) is notational shorthand for ( r k ; t ) and w e also tak e in to consideration the ane part of the mapping (not written out in the ab o v e equation) when w e use the thin-plate k ernel with matrix en try K ij = r 2 ij log r ij for 2D and K ij = r ij for 3D, with r ij = k x i x j k After discretizing in time t the ob jectiv e in ( 3.7 ) is expressed as

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51 E ( M x ; M y ; r ; s ; ( t ) ; ( t )) (3.10) = N 1 X i =1 N X k =1 M x ik jj x i r k jj 2 + T N 1 X i =1 N X k =1 M x ik log M x ik + N 2 X j =1 N X k =1 M y j k jj y j s k jj 2 + T N 2 X j =1 N X k =1 M y j k log M y j k + N X k =1 k s k r k N X l =1 S X t =0 [ P ( t ) d l ( t ) + l ( t ) K ( k ( t ) ; l ( t ))] k 2 + T N X k =1 N X l =1 S X t =0 < k ( t ) ; l ( t ) > K ( k ( t ) ; l ( t )) where P ( t ) = 266666666664 1 11 ( t ) 21 ( t ) 31 ( t ) : : : : : : : : : : : : 1 1N ( t ) 2N ( t ) 3N ( t ) 377777777775 (3.11) and d is the ane parameter matrix. After w e p erform a QR decomp osition on P P ( t ) = ( Q 1 ( t ) : Q 2 ( t )) 0B@ R ( t ) 0 1CA : (3.12) W e iterativ ely solv e for k ( t ) and k ( t ) using an alternating algorithm. When k ( t ) is held xed, w e solv e for k ( t ). The solutions are d ( t ) = R 1 ( t ) [ Q 1 ( t ) ( t + 1) Q 1 ( t ) K ( ( t )) Q 2 ( t ) r ( t )] (3.13) ( t ) = Q 2 ( t ) r ( t ) (3.14)

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52 where K ( ( t )) denotes the thin-plate k ernel matrix ev aluated at ( t ) def = f ( r k ; t ) j k = 1 ; : : : ; N g and r ( t ) = ( Q T2 ( t ) K ( ( t )) Q 2 ( t ) + T ) 1 Q T2 ( t ) ( t + 1) : (3.15) When k ( t ) is held xed, w e use gradien t descen t to solv e for k ( t ): @ E @ k ( t ) = 2 N X l =1 < k ( t ) ; l ( t ) 2 W l > 5 1 K ( k ( t ) ; l ( t )) (3.16) where W l = s l r l P Nm =1 R 1 0 m ( t ) K ( m ( t ) ; l ( t )) dt The clustering of the t w o p oin t sets is handled b y a deterministic annealing EM algorithm whic h iterativ ely estimates the cluster mem b erships M x and M y and the cluster cen ters r and s The up date of the mem b erships is the v ery standard E-step of the EM algorithm [ 17 ] and is p erformed as sho wn b elo w. M x ik = exp( k x i r k k 2 ) P Nl =1 exp( k x i r l k 2 ) ; 8 ik (3.17) M y j k = exp( k y j s k k 2 ) P Nl =1 exp( k y j s l k 2 ) ; 8 j k : (3.18) The cluster cen ter up date is the M-step of the EM algorithm. This step is not the t ypical M-step. W e use a closed-form solution for the cluster cen ters whic h is an appro ximation. F rom the clustering standp oin t, w e assume that the c hange in the dieomorphism at eac h iteration is suciently smal l so that it c an b e ne gle cte d After making this appro ximation, w e get r k = P N 1 i =1 M x ik x k + s k P Nl =1 R 1 0 l ( t ) K ( l ( t ) ; k ( t )) dt 1 + P N 1 i =1 M x ik ; (3.19) s k = P N 2 j =1 M y j k y j + ( r k ; 1) 1 + P N 2 j =1 M y j k ; 8 k : (3.20) The o v erall algorithm is describ ed b elo w.

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53 Initialization: Initial temp erature T = 0 : 5(max i k x i x c k 2 + max j k y j y c k 2 ) where x c and y c are the cen troids of X and Y resp ectiv ely Begin A: While T > T nal Step 1 : Clustering Up date mem b erships according to ( 3.17 ), ( 3.18 ). Up date cluster cen ters according to ( 3.19 ), ( 3.20 ). Step 2: Dieomorphism Up date ( ; v ) b y minimizing E di ( ; v ) = C X k =1 jj s k ( r k ; 1) jj 2 + T Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt according to ( 3.13 )( 3.14 ) and ( 3.16 ). Step 3: Annealing. T r T where r < 1. End 3.5 Applications to 2D Corpus Callosum Shap es W e applied the algorithm to nine sets of 2D corpus callosum slices. The feature p oin ts w ere extracted with the help of a neuroanatomical exp ert. Figure 3{3 sho ws the nine corpus callosum 2D images, lab eled CC1 through CC9. In our exp erimen ts, w e rst did the sim ultaneous clustering and matc hing with the corpus callosum p oin t sets CC5 and CC9. The clustering of the t w o p oin t sets is sho wn in Figure 3{4 There are 68 cluster cen ters. The circles represen t the cen ters and the dots are the data p oin ts. The t w o cluster cen ters induce the dieomorphic mapping of the 2D space. The w arping of the 2D grid under this dieomorphism is sho wn in Figure 3{5 Using this dieomorphism, w e calculated the after-image of original data p oin ts and compared them with the target data p oin ts. Due to the

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54 large n um b er of cluster cen ters, the cluster cen ters nearly coincide with the original data p oin ts and the w arping of the original data p oin ts is not sho wn in the gure. The corresp ondences (at the cluster lev el) are sho wn in Figure 3{6 The algorithm allo ws us to sim ultaneously obtain the dieomorphism and the corresp ondence. Using our form ulation, w e are able to calculate the geo desic distances b et w een the t w o sets of cluster cen ters. This is done on the shap e manifold. Eac h p oin t q on the shap e manifold M represen ts a set of N cluster cen ters x 1 x 2 ... x N 2 R 2 and has a co ordinate q = ( x 11 ; x 21 ; x 12 ; x 22 ; :::; x 1N ; x 2N ) where x i = ( x 1i ; x 2i ), i = 1 ; 2 ; :::; N Let q ( t ) b e the geo desic path connecting t w o p oin ts q 1 and q 2 on the manifold. Using the norm dened for the tangen t v ector in Camion and Y ounes [ 7 ], the geo desic distance b et w een q 1 and q 2 is D geo desic ( q 1 ; q 2 ) = Z 1 0 q q T Q 2 ( Q T2 K ( q ) Q 2 + ) 1 Q T2 q dt: (3.21) where K ( q ) is the k ernel of the L op erator ev aluated at q ( t ) and as men tioned previously the thin-plate spline k ernel is used. Q 2 comes from the QR decomp osition of P P = ( Q 1 : Q 2 ) 0B@ R 0 1CA (3.22) and P = 0BBBBBBBBBB@ 1 x 11 x 21 : : : : : : : : : 1 x 1N x 2N 1CCCCCCCCCCA (3.23) W e also exp erimen ted with dieren t n um b er of cluster cen ters. T able 3{1 sho ws a mo died Hausdor distance as rst in tro duced [ 20 ] b et w een the image

