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## Material Information- Title:
- Shape Based Supervised Classification: Application to Epilepsy
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- 2008
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- Computer vision ( jstor )
Conceptual lattices ( jstor ) Datasets ( jstor ) Epilepsy ( jstor ) Image analysis ( jstor ) Magnetic resonance imaging ( jstor ) Signed distance function ( jstor ) Sine function ( jstor ) Tours ( jstor ) Uniform resource identifiers ( jstor )
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- 8/8/2002
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PAGE 1 SHAPE BASED SUPER VISED CLASSIFICA TION: APPLICA TION TO EPILEPSY By NEETI V OHRA A THESIS 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 MASTER OF SCIENCE UNIVERSITY OF FLORID A 2002 PAGE 2 Cop yrigh t 2002 b y NEETI V OHRA PAGE 3 T o m y paren ts, brother, Nitij and Dr. S.D. Joshi PAGE 4 A CKNO WLEDGEMENTS I w ould lik e to express m y gratitude to all the p eople who ha v e help ed me during the course of this pro ject without whic h this w ork could not ha v e b een accomplished. First and foremost, I w ould lik e to thank m y advisor and committee c hairman, Dr. Baba C. V em uri, for his guidance, constan t supp ort and encouragemen t during the en tire course of the w ork. It w as m y pleasure to get an opp ortunit y to w ork under his sup ervision. I ha v e greatly b eneted from his approac h to researc h, complete dedication to w ork and discussions and commen ts on the thesis. I w ould also lik e to thank Dr. Anand Rangara jan and Dr. John G. Harris for their willingness to serv e on m y committee. I highly appreciate their in terest and imp ortan t suggestions and commen ts on the v arious asp ects of the thesis. My appreciation and sp ecial thanks also go to Dr. Christiana M. Leonard for pro viding the medical image data and giving useful commen ts with regards to understanding the medical asp ects of the w ork. I w ould lik e to extend a sp ecial thank to m y colleagues in the Computer Vision, Graphics and Medical Imaging Group at the CISE departmen t of the Univ ersit y of Florida, Jundong Liu and Zhizhou W ang, for their v aluable commen ts, suggestions and constan t supp ort. I w ould also lik e to thank m y friends Nitij, Bharati, Mansi, An u and Anan th for their supp ort and encouragemen t during the course of m y w ork. Finally , I w ould lik e to thank m y family for their endless lo v e and supp ort whic h has made ev erything p ossible for m y success. I w ould lik e to dedicate this iv PAGE 5 thesis to m y paren ts, brother, Nitij and Dr. S.D. Joshi, who shared all the ups and do wns exp erienced during the w ork of the thesis. This researc h w as in part funded b y the NSF gran t I IS-9811042 and NIH R O1-RR13197. v PAGE 6 T ABLE OF CONTENTS A CKNO WLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : iv LIST OF T ABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ix ABSTRA CT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xiii CHAPTERS 1 INTR ODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Problem Defnition . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Shap e Reco v ery . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Shap e Registration . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Shap e Classifcation . . . . . . . . . . . . . . . . . . . . . 5 1.3 Con tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 SHAPE RECO VER Y : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 Ov erview of Mo deling Sc heme using Snak e P edals . . . . . . . . 11 2.1.1 P edal Curv es . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Snak e P edals . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 P edal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Mo del Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 SHAPE REGISTRA TION : : : : : : : : : : : : : : : : : : : : : : : : 16 3.1 Rigid Registration . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1 Rotation and T ranslation Estimation . . . . . . . . . . . 17 3.1.2 Scale Estimation . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Signed Distance Image . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 F ast Marc hing Lev el Set Metho d . . . . . . . . . . . . . . 19 3.2.2 T agging Algorithm . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Non-Rigid Registration . . . . . . . . . . . . . . . . . . . . . . . 23 3.3.1 Numerical Implemen tation . . . . . . . . . . . . . . . . . 24 3.3.2 Adaptiv e Time Step t . . . . . . . . . . . . . . . . . . . 25 vi PAGE 7 4 SHAPE CLASSIFICA TION : : : : : : : : : : : : : : : : : : : : : : : 26 4.1 Fisc her Linear Discriminan t . . . . . . . . . . . . . . . . . . . . . 27 4.1.1 Measures of Pro jected Data Class Separation . . . . . . . 27 4.1.2 Connection to theOptimalLinearBayesClassifer.....29 4.2 Supp ort V ector Mac hine (SVM) . . . . . . . . . . . . . . . . . . 30 4.3 Fisher Discriminan t Analysis with Kernels . . . . . . . . . . . . . 32 4.4 Kernel Fisher vs SVM . . . . . . . . . . . . . . . . . . . . . . . . 34 5 IMPLEMENT A TION RESUL TS : : : : : : : : : : : : : : : : : : : : : 36 5.1 P art I: Mo del Fitting . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 P art I I: Shap e Registration . . . . . . . . . . . . . . . . . . . . . 37 5.3 P art I I I: Shap e Classifcation . . . . . . . . . . . . . . . . . . . . 45 5.3.1 Health y V olun teers vs. P atien ts withEpilepsy.......46 5.3.2 LA TL vs RA TL . . . . . . . . . . . . . . . . . . . . . . . 50 5.4 In terpretation of theR esults.....................53 6 CONCLUSION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63 REFERENCES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 64 BIOGRAPHICAL SKETCH : : : : : : : : : : : : : : : : : : : : : : : : : : : : 68 vii PAGE 8 LIST OF T ABLES 5.1 Con trols vs Rest, Fv ec: V olume (Con trols=24, Rest=25) . . . . . . . 49 5.2 Con trols vs Rest, Fv ec: Sign of Displacemen t (Con trols=24, Rest=25) 49 5.3 Con trols vs Rest, Fv ec: direction v ector (Con trols=24, Rest=25) . . . 49 5.4 LA TL vs RA TL, Fv ec: V olume (LA TL=11, RA TL=14) . . . . . . . . 52 5.5 LA TL vs RA TL, Fv ec: Sign of displacemen t (LA TL=11, RA TL=14) . 52 5.6 LA TL vs RA TL, Fv ec: Direction v ector (LA TL=11, RA TL=14) . . . 53 viii PAGE 9 LIST OF FIGURES 2.1 f is on the p edal curv e of with resp ect to the p edal p oin t p . . . . . 12 2.2 Pro cess of generating a snak e p edal with an ellipse as an generator . . 13 3.1 Expanding 2D fron t at time T=0 and T=1 . . . . . . . . . . . . . . . 19 3.2 Stencil for the fast marc hing neigh b orho o d . . . . . . . . . . . . . . . 21 3.3 Up date pro cedure for F ast Marc hing Metho d . . . . . . . . . . . . . . 22 5.1 Mo del Fitting using Snak e P edal. First ro w: MRI scan of a health y v olun teer; p oin ts placed b y a neuroscien tist on a slice (in red) and initialized snak e (left), ftted mo del (cen ter), the particular slice sho wing ftted mo del. Second ro w: MRI scan of a patien t with left medial temp oral lob e fo cus. Third ro w: MRI scan of a patien t with righ t medial temp oral lob e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2 Rigid registration of left(in red) and righ t(in blue) hipp o campii using ICP algorithm and scaling. First ro w: P oin t sets for a health y v olunteer; b efore registration (left), after registration (righ t). Second ro w: P oin t sets for a patien t with left medial temp oral lob e fo cus. Third ro w: patien t with righ t medial temp oral lob e fo cus. . . . . . . . . . . 40 ix PAGE 10 5.3 P oin t set o v erla y ed on the zero-set obtained from the corresp onding signed distance image. First ro w: Health y v olun teer; left hipp o campus (left), righ t hipp o campus (righ t). Second Ro w: P atien t with left medial temp oral lob e fo cus. Third Ro w: P atien t with righ t medial temp oral lob e fo cus. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.4 Non-rigid registration of the left and righ t hipp o campii for a health y v olun teer using lev el set form ulation. First ro w: After rigid registration. Second ro w: After non-rigid registration. . . . . . . . . . . . . 42 5.5 Non-rigid registration of the left and righ t hipp o campii for a patien t with left medial temp oral lob e fo cus using lev el set form ulation. First ro w: After rigid registration. Second ro w: After non-rigid registration. 43 5.6 Non-rigid registration of the left and righ t hipp o campii for a patien t with righ t medial temp oral lob e fo cus using lev el set form ulation. First ro w: After rigid registration. Second ro w: After non-rigid registration. 44 5.7 Classifcation results for con trols (red) vs rest (blue) using v olume based feature. First ro w: scatter plot of feature v ector (left), classifcation using linear fsher (righ t). Second ro w: con tour plot with p olynomial basis(left), classifcation using p olynomial basis of degree 2 (righ t). Third ro w: con tour plot using RBF (radius = 0.05) (left), classifcation using RBF (righ t). . . . . . . . . . . . . . . . . . . . . 47 5.8 Classifcation results for con trols (red) vs rest (blue) using sign of displacemen t as feature: linear Fisher (left), k ernel Fisher with P oly . basis (degree=2) (cen ter), k ernel Fisher with radial basis (r = 1000) (righ t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.9 Classifcation results for con trols (red) vs rest (blue) using direction v ector as feature: linear Fisher (left), k ernel Fisher with P oly . basis (degree=2) (cen ter), k ernel Fisher with radial basis (r = 1000) (righ t). 48 x PAGE 11 5.10 Classifcation results for LA TL (red) vs RA TL (blue) using v olume based feature. First ro w: (left) scatter plot of feature v ector, (righ t) classifcation using linear fsher. Second ro w: (left) con tour plot with p olynomial basis, (righ t) classifcation using p olynomial basis of degree 2. Third ro w: (left) con tour plot using RBF (radius = 0.5), (righ t) classifcation using RBF. . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.11 Classifcation results for LA TL (red) vs RA TL (blue) using sign of displacemen t feature; (left) classifcation using linear fsher, (cen ter) classifcation using p olynomial basis of degree 2, (righ t) classifcation using RBF (r=900). . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.12 Classifcation results for LA TL (red) vs RA TL (blue) using direction v ector as feature v ector; (left) classifcation using linear fsher, classifcation using P oly . basis (d=2) (cen ter), (righ t) classifcation using RBF (r=900). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.13 Examples sho wing the colormap corresp onding to the sign of displacemen t as the feature v ec for the 21 x 40 mesh for health y v olun teers . 54 5.14 Examples sho wing the colormap corresp onding to the sign of displacemen t as the feature v ec for the 21 x 40 mesh for patien ts with left medial temp oral lob e fo cus . . . . . . . . . . . . . . . . . . . . . . . 54 5.15 Examples sho wing the colormap corresp onding to the sign of displacemen t as the feature v ec for the 21 x 40 mesh for patien ts with righ t medial temp oral lob e fo cus . . . . . . . . . . . . . . . . . . . . . . . 55 5.16 PCA on sign of displacemen t: k ernel Fisher with d=1 (left), k ernel Fisher with d=2 (cen ter) and k ernel Fisher with RBF (righ t) . . . . 57 5.17 Examples sho wing the colormap corresp onding to the direction v ector as the feature v ec for the 21 x 40 mesh for health y v olun teers . . . . 58 xi PAGE 12 5.18 Examples sho wing the colormap corresp onding to the direction v ector as the feature v ec for the 21 x 40 mesh for patien ts with left medial temp oral lob e fo cus . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.19 Examples sho wing the colormap corresp onding to the direction v ector as the feature v ec for the 21 x 40 mesh for patien ts with righ t medial temp oral lob e fo cus . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.20 PCA on direction v ector: k ernel Fisher with d=1 (left), k ernel Fisher with d=2 (cen ter) and k ernel Fisher with RBF (righ t) . . . . . . . . 61 xii PAGE 13 Abstract of Thesis 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 Master of Science SHAPE BASED SUPER VISED CLASSIFICA TION: APPLICA TION TO EPILEPSY By Neeti V ohra August 2002 Chairman: Dr. B.C. V em uri Ma jor Departmen t: Electrical and Computer Engineering Statistical analysis of the shap e deformations, suc h as those lik ely to o ccur in epilepsy and other neurological disorders, necessitates b oth global and lo cal parameter based c haracterization of the ob ject under study . The most p opular metho d to ac hiev e the same has b een size and v olume based analysis. Ho w ev er, this approac h captures only one of the asp ects necessary for complete c haracterization while shap e based analysis giv es m uc h more information that can b e com bined with the former to help understand the anatomical structures b etter. In this thesis, w e presen t an automatic tec hnique to distinguish b et w een health y v olun teers and t w o classes of patien ts of epilepsy , those with a righ t and those with a left medial temp oral lob e fo cus (RA TL and LA TL), as v alidated b y clinical consensus and subsequen t surgery . The sc heme also indicates the hemispheric lo cation of an epileptic fo cus in the patien ts. The k ey feature is the use of shap e-based features as opp osed to v olume information to ac hiev e the task. Since the data set considered is small, the analysis is done using linear and non-linear Fisher (based on k ernel metho ds), whic h mak es the pro cess xiii PAGE 14 computationally fast and meaningful. Also w e hop e that shap e can pro vide enough information to distinguish b et w een the dieren t classes and hence they can b e linearly separated either in the input space or pro jected space (k ernel feature space). W e demonstrate a high accuracy in distinguishing b et w een con trols and patien ts of epilepsy using shap e in comparison to v olume features. W e further sho w a reasonable impro v emen t for shap e based recognition among patien ts of pathology in con trast to v olume based analysis. xiv PAGE 15 CHAPTER 1 INTR ODUCTION Computational shap e analysis has b ecome an in tegral part of computer vision and computer graphics. The pro cess can b e broadly categorized in to three classes, namely shap e r e c overy , shap e tr ansformation and shap e classic ation . The task of shap e reco v ery in v olv es segmen ting a region of in terest in a giv en image whic h ma y b e done b y a h uman op erator or b y template matc hing. The main requiremen t is that the shap e mo del b e capable of represen ting ne details, necessitating a large n um b er of degrees of freedom. Shap e transformation deals with the normalization op erations required for comparison b et w een t w o shap es. The normalization ma y b e in terms of scale, translation and rotation. Shap e w arping, registration and morphing are common examples of this pro cess. Finally , classication algorithms are applied in order to assign a class to eac h considered shap e. This requires a compact represen tation of the shap e mo del for easy storage and retriev al. Recen tly , shap e analysis has found an imp ortan t application in medical image analysis. Tw o-dimensional and threedimensional shap e analysis tec hniques ha v e b een used in X-ra y , magnetic resonance (MR), computer tomograph y (CT) and ultrasound images to segmen t, register, trac k and recognize a v ariet y of anatomic structures including brain, heart and cerebral. More emphasis is put on automatic metho ds as they can p oten tially lead to more consisten t quan tication. Driving applications of automated segmen tation and brain morphometry for sc hizophrenia, m ultiple sclerosis and epilepsy and non-rigid registration for trac king deformations ha v e b een the fo cus of man y researc hers in recen t y ears in the computer vision comm unit y . 1 PAGE 16 2 1.1 Problem Denition Statistical analysis of the shap e deformations, suc h as those lik ely to o ccur in epilepsy and other neurological disorders, necessitate b oth global and lo cal parameterbased c haracterization of the ob ject under study . The most p opular sc heme to ac hiev e the same has b een size-based and v olume-based analysis. Ho w ev er, this approac h captures only one of the asp ects necessary for complete c haracterization while shap e based analysis giv es m uc h more information, whic h can b e com bined with the former to help understand the anatomical structures b etter. In this thesis, w e are concen trating on studying the shap e dierences in the hipp o campal shap es of t w o classes of patien ts in epilepsy , those with a righ t and those with a left medial temp oral lob e fo cus (RA TL and LA TL), as v alidated b y clinical consensus and subsequen t surgery and a set of age and sex matc hed health y v olun teers (con trols). The aim is to dev elop an automatic diagnostic tec hnique whic h can distinguish b et w een con trols and patien ts with epilepsy and can indicate the hemispheric lo cation of an epileptic fo cus in the patien ts. In the remainder of this c hapter, w e will rst review related shap e analysis tec hniques in literature with an emphasis on the metho ds used in shap e registration and shap e classication. This is follo w ed b y a summary of the main con tributions of this thesis. Finally , the organization of the thesis is presen ted. 1.2 Literature Review Numerous shap e analysis sc hemes can b e found in the literature of computer vision and computer graphics. The follo wing sub-sections giv e a brief discussion of the metho ds uses in the three broad categories of shap e analysis. PAGE 17 3 1.2.1 Shap e Reco v ery Use of lo cal and global