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55 set of p oin ts CC5 after dieomorphism and the target set of p oin ts CC9 when the n um b er of clusters v ary The reason for using the mo died Hausdor distance instead of the Hausdor distance is that the latter is to o sensitiv e to outliers. The denition of the mo died Hausdor distance is H mo d ( A; B ) = max ( h mo d ( A; B ) ; h mo d ( B ; A )) ; (3.24) where A and B are nite p oin t sets and h mo d ( A; B ) = 1 j A j X a 2 A min b 2 B k a b k (3.25) is the a v erage of the minim um distances instead of the maxim um of the minim um distances. It is easy to see that when the n um b er of clusters increases, the matc hing impro v es as the mo died Hausdor distance decreases. In the third column in T able 3{2 w e list the geo desic distances b et w een the t w o sets of cluster cen ters after pair-wise w arping and clustering all the pair of corpus callosum p oin t sets. Using the cluster cen ters as landmarks, a dieomorphic mapping of the space is induced. With this induced dieomorphism, w e mapp ed the original data sets and compared the image of the original p oin t set under the dieomorphism and the target p oin t set using the mo died Hausdor distance. The mo died Hausdor distances b et w een the pairs are listed in the fourth column in T able 3{2 Finally from the original nine corpus callosum p oin t sets, w e w arp ed the rst eigh t p oin t sets on to the nin th set and Figure 3{7 displa ys the o v erla y of all p oin t sets after dieomorphic w arping. 3.6 Applications to 3D Shap es W e applied our form ulation and algorithm to the 3D p oin t data of hipp o campal shap es. W e rst applied the algorithm to syn thetic data, where w e ha v e the

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56 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0 0.2 Figure 3{3: P oin t sets of nine corpus callosum images. 0.4 0.6 0.8 1 1.2 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.4 0.6 0.8 1 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 3{4: Clustering of the t w o p oin t sets. 0.2 0.4 0.6 0.8 1 1.2 0.2 0.1 0 0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1 1.2 0.2 0.1 0 0.1 0.2 0.3 0.4 Figure 3{5: Dieomorphic mapping of the space.

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57 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Figure 3{6: Matc hing b et w een the t w o p oin t sets. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Figure 3{7: Ov erla y of the after-images of eigh t p oin t sets with the nin th set. T able 3{1: Mo died Hausdor distance of the matc hing p oin t sets. Num b er of Clusters Mo died Hausdor Distance 10 0.0082 20 0.0082 30 0.0057 40 0.0050 50 0.0043 60 0.0035 68 0.0027

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58 T able 3{2: Geo desic distances b et w een t w o sets of cluster cen ters and mo died Hausdor distances of matc hing p oin ts. F rom T o Geo desic M.H. CC1 CC2 0.0264 0.0055 CC1 CC3 0.0132 0.0014 CC1 CC4 0.0289 0.0048 CC1 CC5 0.0269 0.0056 CC1 CC6 0.0250 0.0097 CC1 CC7 0.0323 0.0054 CC1 CC8 0.0256 0.0043 CC1 CC9 0.0241 0.0041 CC2 CC3 0.0277 0.0059 CC2 CC4 0.0342 0.0063 CC2 CC5 0.0308 0.0057 CC2 CC6 0.0211 0.0100 CC2 CC7 0.0215 0.0040 CC2 CC8 0.0271 0.0044 CC2 CC9 0.0258 0.0093 CC3 CC4 0.0443 0.0059 CC3 CC5 0.0294 0.0047 CC3 CC6 0.0181 0.0032 CC3 CC7 0.0256 0.0060 CC3 CC8 0.0153 0.0018 CC3 CC9 0.0305 0.0044 CC4 CC5 0.0231 0.0046 CC4 CC6 0.0304 0.0056 CC4 CC7 0.0324 0.0056 CC4 CC8 0.0311 0.0054 CC4 CC9 0.0434 0.0090 CC5 CC6 0.0266 0.0056 CC5 CC7 0.0325 0.0053 CC5 CC8 0.0225 0.0037 CC5 CC9 0.0305 0.0069 CC6 CC7 0.0244 0.0050 CC6 CC8 0.0186 0.0026 CC6 CC9 0.0274 0.0056 CC7 CC8 0.0212 0.0044 CC7 CC9 0.0241 0.0102 CC8 CC9 0.0196 0.0050

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59 200 220 240 260 280 300 320 340 360 120 140 160 180 200 220 240 Figure 3{8: Tw o p oin t sets of hipp o campal shap es. The set with crosses is the original set and the set with dots is the one after GRBF w arping. kno wledge of ground truth and this serv es as the v alidation of the algorithm. W e then exp erimen ted with real data and ev aluated the results using v arious measures. 3.6.1 Exp erimen ts on Syn thetic Data W e selected one hipp o campal p oin t set and w arp ed it with a kno wn dieomorphism using the Gaussian Radial Basis F unction (GRBF) k ernel. W e c ho ose = 60 for the GRBF b ecause with this large v alue of ,w e are able to generate a more global w arping. Figure 3{8 sho ws the t w o p oin t sets of hipp o campal shap es. The set with crosses is the original set and the set with dots is the one after GRBF w arping. First, w e ha v e no noise added. W e used the TPS k ernel to reco v er the dieomorphism via join t clustering using our algorithm. The reason w e use dieren t k ernels for w arping and reco v ering is the ob jectiv eness. It is trivial to reco v er the

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60 T able 3{3: Matc hing errors on syn thetic data with dieren t noise lev els Noise lev el n No. clusters 100 200 300 400 500 0 0.21 0.17 0.16 0.19 0.20 0.1 0.30 0.13 0.28 0.09 0.26 0.08 0.29 0.09 0.31 0.11 0.2 0.41 0.16 0.39 0.12 0.35 0.11 0.37 0.13 0.39 0.14 0.3 0.44 0.17 0.41 0.13 0.39 0.15 0.40 0.16 0.42 0.19 0.4 0.61 0.23 0.54 0.19 0.52 0.18 0.55 0.20 0.59 0.21 0.5 0.68 0.24 0.62 0.25 0.59 0.24 0.63 0.22 0.65 0.28 0.6 0.82 0.38 0.75 0.35 0.72 0.33 0.76 0.37 0.80 0.36 0.7 0.96 0.49 0.90 0.42 0.86 0.42 0.90 0.44 0.94 0.46 0.8 1.21 0.54 1.13 0.51 0.92 0.48 1.09 0.49 1.18 0.51 0.9 1.63 0.72 1.48 0.66 1.45 0.62 1.49 0.61 1.52 0.68 1.0 1.82 0.78 1.70 0.71 1.64 0.67 1.69 0.73 1.77 0.75 deformation that is w arp ed with the same k ernel. Since the reference data are syn thesized, w e kno w the ground truth and w e are able to compare our result with the ground truth. After un w arping the p oin t set with our reco v ered dieomorphism, w e nd the squared distances b et w een the corresp onding data p oin ts, and nd the a v erage and then tak e the square ro ot. This is the standard error for our reco v ered dieomorphism. W e ha v e t w o free parameters, and T f inal T f inal is determined b y the limiting v alue of T whic h is in turn determined b y the n um b er of clusters. W e c ho ose a v alue suc h that the whole optimization pro cess is stable in the temp erature range from initial T to T f inal W e exp erimen ted with dieren t n um b ers of clusters and listed the corresp onding standard errors in the rst ro w of T able 3{3 It is easy to see that the standard error go es do wn as the n um b er of clusters go es up from 100 to 300 and go es up again when the n um b er of clusters increases further. This is b ecause when w e ha v e to o few clusters, the p oin ts are not w ell represen ted b y the cluster cen ters. On the other hand, if w e ha v e to o man y clusters, the v ariance b et w een the t w o shap es is to o big and the deformation increases dramatically F or the standard error, there is an optimal n um b er of clusters and in this case w e nd it to b e 300. W e need to estimate the macroscopic and microscopic dimensions of the shap e in order to see ho w big the standard error is. W e calculated the co v ariance matrix of the original data set. W e nd their