parameters for represen ting shap e has b een the fo cus of researc h in 3D mo deling. More recen tly , a new class of h ybrid mo dels com bining b oth ends is coming in to fo cus. T erzop oulos and Metaxas [48] prop osed a h ybrid mo del based on deformable sup erquadrics. This mo del com bines the tigh tly constrained global parameters of a con v en tional sup erellipsoid with the rexibilit y of a mem brane spline. This approac h allo ws a smo oth transition b et w een the t w o ends of vision o ccupied b y lump ed and distributed mo dels. Since then, man y mo dications/enhancemen ts of the basic mo del ha v e b een prop osed in literature [53, 31]. Bardinet et al. [4], Heb ert et al. [23] and Sederb erg and P arry [43] prop osed sup erimp osing free form deformations (FFD) on a core sup erquadric whic h allo ws represen tation of a v ariet y of global shap es with lo cal detail. The idea is to enclose the sup erquadric in a \rubb er b o x" whic h has a small n um b er of con trol p oin t. The sup erquadric is deformed b y applying deformations on this \rubb er b o x" and they , in turn, c haracterize the lo cal detail. Th us, relativ ely few er parameters are used to represen t lo cal details. F or global deformations suc h as b ending, tap ering and t wisting, sp ecial parameters are required. These parameters, though small in n um b er, result in a highly non-linear mo del equation and can ha v e a destabilizing eect on the n umerical asp ects of the mo del tting algorithm emplo y ed. A no v el solid shap e mo deling sc heme w as prop osed b y O'Donnel et al. [35 ]. The mo del includes built-in osets from a core base mo del whic h help to form an exp ected mo del shap e and scales the base mo del in case of large global deformations. Displacemen t v ectors attac hed to the base mo del allo w the lo cal deformations o v er the scaled-oset mo del. Using a dynamic framew ork as in [48 ], a mo del can b e tted to a giv en data. The sc heme is suitable for tasks suc h as cardiac motion reco v ery from tagged MR images. PAGE 18 4 1.2.2 Shap e Registration Shap e based registration metho ds can b e broadly categorized in to t w o classes namely , feature-based and direct metho ds. F eature-based metho ds require detection of features/surface/con tours in the images and hence, the accuracy of the metho d dep ends on the accuracy of the feature detector. This, in turn, also increases the time required to detect the features b efore the actual registration can tak e place. The alternativ e class of registration sc hemes uses the direct approac h to determine the registration transformation. The en tire pro cess is p erformed on the image data and the prepro cessing of feature extraction is completely eliminated. In this section w e will briery review some of the direct metho ds as our approac h falls in to this category . Okuomi and Kanade [36] prop osed a lo cally adaptiv e correlation windo w based sc heme whic h can ac hiev e high accuracy . Ho w ev er, Szeliski and Coughlan [47 ] sho w ed that this sc heme is computationally exp ensiv e. A more general approac h is using an optical ro w form ulation, where the problem of registration is seen as computing the ro w b et w een the data sets. Numerous tec hniques can b e found in the literature based on this approac h [24 , 6, 14, 27, 8 ]. An in teresting implemen tation has recen tly b een presen ted b y V em uri et al. [52] for registering in tra-mo dal data sets. They use a hierarc hical optical ro w motion mo del that allo ws for b oth lo cal and global ro w estimation. The ro w eld is represen ted b y a B-spline basis whic h implicitly incorp orates smo othness constrain ts on the eld. Using a mo died Newton iteration sc heme, the motion is computed b y minimizing the exp ectation of a sum of squared dierences energy function. The Newton metho d w as mo died using the idea of precomputing the Hessian of the energy function at the optim um without explicitly kno wing the optim um. This has b een used for b oth lo cal and global motion estimation. Ho w ev er, this tec hnique cannot handle v ery large deformations due to the smo othness assumption on the ro w v ectors. PAGE 19 5 Another p opular approac h is based on the idea of maximizing m utual information b et w een the mo del and the image [55, 57, 10] using the statistical analog of the gradien t decen t metho d in W ells et al. [57]. The main computational task in this approac h is the estimation of the probabilit y densit y functions and their deriv ativ es from the giv en data sets. This ma y not b e meaningful if appropriate restrictiv e assumptions are not made. Most of the literature based on this metho d has its fo cus on global registration. Recen tly , Mey er et al. [32] rep orted b oth global and lo cal transformations. Though, the rep orted results are quite impressiv e in nature the main dra wbac k is high computational complexit y whic h is an activ e area of researc h. Ba jscy and Ko v acic [3] prop osed a non-rigid registration sc heme using v olumetric deformations based on elasticit y theory of solids. F urther, Christensen et al. [9] in tro duced a \ruid registration" approac h, where the registration is mo deled using a viscous ro w mo del expressed as a partial dieren tial equation (PDE). An in teresting demon-based approac h w as prop osed b y Thirion [49]. This can b e view ed as the v ariation of the basic ruid-based registration, whereb y the elastic lter has b een replaced b y a Gaussian. All these metho ds suer from high computational requiremen ts for 2D and 3D data set registration. 