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61 T able 3{4: Limiting v alue of determined b y the n um b er of clusters Num b er of clusters 100 200 300 400 500 Limit 3.3 1.9 1.2 0.9 0.6 eigen v alues to b e [48.1, 11.8, 4.1]. This giv es us an estimate of the macroscopic dimensions to b e ab out 100, 24 and 8, namely t wice the eigen v alues. W e then nd out the a v erage distance b et w een the nearest neigh b ors to b e 2.65. This giv e us the microscopic dimension of the shap e. As w e can see from the table, our matc hing is v ery accurate. Next w e add noise to the w arp ed data and test the robustness of our algorithm to noise. After GRBF w arping, w e add Gaussian noise to the w arp ed data with dieren t v ariances W e exp erimen ted with ten trials for eac h noise lev el from 0.1 to 1.0 and for eac h cluster lev el from 100 to 500. The standard errors and the deviation are sho wn in T able 3{3 W e can see the standard error increase with the increasing noise lev el but it appro ximately sta ys in the range of the noise. Stronger noise do es not increase the matc hing error dramatically and this sho ws the algorithm is robust against noise. This is easier to see when plotted in Figure 3{9 with error bars. W e split the v e lev els of clusters in to t w o plots b ecause it lo oks messy if they w ere put together in a single plot Figure 3{9 (a) has the errors for 100, 200 and 300 clusters and Figure 3{9 (b) has the errors for 300, 400 and 500 clusters. W e can see that at the 300 cluster lev el, w e obtain the b est matc hing. 3.6.2 Exp erimen ts on Real Data W e applied the algorithm on dieren t real hipp o campal data sets. Figure 3{10 sho ws t w o hipp o campal shap es. Figure 3{11 sho ws the annealing pro cess. The x axis is the iteration step. The dashed line is the scaled temp erature p T = 2 or T The solid line is the actual v ariance W e can see when the temp erature go es do wn, it driv es the do wn. W e observ e a phase transition at temp erature

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62 T = 3 : 37 10 3 W e observ e that there is a lo w er limit for When the temp erature gets v ery lo w, the b ecomes a constan t, whic h is 1.2, and no matter ho w m uc h lo w er the temp erature gets, the sta ys constan t. This constan t is determined b y the n um b er of clusters. In Figure 3{12 a through Figure 3{12 e, w e sho w ho w this limit c hanges with the n um b er of clusters. When the n um b er of clusters equals or exceeds the n um b er of data p oin ts, the limit approac hes zero. T able 3{4 displa ys the limits of c hanging with the n um b er of clusters. Because of noise and sampling error, w e should not allo w this limit to go to zero. Again w e observ e when w e ha v e 300 clusters, w e ha v e a reasonable = 1 : 2 as w e recall the a v erage distance b et w een the nearest neigh b ors is ab out 2.65. Figure 3{13 sho ws the clustering of the t w o shap es. W e then did the matc hing for all the pairs out of ten hipp o campal shap es. T able 3{5 and T able 3{6 sho ws three measures for the matc hing results with dieren t clusters: Jensen-Shannon div ergence, Hausdor distance and mo died Hausdor distance. The Jensen-Shannon div ergence (a sp ecial case with = 1 = 2) is dened as [ 25 ] D = Z n ( p ( x ) log 2 p ( x ) p ( x ) + q ( x ) + q ( x ) log 2 q ( x ) p ( x ) + ( x ) ) d x ; (3.26) where x is the random v ariable while p ( x ) and q ( x ) are the t w o probabilit y densities. Notice this measure is highly non-linear. When p ( x ) and q ( x ) are completely indep enden t, namely in our matc hing case, when the t w o shap es are completely dieren t, D has a maxim um of 2 log 2 = 1 : 39. In practice, w e use the follo wing tec hnique to compute D W e observ e in ( 3.26 ), the in tegral can b e expressed as the exp ectation v alues of some functions under t w o dieren t probabilit y distributions: D = < log 2 p ( x ) p ( x ) + q ( x ) > p + < log 2 q ( x ) p ( x ) + ( x ) > q ; (3.27)

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63 where < log 2 p ( x ) p ( x )+ q ( x ) > p is the exp ectation v alue of function log 2 p ( x ) p ( x )+ q ( x ) under probabilit y distribution p ( x ) and < log 2 q ( x ) p ( x )+( x ) ) > q is the exp ectation v alue of function log 2 q ( x ) p ( x )+( x ) ) under probabilit y distribution q ( x ). In our Gaussian mixture mo del, w e see the data p oin ts as samples from a Gaussian mixture probabilit y distribution with kno wn cluster cen ters. Here p ( x ) = 1 N 1 (2 2 ) 3 = 2 N X k =1 exp( jj x r k jj 2 2 2 ) ; (3.28) and q ( x ) = 1 N 1 (2 2 ) 3 = 2 N X k =1 exp( jj x s k jj 2 2 2 ) ; (3.29) where x is the random v ariable, namely the space lo cation and f r k g is the rst set of cluster cen ters and f s k g is the second set of cluster cen ters. W e use the a v erage of nite samples as an appro ximation of the exp ectation v alues and w e ha v e D = 1 N 1 N 1 X i =1 log 2 p ( x i ) p ( x i ) + q ( x i ) + 1 N 2 N 2 X j =1 log 2 q ( y j ) p ( y j ) + ( y j ) ; (3.30) where f x i g is the rst set of data p oin ts; N 1 is the n um b er of p oin ts in the rst set; f y j g is the second set of data p oin ts; and N 2 is the n um b er of p oin ts in the second set. W e ha v e seen that Jensen-Shannon div ergence is v ery useful in estimating the v alidit y of the registration of t w o p oin t shap es without kno wing the ground truth. The Hausdor distance is dened as H ( A; B ) = max( h ( A; B ) ; h ( B ; A )) ; (3.31) where A and B are nite p oin t sets and h ( A; B ) = max a 2 A min b 2 B k a b k : (3.32) The Hausdor distance measures the w orst case dierence b et w een the t w o p oin t sets. F rom T able 3{5 and T able 3{6 w e can see that when w e ha v e 300

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64 clusters, w e ha v e the minim um Jensen-Shannon div ergence and the Hausdor distance. Ho w ev er, the Hausdor distance is to o sensitiv e to outliers. W e also calculated the mo died Hausdor distance as rst in tro duced in Dubuisson and Jain [ 20 ]. The denition of the mo died Hausdor distance w as giv en b efore in ( 3.24 ) and ( 3.25 ). It is the a v erage of the minim um distances instead of the maxim um of the minim um distances. It is easy to see that when the n um b er of clusters increases, the mo died Hausdor distance decreases.

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65 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 N=100 N=200 N=300 (a) 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 N=500 N=400 N=300 (b) Figure 3{9: Matc hing errors on syn thetic data for dieren t n um b er of clusters

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66 220 240 260 280 300 320 340 360 380 120 140 160 180 200 220 240 0 10 20 Figure 3{10: Tw o hipp o campal shap es

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67 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 iteration steps s T s x100 Figure 3{11: Deterministic annealing in the clustering pro cess: the dashed line is the scaled temp erature p T = 2 or T The solid line is the actual v ariance When the temp erature go es do wn, it driv es the do wn. There is a phase transition at temp erature T = 3 : 37 10 3 and there exists a lo w er limit 1.2 for

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68 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 10 11 12 iteration steps s s T (a) 100 clusters 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 10 11 12 iteration steps s s T (b) 200 clusters 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 10 11 12 iteration steps s s T (c) 300 clusters 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 10 11 12 iteration steps s s T (d) 400 clusters 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 10 11 12 iteration steps s s T (e) 500 clusters Figure 3{12: Limiting v alue of determined b y the n um b er of clusters

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69 (a) Clustering of the template hipp o campal shap e (b) Clustering of the reference hipp o campal shap e Figure 3{13: Clustering of the t w o hipp o campal shap es

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70 T able 3{5: Jensen-Shannon div ergence for v arious pairs of shap es Jensen-Shannon div. T rial no. n No. clusters 100 200 300 400 500 1 0.87 0.31 0.03 0.13 0.21 2 0.93 0.62 0.47 0.05 0.24 3 0.76 0.27 0.04 0.16 0.32 4 0.98 0.52 0.34 0.09 0.45 5 0.69 0.41 0.14 0.18 0.36 6 0.57 0.23 0.43 0.78 0.97 7 0.66 0.21 0.05 0.14 0.30 8 0.99 0.70 0.25 0.19 0.63 9 0.85 0.42 0.11 0.68 0.74 10 0.97 0.62 0.10 0.18 0.55 11 0.70 0.33 0.06 0.13 0.26 12 1.02 0.64 0.08 0.44 0.71 13 0.89 0.54 0.20 0.31 0.65 14 0.57 0.09 0.15 0.66 0.80 15 0.88 0.30 0.05 0.29 0.36 16 0.90 0.75 0.12 0.17 0.44 17 0.61 0.16 0.28 0.53 0.72 18 0.91 0.37 0.18 0.40 0.88 19 1.12 0.80 0.47 0.09 0.28 20 0.96 0.54 0.33 0.60 0.74 21 0.65 0.23 0.51 0.78 1.04 22 0.93 0.46 0.22 0.51 0.68 23 0.92 0.60 0.28 0.15 0.34 24 0.80 0.26 0.57 0.69 0.86 25 1.10 0.62 0.44 0.78 0.97 26 0.90 0.39 0.05 0.21 0.47 27 0.58 0.07 0.20 0.56 0.77 28 0.93 0.51 0.09 0.40 0.63 29 0.99 0.26 0.18 0.37 0.70 30 0.60 0.06 0.17 0.54 0.57 31 0.83 0.19 0.08 0.37 0.76 32 1.22 0.42 0.57 0.70 0.95 33 0.80 0.59 0.30 0.86 0.92 34 0.89 0.76 0.35 0.28 0.67 35 1.05 0.42 0.37 0.81 1.13 36 0.92 0.25 0.31 0.60 0.85 37 0.79 0.35 0.08 0.24 0.40 38 0.90 0.42 0.16 0.35 0.68 39 0.86 0.27 0.38 0.50 0.71 40 0.55 0.04 0.19 0.36 0.67 41 1.02 0.30 0.47 0.81 0.98 42 0.43 0.07 0.22 0.56 0.86 43 0.78 0.56 0.18 0.39 0.61 44 0.61 0.09 0.25 0.70 0.82 45 0.44 0.15 0.26 0.53 0.93

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71 T able 3{6: Hausdor and mo died Hausdor distance for v arious pairs of shap es Hausdor distance mo died Hausdor T rial no. n No. clusters 100 200 300 400 500 100 200 300 400 500 1 7.1 7.4 5.7 6.2 7.3 2.8 2.0 1.4 1.2 1.1 2 9.3 8.9 7.2 8.3 8.7 3.5 3.1 2.8 2.4 2.3 3 7.2 6.1 4.9 5.6 6.4 2.0 1.7 1.4 1.3 1.2 4 8.4 7.8 7.2 5.2 6.5 2.7 2.4 2.3 1.7 1.4 5 9.6 9.7 8.0 8.4 8.9 3.9 3.6 3.1 2.8 2.7 6 9.2 6.3 7.1 7.8 8.6 3.1 2.8 2.5 2.2 2.1 7 6.9 5.8 4.4 6.0 7.3 2.4 2.2 2.1 1.7 1.5 8 8.9 8.5 7.0 6.4 8.2 3.0 2.6 2.4 2.2 1.9 9 9.3 8.0 5.9 7.6 9.1 2.9 2.7 2.3 2.1 1.6 10 7.8 7.3 4.7 6.7 8.1 3.2 2.8 2.3 1.8 1.4 11 8.7 7.7 5.8 7.4 9.0 2.5 2.1 1.6 1.4 1.2 12 9.1 8.3 6.1 7.4 8.6 3.3 3.0 2.5 2.2 2.0 13 9.3 8.4 6.5 7.0 8.7 3.6 3.4 3.1 2.4 2.2 14 7.4 5.1 5.5 6.8 8.3 3.0 2.7 2.3 2.0 1.8 15 8.8 7.1 4.9 6.3 7.8 2.6 2.0 1.5 1.3 1.2 16 9.4 9.0 6.1 7.3 8.5 3.2 3.0 2.4 2.1 1.9 17 8.6 6.8 7.9 9.0 9.9 3.4 3.5 3.1 2.7 2.4 18 9.5 8.2 6.5 7.4 8.0 3.7 3.2 2.9 2.6 2.1 19 9.2 7.8 7.2 5.1 6.4 2.8 2.6 2.3 2.2 2.0 20 9.6 8.0 7.3 8.7 9.3 3.9 3.5 3.3 3.1 2.7 21 8.4 6.1 6.9 7.8 9.5 3.3 3.1 2.8 2.4 2.3 22 9.7 8.5 7.0 8.1 9.0 2.9 2.7 2.6 2.3 2.1 23 9.6 8.2 7.3 6.5 7.7 2.4 2.1 1.7 1.6 1.5 24 7.8 6.6 7.2 8.9 9.6 3.1 2.7 2.5 2.2 1.9 25 9.8 7.9 7.6 8.8 9.2 3.8 3.4 3.0 2.8 2.7 26 9.0 7.3 5.8 7.0 8.7 2.9 2.4 2.0 1.4 1.2 27 7.8 6.0 6.5 7.2 8.3 3.2 2.9 2.5 2.1 1.9 28 9.5 8.1 6.1 7.4 8.8 3.0 2.8 2.5 2.1 1.8 29 9.7 8.3 6.7 7.0 8.5 3.4 2.7 2.2 1.9 1.7 30 7.1 5.6 6.2 7.3 7.8 2.5 2.2 1.8 1.3 1.2 31 9.3 6.7 5.8 7.5 8.4 2.7 2.4 2.2 1.6 1.4 32 9.9 7.2 7.8 8.5 9.2 3.5 3.0 2.8 2.5 2.3 33 8.9 8.0 6.3 7.1 8.7 3.3 3.1 2.6 2.4 2.3 34 8.9 8.5 6.7 6.4 7.0 3.0 2.8 2.5 2.2 2.1 35 9.6 8.4 7.0 8.1 9.9 3.7 3.4 3.0 2.3 2.0 36 8.7 5.9 6.2 7.5 8.4 3.2 2.9 2.5 2.3 2.2 37 7.7 6.3 5.4 6.1 7.6 2.4 2.2 1.9 1.6 1.5 38 9.5 7.1 5.7 6.6 7.9 2.8 2.4 2.3 2.0 1.8 39 9.2 7.3 8.2 8.4 9.0 3.2 3.0 2.7 2.5 2.4 40 6.5 5.2 5.8 6.7 7.3 2.0 1.6 1.3 1.1 0.9 41 9.4 7.5 7.9 8.6 9.2 3.8 3.2 2.5 2.4 2.3 42 7.2 5.8 6.7 7.3 7.9 2.3 1.9 1.6 1.5 1.4 43 8.2 7.4 6.0 6.7 7.8 2.9 2.4 2.2 2.0 1.8 44 7.1 5.9 6.6 7.5 8.0 2.2 1.9 1.7 1.5 1.4 45 7.0 6.5 7.2 7.7 9.1 2.5 2.1 1.9 1.7 1.6