1.2.3 Shap e Classication The problem of shap e classication can b e addressed in t w o w a ys. Giv en an input image, decide whether it b elongs to some sp ecied class or dene or iden tify the in v olv ed classes in a p opulation of previously unclassied shap es. The former tec hnique is called sup ervise d classic ation and the later is called unsup ervise d classic ation . Both the sc hemes in v olv e comparing shap es, whic h in man y situations, is done b y matc hing sp ecic corresp onding p oin ts of them or using some features PAGE 20 6 from them. The classication sc hemes can b e ha v e a line ar or non-line ar discriminan t function. Using these sc hemes, the problem of statistical analysis of anatomical shap e dierences b et w een dieren t groups can b e reduced to a classication problem. Gerig et al. [17] prop osed the use of b oth shap e-based features and v olume measuremen ts to detect group dierences in hipp o campal shap es in sc hizophrenia. The lo cal deformations are accoun ted b y ripping one ob ject across the mid-sagittal plane, aligning the reference and the mirrored ob ject using the rst ellipsoid and b y calculating the MSD b et w een the t w o surfaces. The dierence b et w een the t w o classes is c haracterized b y using a supp ort v ector mac hine and the p erformance of classications is ev aluated using lea v e-one-out tests. The o v erall p erformance using b oth global and lo cal features is sho wn to b e 87% as compared to 73% b y using only v olume analysis. F rom the results rep orted, it can b e concluded that shap e alone could not capture the class dierences whic h ma y b e b ecause of w eak shap e features or the nature of the groups under study is suc h that shap e alone cannot represen t the en tire class c haracter. Csernansky et al. [12] used high dimensional transformations of a brain template to compare the hipp o campal v olume and shap e c haracteristics in sc hizophrenia and con trol sub jects. The transformations are carried out in t w o steps namely , course registration using previously placed landmarks follo w ed b y lo cal registration using ruid transformation. Optimal shap e represen tations are obtained b y computing the transformation v ector elds from the triangulated graph of p oin ts sup erimp osed on the hipp o campus surface. Linear discriminan t analysis is then used for classication based on the rst six v ectors with the largest eigen v alues. A more optimal solution is sho wn to b e obtained b y using a step-wise pro cedure. Based on b oth the metho ds log lik eliho o d ratios w ere calculated for all the sub jects and Wilk's Lam b da w as used to test the statistical signicance of group dierences. On the other hand, v olume PAGE 21 7 estimation is done b y measuring the v olume enclosed b y the transformed hipp o campal surface. This w as then analyzed using a t w o-w a y , rep eated measures ANO V A, with diagnostic group and hemisphere as the factors. Ho w ev er, the sc heme suers from high time complexit y due to use of ruid transformation to determine the lo cal v ector displacemen ts. An in teresting implemen tation of the unsup ervised clustering has b een prop osed b y Duta et al. [15 ]. The idea is to automatically cluster the shap e training set according to the PSS (Pr o crustes sum of squar es) distance in the original shap e space and then p erform Pro crustes analysis on eac h cluster for its protot yp e and information ab out shap e v ariation. The mean shap es obtained can then b e used to detect ob ject instances in the new image. The sc heme has the scop e of impro v emen t b y using an incremen tal learning pro cess and needs to b e tested for 3D anatomic structures. Recen tly , Golland et al. [18, 19, 20] ha v e in v estigated the use of a Supp ort V ector Mac hine for shap e-based classication in anatomical structures. In Golland et al. [19], distance transforms w ere b eing used as the shap e descriptor and shap e dierences are learn t in the original high dimensional space using Gaussian k ernel while con trolling the capacit y (generalization error) of the classier. The results of the sc heme are demonstrated b y nding the classication b et w een sc hizophrenia patien ts and normal con trols and an impro v emen t from 63% to 73% has b een sho wn as compared to using only v olume information. The tec hnique can b e further impro v ed b y com bining the global and lo cal features for classication and incorp orating in v arian ts in to the learning pro cess. 1.3 Con tributions In this thesis, w e presen t the application of the k ernel Fisher algorithm in the statistical analysis of shap e deformations that migh t indicate the hemispheric lo cation of an epileptic fo cus. The scans of t w o classes of patien ts with epilepsy , PAGE 22 8 those with a righ t and those with a left medial temp oral lob e fo cus (RA TL and LA TL), as v alidated b y clinical consensus and subsequen t surgery , are compared to a set of age and sex matc hed health y v olun teers. The k ey feature of the w ork is the use of shap e based features as opp osed to v olume based analysis in ac hieving the task. The c hoice of k ernel Fisher as the classier has b een motiv ated from the fact that giv en go o d features, it is p ossible to separate the classes linearly either in the input space of the giv en feature v ector or a v ery high or innite-dimensional space. The training algorithm used in this w ork do es not require non-linear optimization unlik e SVM and hence is simple to implemen t and computationally more ecien t. Note that the optimization that the SVM tries to solv e is a quadratic programming problem with constrain ts and is kno wn to b e NP-complete. Kernel Fisher has sho wn results comparable to SVM in v arious other applications [33, 34 ]. The complete analysis includes shap e mo deling, shap e registration and shap e classication. Quan titativ e and qualitativ e results of the implemen tation are sho wn in Chapter 5. 