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CHAPTER 4 TOPOLOGICAL CLUSTERING AND MA TCHING In this c hapter w e extend our dieomorphic p oin t matc hing theory and algorithm to include the kno wn information of the top ology of the underlying shap es. In Section 4.1 w e pro vide a brief in tro duction to the basic concepts of top ological spaces. In Section 4.2 w e review the Kohonen Self-Organizing F eature Map (SOFM) whic h w as rst in tro duced b y Kohonen in the con text of neural net w orks in 1980s. Our top ological clustering and matc hing is related to SOFM b ecause the essence of SOFM is top ology preserving. Ho w ev er, our top ological clustering and matc hing is dieren t from SOFM in man y w a ys and as part of Section 4.3 w e discuss these dierences. In Section 4.3 w e discuss the motiv ation and metho ds of top ological clustering and matc hing. W e do this with graph top ology assigned to the set of cluster cen ters. The graph top ology can b e prescrib ed if w e ha v e prior kno wledge of the shap e top ology or it can b e arbitrary in the case w e don't kno w the shap e top ology in adv ance. In Section 4.5 w e presen t results for clustering and matc hing with prescrib ed top ology with example of c hain top ology ring top ology and gen us zero closed surface top ology or S 2 top ology Section 4.6 describ es ho w w e can appro ximate the top ology if w e do not ha v e the prior kno wledge ab out the top ology of the shap e in adv ance. 4.1 F undamen tals of T op ological Spaces The heart of top ology is the concept of \nearness," describ ed b y \neigh b orho o d". W e rst in tro duce the concept of top ological space. Readers can confer the Encyclop edic Dictionary of Mathematics compiled b y Mathematical So ciet y of Japan and translated b y Massac h usetts Institute of T ec hnology [ 47 ]. W e also cite from a b o ok b y Bourbaki [ 6 ] and a b o ok b y Rosenfeld [ 61 ]. 72

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73 F elix Hausdor in his F oundations of Set The ory (Grundz uge der Mengenlehre. Leipzig, 1914) [ 35 36 ] dened his concept of a top ological space based on the four axioms. Let X b e a set. A neigh b orho o d system for X is a function U that assigns to eac h p oin t x of X a family U ( x ) of subsets of X sub ject to the follo wing axioms ( U ): ( U 1) x 2 U for eac h U in U ( x ). ( U 2) If U 1 U 2 2 U ( x ), then U 1 \ U 2 2 U ( x ). ( U 3) If U 2 U ( x ) and U V then V 2 U ( x ). ( U 4) F or eac h U in U ( x ), there is a mem b er W of U ( x ) suc h that U 2 U ( y ) for eac h y in W U ( x ) is in terpreted as the family of all neigh b orho o ds of p oin t x An elemen t U 2 U ( x ) is called a neigh b orho o d of p oin t x The in tuitiv e translation of the ab o v e axioms is as follo ws. ( U 1) x is in eac h neigh b orho o d of x ( U 2) The in tersection of t w o neigh b orho o ds of x is a neigh b orho o d of x ( U 3) If a set V con tains a neigh b orho o d of x then V is itself a neigh b orho o d of x ( U 4) F or eac h neigh b orho o d U of x there is another neigh b orho o d W of x suc h that U is the neigh b orho o d of eac h p oin t y in W P a v el Sergeevi c Aleksandro v [Alexandro] prop osed in the pap er On the foundation of n-dimensional top olo gy (Zur Begr undugn der n -dimensionalen T op ologie. Leipzig, 1925) [ 1 ]: A system of op en sets for a set X is a family O of subsets of X satisfying the follo wing axioms ( O ): ( O 1) X ; 2 O ( O 2) If O 1 O 2 2 O then O 1 \ O 2 2 O

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74 ( O 3) If O 2 O ( 2 ), then S 2 O 2 O The elemen ts in O are called op en sets. An easy in tuitiv e in terpretation of this set of axioms is ( O 1) The empt y set is an op en set. The en tire space X is an op en set. ( O 2) The in tersection of t w o op en sets is an op en set. ( O 3) The union of arbitrarily man y op en sets is an op en set. Using the set complemen t and DeMorgan's la w w e get a system of closed sets. A system of closed sets for a space X is a family F of subsets of X satisfying the follo wing axioms ( F ): ( F 1) X ; 2 F ( F 2) If F 1 F 2 2 F then F 1 [ F 2 2 F ( F 3) If F 2 F ( 2 ), then T 2 F 2 F The elemen ts in F are called close d sets. An easy in tuitiv e in terpretation of this set of axioms is ( F 1) The en tire space X is a closed set. The empt y set is a closed set. ( F 2) The union of t w o closed sets is a closed set. ( F 3) The in tersection of arbitrarily man y closed sets is a closed set. Kazimierz Kurato wski in the pap er The op er ation A of analysis situs (L'op eration A de l'analysis situs. W arsa w, 1922) [ 44 ] prop osed: A closure op erator for a space X is a function that assigns to eac h subset A of X a subset A a of X satisfying the follo wing axioms( C ): ( C 1) ; a = ; ( C 2) ( A [ B ) a = A a [ B a ( C 3) A A a ( C 4) A a = A aa The in tuition of the closure of a set A is the union of A and its b oundary This set of axioms explained in w ords is as follo ws.

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75 ( C 1) The closure of the empt y set is the empt y set. ( C 2) The closure of ( A [ B ) is the union of closure of A and the closure of B ( C 3) The closure of A con tains A as a subset. ( C 4) The closure of the closure of A is the same as the closure of A This is sa ying that the closure op erator is idemp oten t. Related to the closure op erator is an in terior op erator. An in terior op erator for a space X is a function that assigns to eac h subset A of X a subset A i of X satisfying the follo wing axioms ( I ): ( I 1) X i = X ( I 2) ( A \ B ) i = A i \ B i ( I 3) A i A ( I 4) A ii = A i The in tuition of the in terior of a set A is A min us its b oundary This set of axioms explained in w ords is as follo ws. ( I 1) The in terior of the en tire space X is itself. ( I 2) The in terior of ( A \ B ) is the in tersection of the in terior of A and the in terior of B ( I 3) The in terior of A is a subset of A ( I 4) The in terior of the in terior of A is the same as the in terior of A This is sa ying that the in terior op erator is idemp oten t. All these systems equiv alen tly dene the top ological space. W e can in terpret one easily in the language of another. W e can dene op en set using neigh b orho o d: A set A X is an op en set, if 8 x 2 A ther e exists a neighb orho o d U of x such that U A then A is an op en set. Dene closed set using op en set: A set A X is a close d set, if X A is an op en set.

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76 Dene closure of A using closed set: If A X the closur e of A is the set A a T B wher e B is a close d set and A B In other wor ds, the closur e of A is the smal lest close d set that c ontains A Dene in terior using closure: A i X ( X A ) a In other wor d, for set A X rst nd the c omplement of A B = X A Then nd B a the closur e of B The the interior of A is the c omplement of B a Dene neigh b orho o d using in terior: A set U X is the neighb orho o d of p oint x 2 X if x 2 U i namely x is in the interior of U Figure 4{1 sho ws some examples of dieren t top ological spaces. Another imp ortance concept in top ological spaces is the separation axioms, dictating the exten t the p oin ts are separated from eac h other. Of our in terest is one particular space, called Hausdor space. Denition 4. A top olo gic al sp ac e is c al le d Hausdor sp ac e if any two distinct p oints have disjoint neighb orho o ds. 4.2 Kohonen Self-Organizing F eature Map (SOFM) Kohonen dev elop ed the Self-Organizing F eature Map algorithm. He describ ed the SOFM in the con text of neural net w orks with an aim to understand the the brain cortex mapping of sensory organs, lik e retina [ 41 40 42 ]. In the Kohonen net w ork, there are t w o la y ers of neurons, with the rst la y er the input and the second la y er the output. The output neurons are arranged in a rectangular grid. Eac h input neuron is connected with eac h output neuron. Eac h output neuron is asso ciated with d -w eigh ts, with d b eing the n um b er of the input neurons. Kohonen describ es a pro cedure of the initialization and up date of the w eigh ts. The imp ortan t part of the algorithm is that eac h neuron on the output grid has a neigh b orho o d and when eac h neuron up dates itself, the neurons in the

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77 Figure 4{1: Dieren t top ological spaces neigh b orho o d up date themselv es accordingly This is describ ed in man y b o oks as \top ologically ordered map", \top ology preserving map" or \top ographical map". There are man y misconceptions and misnomer here. First, the neural net w ork is not suc h a map but it sim ulates suc h a map b ecause the co domain is a discrete space, namely the rectangular grid. Second, \top ographical map" is a misnomer for \top ological map." Third, what the net w ork sim ulates is a con tin uous map. That is the exact in terpretation of the idea that \if the t w o outputs are in a neigh b orho o d then the t w o inputs are also in a neigh b orho o d." \T op ology preserving" is not guaran teed. In fact this is ob vious when the input domain is t w o dimensional and the output domain is one dimensional. What w e get is a plain lling P eano curv e. As it is w ell kno wn that a plain region cannot b e top ologically equiv alen t to a curv e segmen t and the P eano curv e is a con tin uous map but not a top ological map, or homeomorphism. Kohonen only giv es a pro cedure but do es not giv e an ob jectiv e function that this pro cedure optimize. P eople so on nd an in terpretation of SOFM in the con text

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78 of clustering and dimension reduction of data, with mathematical abstraction. Ritter et al. giv e an ob jectiv e function [ 58 ] E f w g = 1 2 X ik M k ( i; k ) jj k w i jj 2 ; where M k is the mem b ership matrix elemen t, equal to 1 if patter is in cluster k and 0 otherwise. ( i; k ) is the neigh b orho o d function b et w een cluster i and cluster k ( i; k ) = 1 if i = k and falls o with distance jj w i w k jj A t ypical c hoice for ( i; k ) is ( i; k ) = e jj w i w k jj 2 2 2 : The Kohonen pro cedure is actually the clustering pro cess of high dimensional data. It tries to mo del the dimensionalit y reduction. The data p oin ts are in h dimensional Euclidean space but they ma y appro ximately lie on a t w o dimensional manifold. The clusters are constrained with a 2D rectangular grid. During the clustering it is required that neigh b oring cluster cen ters sta y close to eac h other. Th us this pro cedure pro vides an appro ximate discrete 2D patc h whic h is a map from a 2D rectangular region with grid to the 2D manifold em b edded in h dimensional Euclidean space, where the data dw ell. This pro vides an in trinsic co ordination (2D) for the data and it extracts 2D features from the data. Because of this clustering pro cess, the co ordinates are only appro ximate and the co ordinates are discrete (the i; j indices). In the general case, eac h data p oin t has h co ordinate comp onen ts and th us liv e in R h but the set of data only p opulate a l -dimensional submanifold M l of R h Supp ose w e ha v e a k -dimensional grid of cluster cen ters, U R k Only when k = l can w e ha v e a homeomorphism from U to an op en neigh b orho o d of M l One example is that l = 3 and k = 2. If the em b edded manifold is roughly a 2D sheet with some non-negligible thic kness in the third

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79 dimension, the clustering result in an appro ximation in whic h w e represen t this 3D curv ed thic k sheet with a 2D thic k-less sheet. If instead of a thic k sheet, w e ha v e a solid 3D manifold and w e still w an t to use a 2D sheet to appro ximate it, it will result in that the 2D sheet will wrinkle and scram ble so that it tries to ll the 3D space. It is easier to visualize when l = 2 and k = 1 and then w e ha v e a space lling P eano curv e, resulting a con tin uous map but not an homeomorphism. 4.3 T op ological Clustering and Matc hing 4.3.1 Wh y: the Need for T op ology When the t w o shap e dier b y a large deformation, complications do o ccur. Figure 4{2 sho ws the con tours of t w o hand shap es. The hand on the left has the th um b and the fore nger p oin ting out while the hand on the righ t has the fore nger folded. These t w o shap es dier b y a large deformation. W e apply our dieomorphic p oin t matc hing algorithm to these t w o shap es and Figure 4{3 sho ws the clustering of the t w o shap es. While the clustering lo oks prett y go o d, a closer examination of the corresp ondence as sho wn in Figure 4{4 indicates that there are incorrect corresp ondences. W e ha v e discussed in Chapter 1 the issues that there is no w a y to clearly dene what the correct corresp ondence is, with our visual in tuition, w e kno w this is not the corresp ondence w e w an t. This is the case where nearb y p oin ts corresp ond to p oin ts that are not nearb y a violation of top ological prop ert y of the mapping. W e w an t to enforce the constrain t that nearb y p oin ts are mapp ed to nearb y p oin ts and in tro duce the top ology constrain t to the matc hing. This has signicance in t w o scenarios: 1. The problem denition of the shap e matc hing do es not ha v e top ology constrain t and the ob jectiv e function do es not ha v e the top ology term. Ho w ev er, the n umerical pro cedure of solving this problem ma y b e caugh t up in a lo cal minim um, whic h giv es incorrect corresp ondence. In this case,

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80 Figure 4{2: Image con tours of t w o hands in tro ducing the top ological constrain t will help a v oid the lo cal minim um and nd the correct corresp ondence as dened b y the ob jectiv e function. 2. The n umerical pro cedure do es nd the global minim um dened b y the objectiv e function without top ological constrain t. Ho w ev er, the corresp ondence is still not what w e in tended. In this case, the p oin t set as a set of p oin t without an y top ological structure is not sucien t to describ e the underlying shap e. F or example of the t w o hand shap es as in Figure 4{2 w e kno w that the p oin ts lie on a con tour curv e. By adding this top ology requiremen t, w e are actually dening a dieren t problem from that without top ology structure and of course the solutions of the corresp ondence should b e dieren t with and without top ology since they are t w o dieren t problems. 4.3.2 Ho w: Graph T op ology It is clear that the information of the top ological structure of the underlying shap e of the p oin t set helps dene and solv e for the correct corresp ondence. Then it is natural to in tro duce some top ological structure in the set of cluster cen ters and a graph is the easiest w a y to represen t this top ology W e mak e the cluster cen ters the v ertices of the graph and assign edges b et w een the v ertices. An in tuitiv e thinking of the top ology construction on the graph ma y b e that mak e the set of v ertices the supp ort set and mak e the adjacen t v ertices