1.4 Thesis Organization The rest of the thesis is organized as follo ws. Chapter 2 discusses the shap e reco v ery sc heme used for represen tation of global and lo cal c haracteristics of hipp o campal shap es in health y v olun teers/patien ts. The sc heme has b een prop osed b y V em uri and Guo [51]. Chapter 3 rst explains the metho d for rigid registration of t w o p oin t sets with equal cardinalit y based on the algorithm b y Besl and McKa y [7]. This is follo w ed b y the discussion of F ast Marc hing Metho d [44, 1 ] used to obtain the signed-distance transform for eac h input p oin t set. Finally , non-rigid registration using a lev el-set form ulation [54 ] is presen ted. Chapter 4 discusses the classication sc hemes used in the statistical analysis of the data whic h included Linear Fisher and Kernel Fisher algorithms. The c hapter PAGE 23 9 also discusses the Supp ort V ector Mac hine as a non-linear classier follo w ed b y a discussion v alidating the use of k ernel Fisher o v er the SVM as the non-linear classier. Chapter 5 presen ts the implemen tation results. First w e sho w the mo del tting exp erimen ts using snak e p edal on the MRI scans of hipp o campal shap es on health y v olun teers/patien ts. This is follo w ed b y results obtained b y rigid and nonrigid shap e registration. Finally w e sho w the p erformance of linear and non-linear Fisher with shap e and v olume based features. The classication problem is done in t w o stages: b et w een con trols and patien ts of epilepsy and then determining the hemispheric lo cation of epileptic fo cus in patien ts. Finally , Chapter 6 presen ts the conclusions dra wn. PAGE 24 CHAPTER 2 SHAPE RECO VER Y The task of shap e reco v ery requires the segmen tation of a region of in terest in a giv en image whic h ma y b e done b y a h uman op erator or b y template matc hing. Shap e mo deling, as an imp ortan t constituen t of computer vision and computer graphics, oers sophisticated tec hniques for shap e represen tation, shap e reconstruction and recognition. The fundamen tal requiremen t in these shap e mo dels is the capabilit y to represen t fne details. Most mo deling sc hemes fall in to t w o ma jor sc hemes, namely , activ e and passiv e mo dels. Activ e mo dels are distributed parameter mo dels and ha v e broad geometric co v erage. Hence, they are go o d for shap e reco v ery and reconstruction but not w ell suited for recognition b ecause of the large n um b er of parameters required. T ypical examples of this class include snak es [25 ]. P assiv e mo dels, on the other hand, are lump ed parameter mo dels suc h as spheres and cylinders [5, 45]. They are go o d for recognition with less parameters but not for represen ting detailed shap es. The gap b et w een the t w o classes of mo dels has motiv ated for the dev elopmen t of h ybrid mo dels. These mo dels represen t b oth ends of the shap e description with an accepted compromise b et w een the t w o represen tations. Ho w ev er, they also in tro duce a lot of non-linearities for represen ting global deformations. This, in turn, aects the stabilit y of the mo del ftting algorithms to image data. In this c hapter, w e will discuss the compact and v ersatile shap e mo deling sc heme with ph ysics-based con trol for shap e design and shap e reco v ery from image data prop osed b y V em uri et al.[51]thathasbeenusedformodelfttinginour thesis. The metho d uses a com bination of a global and lo cal sc heme for mo del ftting whic h is quite stable and computationally fast. 10 PAGE 25 11 2.1 Ov erview of Mo deling Sc heme using Snak e P edals V em uri and Guo et al. [51 ] prop ose a mo deling sc heme that represen ts shap es with p edal curv es and surfaces. P edal curv es/surfaces are defned as the lo ci of the fo ot of the p erp endiculars to the tangen ts of a fxed curv e/surface from a fxed p oin t called the p edal p oin t [51 ]. A large class of shap es can b e syn thesized b y v arying the lo cation of the p edal p oin t. F urther, ph ysics based con trol is in tro duced b y using a snak e to represen t the p osition of the v arying p edal p oin t. Th us the mo del is called \snak e p edal". The mo del allo ws the represen tation of global deformation suc h as tap ering and b ending without in tro ducing additional parameters. Using an y globally parameterized shap e to represen t the generator, ob ject of v arying top ology can b e syn thesized. 2.1.1 P edal Curv es The p edal curv e of a regular curv e : ( c; d ) ! < 2 with resp ect to a fxed p oin t p 2 < 2 is giv en b y: p edal ( t ) = p + ( ( t ) )Tj/T1_2 11.9552 Tf11.9894 0 Td(p ) : J 0 ( t ) k J 0 ( t ) k 2 J 0 ( t ) (2.1) where t is the domain parameter, 0 ( t ) is the tangen t line of the plane curv e ( t ) and J : < 2 ! < 2 is a linear map giv en b y J [ x ; y ] = [ )Tj/T1_1 11.9552 Tf9.2294 0 Td(y ; x ]. The p edal curv e is capable of exhibiting b oth global and lo cal deformations. Hence, b y mo ving the p edal p oin t a v ariet y of top ologies can b e syn thesized. The curv e ( t ) is called the gener ator for the p edal curv e f ( t ) and the pro cess of generating p edal curv e is called as the p e daling op er ation . Figure (2.1.1) sho ws f as the p edal curv e of with resp ect to the fxed p edal p oin t p and at a particular t, ( t ) = g and f ( t ) = f . 2.1.2 Snak e P edals Represen ting the p edal p oin t b y another curv e p ( t ) and applying the p edaling op eration to eac h p oin t on the generator i = ( t i ) with resp ect to the corresp onding PAGE 26 12 Figure 2.1: f is on the p edal curv e of with resp ect to the p edal p oin t p p edal p oin t p i = p ( t i ), more general shap es can b e syn thesized. The curv e generated x ( t ) is called a snake p e dal . Th us, equation (2 : 1) b ecomes: x ( t ) = p ( t ) + ( ( t ) p ( t )) J 0 ( t ) k J 0 ( t ) k 2 J 0 ( t ) (2.2) The corresp ondence b et w een the curv es and p can b e established b y ordered sampling of the t w o curv es. This can b e done b y initializing the curv e p ( t ) to b e concentric with the generator and for eac h p oin t ( t i ), dra w a straigh t line from the cen ter of the generator O to the p oin t ( t i ), and denote b y p ( t i ) the in tersection of the line O ( t i ) and the curv e p ( t ). The