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81 0 50 100 50 0 50 100 150 200 100 150 200 250 50 0 50 100 150 200 Figure 4{3: Clustering of t w o hands 50 100 150 200 0 50 100 150 Figure 4{4: Corresp ondence

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82 B A U 3 U 2 1 U U A nn rr Figure 4{5: Finite top ology neigh b orho o d. A more careful analysis will sho w that this idea should fail. Since the set of the v ertices is a nite set, w e pro v e a prop ert y of nite top ology Theorem 7. L et ( X ; T ) b e a top olo gic al sp ac e and X is a nite set. T is Hausdor i T is discr ete top olo gy. Pr o of. First w e pro v e if T is discrete top ology then T is Hausdor. The pro of is trivial b ecause in discrete top ology ev ery singleton set of a p oin t is an op en set. F or an y t w o distinct p oin ts A 2 X and B 2 X the disjoin t neigh b orho o d U A and U B are U A = f A g and U B = f B g Next w e pro v e if T is Hausdor then T is discrete top ology W e use pro of b y con tradiction. No w w e assume the con trary namely T is Hausdor but T is a top ology other than discrete top ology Then there m ust exist a p oin t A 2 X suc h that f A g is not an op en set. Let U A b e the smallest op en set that con tains A : U A = U 1 \ U 2 \ ::: \ U k ; where U 1 ; U 2 ; :::; U k are all the op en sets that con tains A By assumption, U A m ust ha v e at least another p oin t dieren t from A W e call this p oin t B as sho wn in

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83 Figure 4{5 No w it is ob vious that there is no op en neigh b orho o d of A that do es not con tain B Hence T is not Hausdor. So the original claim is pro v ed. Since the Euclidean space is a Hausdor space and the shap es w e consider as sub-top ological spaces of the Euclidean space are Hausdor spaces, w e are not in terested in graph top ologies that are not Hausdor. Ho w ev er, from the ab o v e theorem w e kno w that the only Hausdor top ology is discrete top ology The discrete top ology is not go o d here b ecause eac h p oin t is an op en set and eac h p oin t can ha v e a op en set consisting of itself. Eac h p oin t is completely discrete, meaning isolated and disconnected. So the discrete top ology in some sense, is no top ology no \nearness" or top ological structure. The correct approac h is through gr aph r e alization or gr aph emb e dding This is the study of a branc h of graph theory top olo gic al gr aph the ory [ 30 ] In tuitiv ely graph realization or graph em b edding is to think the v ertices of the graph as p oin ts in an Euclidean space and the edges of the graphs as the lines or curv es in the Euclidean space connecting the v ertices. With the graph realization or graph em b edding, the graph G is a sub-top ological space of the shap e S as a top ological space. So if the p oin ts in the graph G are in a neigh b orho o d, then they are also in a neigh b orho o d in the shap e top ological space S In the follo wing of this c hapter, w e dev elop top ological clustering tec hniques for the purp ose of dieomorphic p oin t matc hing. It has man y similarities with Kohonen SOFM but there are also man y dierences. First, the purp ose of SOFM is dimensionalit y reduction while our top ological clustering is for p oin t matc hing. Second, the map in SOFM is lo cal while our top ology is global. SOFM only pro vides a single patc h for one op en neigh b orho o d of a p oin t on the submanifold while our clustering allo ws non-trivial top ologies whic h cannot b e co v ered b y a single patc h. Third, the in teraction in SOFM is b et w een one cluster cen ter and

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84 the data p oin ts in the neigh b oring clusters while the in teraction in our mo del in b et w een one cluster cen ter and the neigh b oring cluster cen ters. 4.4 Ob jectiv e F unction and the Algorithm In order to enforce the principle of that p oin ts in a neigh b orho o d should sta y in the neigh b orho o d, w e add the term N X m =1 N X n =1 G mn jj r m r n jj 2 (4.1) to the ob jectiv e function, where G is the adjacency matrix of the graph of the cluster cen ters of one set. The other set has the similar top ological constrain ts. G mn is 1 if there is an edge b et w een r m and r n and 0 otherwise. The new ob jectiv e function no w is E ( M x ; M y ; r ; s ; v ; ) (4.2) = N 1 X i =1 N X k =1 M x ik jj x i r k jj 2 + 2 2 T N 1 X i =1 N X k =1 M x ik log M x ik + X N X k =1 M y j k jj y j s k jj 2 + 2 2 T N 2 X j =1 N X k =1 M y j k log M y j k + N X k =1 jj s k ( r k ; 1) jj 2 + 2 2 T Z 1 0 Z n jj Lv ( x; t ) jj 2 dxdt + 4 N X m =1 N X n =1 G mn jj r m r n jj 2 + 4 N X q =1 N X q =1 G pq jj s p s q jj 2 : The matrix G is a symmetric matrix. The reason for the factor 1 2 in the last line is that eac h edge in the graph is coun ted t wice in the summation. is a new parameter describing the strength of the links b et w een the cluster cen ters. The algorithm to minimize this energy is v ery similar to the algorithm in tro duced in Chapter 3 Ho w ev er, the up date equations for the cluster cen ters ( 3.19 ) and ( 3.20 ) should b e mo died accordingly:

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85 r k = P N 1 i =1 M x ik x k + s k P Nl =1 R 1 0 l ( t ) K ( l ( t ) ; k ( t )) dt + P Nm =1 G mk r m 1 + P N 1 i =1 M x ik + P Nm =1 G mk ; (4.3) s k = P N 2 j =1 M y j k y j + ( r k ; 1) + P Nm =1 G mk s m 1 + P N 2 j =1 M y j k + P Nm =1 G mk ; 8 k : (4.4) The rule of mo dication is, when up dating the cluster cen ter p ositions for r k consider all other cluster cen ters r m ; m = 1 ; 2 ; :::; N in the graph, whenev er there is an edge from r m to r k w e add r m to the n umerator and w e add to the denominator. The mo dication to the up date of s k in the second set is similar. 4.5 Prescrib ed T op ology In some situations, the class of shap es ha v e the same top ology and the top ology is kno wn and the graph represen tation of the top ology is easy In suc h cases, w e can initialize the graph with the prescrib ed top ology In the 2D case, when w e deal with line con tours, t w o t ypical situations are the op en curv es and closed curv es. W e can use c hain top ology and ring top ology for the graphs. 4.5.1 Chain T op ology W e solv ed the matc hing problem again with the hand shap es with c hain top ology The clustering result is sho wn in Figure 4{6 and this w a y w e nd the correct corresp ondence in Figure 4{7 4.5.2 Ring T op ology W e applied the top ological clustering and matc hing algorithm to the corpus callosum data, with ring top ology Figure 4{8 sho ws the top ological clustering while Figure 4{9 sho ws the corresp ondence. 4.5.3 S 2 T op ology W e kno w the hipp o campus shap es ha v e a S 2 top ology W e initialize the graph as a latitude and longitude grid. Figure 4{11 is the top ological clustering of the

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86 0 50 100 50 0 50 100 150 200 100 150 200 250 50 0 50 100 150 200 Figure 4{6: T op ological clustering and matc hing of t w o hands 50 100 150 200 20 0 20 40 60 80 100 120 140 160 Figure 4{7: Corresp ondence with top ology constrain t