generator can b e either a parameterized or an implicit function represen ting a curv e. F or a generator as an ellipse, ( t ) = cos sin sin cos a 1 cos t a 2 sin t + m 1 m 2 (2.3) where a 1 , a 2 are asp ect ratio parameters, is the rotation angle b et w een the w orld co ordinates and the ob ject cen tered co ordinated and m = ( m 1 ; m 2 ) T is the centroid of the generator in the w orld co ordinates. Collecting all in to a v ector g = ( a 1 ; a 2 ; ; m 1 ; m 2 ) T , giv es the glob al shap e p ar ameter ve ctor . PAGE 27 13 Figure 2.2: Pro cess of generating a snak e p edal with an ellipse as an generator The term J 0 ( t ) for an ellipse can b e expressed as: J 0 ( t ) = n 1 n 2 = )Tj/T1_2 11.9552 Tf9.2294 0 Td(a sin t sin )Tj/T1_2 11.9552 Tf11.8694 0 Td(b cos t cos )Tj/T1_2 11.9552 Tf9.2294 0 Td(a sin t cos + b cos t sin (2.4) It can b e seen that the k J 0 k 2 is indep enden t of the rotation angle . The p edal p oin t can b elong to a curv e whic h ma y not b e parameterized in nature. Sampling the curv e at M p oin ts so that there is one-to-one corresp ondence with the generator, giv es the lo c al shap e p ar ameter ve ctor . Th us, a snak e p edal curv e can b e describ ed completely b y global and lo cal parameters. Mo difying equation (2 : 2) b y subtracting the second term from the frst term allo ws the syn thesis of larger class of shap es as sho wn in V em uri and Guo [51]. The snak e p edal curv e allo ws for more lo cal deformations including shrink age and expansion con trolled b y the lo cation of the snak e. Th us the equation (2 : 2) b ecomes: x ( t ) = p ( t ) )Tj/T1_0 11.9552 Tf13.2468 8.04 Td(( ( t ) )Tj/T1_8 11.9552 Tf11.9894 0 Td(p ( t )) J 0 ( t ) k J 0 ( t ) k 2 J 0 ( t ) (2.5) PAGE 28 14 2.1.3 P edal Surfaces The p edal surface of a parameterized surface or a patc h : U ! < 3 with resp ect to a p oin t p 2 < 3 is defned b y x ( u; v ) = p + ( ( u ; v ) )Tj/T1_1 11.9552 Tf11.9894 0 Td(p ) ( u v ) k u v k 2 u v (2.6) where U is an op en subset of < 2 , ( u; v ) is a parameterized generator surface with parameters u and v and u and v are tangen t v ectors in the u and v directions [51 ]. With an ellipsoid generator, w e get ( u; v ) = R 2 4 a 1 cos u cos v a 2 cos u sin v a 3 sin v 3 5 + 2 4 m 1 m 2 m 3 3 5 (2.7) where a 1 , a 2 and a 3 are asp ect ratio parameters, m = ( m 1 ; m 2 ; m 3 ) denote the cen troid of the generator in the w orld co ordinates, R is the rotation matrix b et w een the w orld co ordinates and the ob ject cen tered co ordinates represen ted b y quaternions ( q 1 ; q 2 ; q 3 ; q 4 ) T . Collecting these parameters in to g = ( a; b; c; m 1 ; m 2 ; m 3 ; q 1 ; q 2 ; q 3 ; q 4 ) T giv es the global shap e parameter v ector in 3D. The lo cal shap e parameter v ector is obtained in a similar w a y as explained for 2D. 2.2 Mo del Fitting The snak e p edal mo del giv es an ecien t tec hnique for ftting a mo del to a giv en data set in 2D/3D. This is done b y minimizing the sum of the squared distances from eac h data p oin t D i to the corresp onding closest p oin t x i on the snak e p edal surface. Let f p i g M i =1 b e the set of discrete p oin ts on the parameterized snak e surface p ( u ; v ), f x i g M i =1 b e the corresp onding p oin t on the snak e p edal surface and D = f D i g m i = 1 b e the giv en data set, then the quan tit y to b e minimized is giv en b y E D ( D ; g ; p ) = 1 2 m X i =1 f ( D i )Tj/T1_2 11.9552 Tf11.8694 0 Td(x i ) 2 (2.8) where f relates the uncertain t y in the measuremen t of the data. The minimization is p erformed b y using a fast iterativ e algorithm outlined in V em uri and Guo [51 ]. The PAGE 29 15 tec hnique uses the Lev en b erg-Marquardt (LM) metho d [38] to estimate the global parameters and the Alternating Direction Implicit (ADI) metho d [30 , 29, 16] for lo cal parameters. PAGE 30 CHAPTER 3 SHAPE REGISTRA TION A general problem denition for shap e registration can b e p osed as follo ws. Giv en a pair of shap es S 1 (X) and S 2 (X'), where X 2 < 3 and X' 2 < 3 , estimate the optimal transformation T b et w een them suc h that the matc hing criteria dened on the shap es (for example distance metric) is minimized. Equiv alen tly min T M( S 1 (X), S 2 (T(X'))) where T is the unkno wn transformation. In general, this transformation T is b oth global and lo cal in nature. The global transformation deals with the estimation of the rigid parameters suc h as rotation, translation and scaling to align the co ordinate systems of the t w o shap es under consideration. On the other hand, the lo cal transformation deals with the problem of nding the displacemen t v ector for the corresp onding p oin ts in the source and target image individually . In this c hapter, w e will briery discuss the sc hemes used for rigid and non-rigid registration of the shap es under consideration. 3.1 Rigid Registration Giv en t w o digitized unxtured 3D data sets, the transformation parameter T that relates the co ordinate frames of the data under study b y rotation and translation can b e describ ed as follo ws: B = RA + V (3.1) where A is a 3D p oin t in the source co ordinate system, R is the rotation matrix, V is the translation matrix and B is the corresp onding 3D p oin t in the target co ordinate 16 PAGE 31 17 system. The shap es ma y also ha v e a global scaling factor in eac h dimension whic h is captured, in general, using an ane transformation. In the presen t study , the shap e under consideration, hipp o campus, in general has a natural scale factor b et w een the corresp onding left and righ t part. Also the v olume data for the left and righ t hipp o campii are estimated in dieren t co ordinate systems. This necessitates rigid registration under scaling, rotation and translation. The follo wing subsections summarize the tec hniques used for the estimation of these parameters. 