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87 0.4 0.6 0.8 1 1.2 0.2 0 0.2 0.4 0.4 0.6 0.8 1 0.2 0 0.2 0.4 Figure 4{8: T op ological clustering of corpus callosum shap es

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88 0.5 0.6 0.7 0.8 0.9 1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Figure 4{9: Corresp ondence in top ological clustering of corpus callosum shap es

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89 292 294 296 192 194 196 4 5 6 7 8 9 Figure 4{10: Sphere top ology rst hipp o campus set and Figure 4{12 is the top ological clustering of the second hipp o campus set. 4.6 Arbitrary T op ology W e can see the limitations with prescrib ed top ology First, the class of shap es ma y ha v e dieren t top ologies. Second, ev en if the class of the shap es ha v e the same top ology the top ology ma y b e unkno wn b efore w e run the matc hing algorithm. Third, ev en if the top ology is kno wn, it ma y b e to o complicated to construct a graph appro ximation and the initialization ma y in v olv e h uman in terv en tion. F orth, the edges of the graph is not truly top ological in a sense they are not indenitely rexible and stretc hable strings. The top ological constrain ts in the ob jectiv e function 4.2 are actually iden tical elastic strings. So w e see the geometrical or metric factor in the constrain ts. Sure w e can adjust the co ecien t

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90 250 300 350 160 180 200 220 240 0 5 10 15 Figure 4{11: T op ological clustering of hipp o campus with S 2 top ology: the rst set

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91 250 300 350 120 140 160 180 200 220 0 5 10 Figure 4{12: T op ological clustering of hipp o campus with S 2 top ology: the second set

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92 for dieren t stiness of the strings, but it is dicult to tune the dieren t relativ e stiness connecting the cluster cen ters. Because this elasticit y the graph ma y ha v e resistance to completely t the shap e. This can b e seen in previous hipp o campus examples. The solution is to nd the top ology of the shap e on the ry First w e cluster one of the data set. W e then nd the N-nearest neigh b ors of eac h cluster cen ter. F rom eac h cluster cen ter, w e dra w a directed edge to eac h of its nearest neigh b ors. This w a y w e ha v e a directed graph. The adjacency matrix is not symmetric in general. W e then symmetrize the adjacency matrix, meaning if w e ha v e an edge from no de i to j but not an edge j to i w e then add an edge from j to i This w a y w e get an undirected graph. W e will use this graph top ology for the clustering. Since in the matc hing problem, w e assume the t w o data sets ha v e the same top ology w e will use the same graph for b oth data sets. W e shrink the graph to the cen troid of the data sets in the initialization and w e con tin ue with the join t clustering and matc hing algorithm. Figure 4{13 is the appro ximate graph top ology of hipp o campus data sets with 4 nearest neigh b ors. Figure 4{14 sho ws the top ological clustering of the rst hipp o campus set and Figure 4{15 of the second hipp o campus set. Figure 4{16 a and b sho ws t w o 2D p oin t shap es. Figure 4{18 is the graph top ology w e ha v e learned using 4 nearest neigh b ors after clustering. Figure 4{18 sho ws the top ological clustering of the t w o shap es.

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93 220 240 260 280 300 320 340 360 160 180 200 220 240 2 4 6 8 10 12 14 Figure 4{13: Graph top ology of hipp o campus shap e through learning

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94 250 300 350 160 180 200 220 240 0 5 10 15 Figure 4{14: T op ological clustering of hipp o campus shap e: the rst p oin t set

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95 250 300 350 120 140 160 180 200 220 0 5 10 Figure 4{15: T op ological clustering of hipp o campus shap e: the second p oin t set 100 0 100 200 300 400 500 0 50 100 150 200 250 300 350 400 450 500 100 0 100 200 300 400 500 0 50 100 150 200 250 300 350 400 450 500 (a) (b) Figure 4{16: Fish shap es

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96 0 100 200 300 400 500 50 100 150 200 250 300 350 400 450 Figure 4{17: Graph top ology for the sh shap e with 4 nearest neigh b ors

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97 0 100 200 300 400 500 200 100 0 100 200 300 400 500 600 700 0 100 200 300 400 100 0 100 200 300 400 500 600 700 Figure 4{18: T op ological clustering and matc hing of the sh shap es

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CHAPTER 5 CONCLUSIONS 5.1 Con tributions The need for a go o d p oin t feature matc hing algorithm arises in v arious application areas of medical image analysis. T o m y kno wledge, this is one of the rst attempts at dieomorphic p oin t matc hing in the circumstances of unkno wn corresp ondence. W e ha v e designed an ob jectiv e function and an algorithm to sim ultaneously nd the b est clustering of t w o p oin t sets and a mapping with the least deformation of space. W e require the space deformation to b e a dieomorphic mapping b ecause it is smo oth and homeomorphic at the same time. The essence of this requiremen t is that in a homeomorphic mapping, neigh b oring p oin ts are mapp ed to neigh b oring p oin ts and the same is true for the in v erse mapping. A dieomorphic mapping can preserv e the features of shap es. The dieomorphism parameterization allo ws us to reco v er large deformations while sim ultaneously ac hieving go o d corresp ondence. W e ha v e demonstrated a join t clustering and dieomorphism algorithm and applied it to 2D corpus callosum shap es and 3D hipp o campal p oin t sets in medical imaging. After dieomorphism estimation, the shap e distance, dened as the geo desic distance on the shap e manifold is computed. Since the p oin t sets ha v e dieren t cardinalities and since the shap e distance is only dened w.r.t. the cluster cen ters, w e also computed a mo died Hausdor distance b et w een one original p oin t set and the after-image of a second p oin t set. W e conclude that when the n um b er of cluster cen ters increases, the mo died Hausdor distance decreases. In the pro cess of careful v alidation, w e in v estigated the role of the dieren t n um b ers of clusters using Jensen-Shannon div ergence in the join t clustering and dieomorphism optimization pro cess. 98

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99 W e further prop ose to use a graph represen tation for the shap e top ology information. Results are giv en for prescrib ed top ologies lik e c hain top ology ring top ology { whic h are v ery common in dealing with 2D con tour shap es { and gen us zero closed surface top ology in 3D. W e also in v estigate the top ology problem in general and the learning of top ology with a nearest neigh b or graph. 5.2 F uture W ork In the curren t form ulation, w e still ha v e a free parameter whose v alue has to b e determined. The immediate future goal is to further address (theoretically and exp erimen tally), the role of free parameters. The same framew ork can b e used for atlas estimation. F urthermore, once w e ha v e a turnk ey 3D dieomorphic feature matc hing algorithm, w e plan to use it for hipp o campal shap e classication of epilepsy patien ts [ 69 ]. W e realize with top ological clustering and matc hing, it is a compromise and comp etition b et w een clustering and top ology preserv ation, in the future, w e w an t to separate the t w o phases of corresp ondence and dieomorphism and hop e that ma y further impro v e the matc hing accuracy

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BIOGRAPHICAL SKETCH Hongyu Guo w as b orn in China. He receiv ed BS and MS degree in ph ysics in Nank ai Univ erisit y He receiv ed his MS degree in computer sciece from the Univ ersit y of Florida. Hongyu Guo has taugh t in a univ ersit y He has also w ork ed in soft w are industry in San F rancisco, CA. His curren t researc h in terests are in computer vision and biomedical imaging. 106