3.1.1 Rotation and T ranslation Estimation T o estimate the rotation R and translation V matrices defned in (3.1) under the assumption that the cardinalit y of the t w o data sets is same, w e are using the Iterativ e Closest P oin t (ICP) algorithm prop osed b y Besl and McKa y[7].The metho d allo ws the determination of the equiv alence of the shap es b y mean-square distance metric and is quaternion-based [7]. The unit quaternion is a four v ector )Tj0.2294 0 Td(! q R = [ q 0 q 1 q 2 q 3 ] t , where q 0 0, and q 2 0 + q 2 1 + q 2 2 + q 2 3 = 1. Defning )Tj-0.0106 0 Td(! q T = [ q 4 q 5 q 6 ] t as the translation v ector, the complete registration v ector is giv en b y )Tj-0.0106 0 Td(! q = [ )Tj0.1094 0 Td(! q R j )Tj-0.0107 0 Td(! q T ] t [7]. Let P = f )Tj-0.0106 0 Td(! p i g b e a measured shap e p oin t set to b e aligned with a reference shap e p oin t set X = f )Tj-0.0106 0 Td(! x i g , where N x = N p . The mean square ob jectiv e function to b e minimized is giv en b y f ( )Tj-0.0106 0 Td(! q ) = 1 N p N p X i =1 k )Tj-0.0106 0 Td(! x i )Tj/T1_11 11.9552 Tf11.9894 0 Td(R ( )Tj0.1094 0 Td(! q R ) )Tj-0.0106 0 Td(! p i )Tj12 4.2 Td()Tj-0.0106 0 Td(! q T k 2 (3.2) Based on the pro cedures for fnding the closest p oin t and least squares quaternion op eration explained in [7], the ICP algorithm computes the registration v ector as follo ws: 1. Initialize P 0 = P , )Tj-0.0106 0 Td(! q 0 = [1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0] t and k = 0. 2. Compute the closest p oin ts: Y k = C ( P k , X ). PAGE 32 18 3. Compute the registration: ( )Tj-0.0106 0 Td(! q k , d k ) = Q ( P 0 , Y k ). 4. Apply the registration: P k +1 = )Tj-0.0106 0 Td(! q k ( P 0 ). 5. Rep eat steps 3 through 5 un til the c hange in mean square error falls b elo w a user defned threshold 0. 3.1.2 Scale Estimation Once the source and target shap es ha v e b een registered via rotation and translation, the next task is to eliminate the global scale factor b et w een them. This is done b y using the appro ximation of the shap e of the hipp o campus with the smallest ellipsoid that can enclose it. The pro cedure can b e summarized as follo ws: 1. Find the eigen v alues and eigen v ectors for the corresp onding data sets. 2. Find the ratio of the eigen v alues of the corresp onding axes in the new co ordinate system. 3. Pro ject the data sets in the new co ordinate system and scale the source using the ratios obtained in step 2. 4. Pro ject the scaled data set and the target data set bac k in to their original co ordinate system. 3.2 Signed Distance Image T o estimate the non-rigid registration b et w een the source and target shap es, w e are using the lev el-set form ulation prop osed b y V em uri et al.[54].Thedesired lev el-set form ulation is ac hiev ed b y using the F ast Marc hing Metho d (FMM) describ ed b y Sethian[44]andthetaggingalgorithm.Theideaistoderiveasigned distance image defned on a 3D lattice from the giv en p oin t set, with the p oin t set represen ted as the zero set of the image. PAGE 33 19 Figure 3.1: Expanding 2D fron t at time T=0 and T=1 3.2.1 F ast Marc hing Lev el Set Metho d Sethian[44]proposedafastmarchingmethodforsolvingtheEikonalequation of the form jr T j F = 1 (3.3) where for monotonically adv ancing fron ts, T is the crossing time and F is a function of p osition only . The ab o v e equation can b e solv ed b y using a fnite dierence op erator on a fxed cartesian grid. Using an up wind, viscosit y-solution appro ximation to the gradien t [44] giv es max( D )Tj/T1_10 7.9701 Tf6.5992 0 Td(x ij k T ; 0) 2 + min( D + x ij k T ; 0) + max( D )Tj/T1_10 7.9701 Tf6.5992 0 Td(y ij k T ; 0) 2 + min ( D + y ij k T ; 0) (3.4) +max( D )Tj/T1_10 7.9701 Tf6.5992 0 Td(z ij k T ; 0) 2 + min ( D + z ij k T ; 0) = 1 F ij k 2 A mo difed up wind sc heme, giv en in [39], results in a more con v enien t equation max( D )Tj/T1_10 7.9701 Tf6.5992 0 Td(x ij k T ; )Tj/T1_8 11.9552 Tf9.2294 0 Td(D + x ij k T ; 0) 2 + max( D )Tj/T1_10 7.9701 Tf6.5992 0 Td(y ij k T ; )Tj/T1_8 11.9552 Tf9.2294 0 Td(D + y ij k T ; 0) 2 +max( D )Tj/T1_10 7.9701 Tf6.5992 0 Td(z ij k T ; )Tj/T1_8 11.9552 Tf9.2294 0 Td(D + z ij k T ; 0) 2 = 1 F ij k 2 (3.5) Since the up wind dierence structure of equation (3 : 4) implies that the information propagates in one direction starting from a small v alue of T , Sethian [44] prop oses PAGE 34 20 to use a narro w band metho dology to solv e (3 : 5). An imp ortan t application of this metho d is the computation of distance elds for surfaces in 3D. This is p ossible b ecause if the fron t ev olv es at unit sp eed, the arriv al time corresp onds to the distance. Considering a lattice, the pro cess b egins with initializing single or more v o xels with the exact distance v alue. These v o xels are then fr ozen and the distances are computed for the neigh b ors using the equation (3 : 5). V o xels that ha v e computed distances but not y et frozen are called narr ow b and v o xels. All the remaining v o xels are called as far-away v o xels. A t eac h iteration, using the min-heap structure prop osed in [44] the narro w band v o xel with the smallest distance v alue is extracted. This v o xel is then frozen and the distance is computed for its neigh b ors whic h are either in a narro w band or a far-a w a y region using (3 : 5) follo w ed b y adding them to the narro w band. T o solv e equation (3 : 5) only frozen v o xels are used. This is done b y considering a stencil of grid p oin t and its six neigh b ors as sho wn in Fig. (3 : 2 : 1). On a grid with unit v o xel distance, considering eac h term of the form max( D x ij k T ; D + x ij k T ; 0) 2 = max( T i;j;k T i 1 ;j;k ; T i;j;k T i +1 ;j;k ; 0) 2 (3.6) it can b e seen that w e need to c ho ose the smaller of the t w o v alues T i 1 ;j;k and T i +1 ;j;k since with T i 1 ;j;k < T i +1 ;j;k ) T i;j;k T i 1 ;j;k > T i;j;k T i +1 ;j;k (3.7) Also if neither of T i 1 ;j;k and T i +1 ;j;k is frozen, then the term needs to b e dropp ed [1]. Th us assuming T i 1 ;j;k < T i +1 ;j;k , T i;j 1 ;k < T i;j +1 ;k and T i;j;k 1 < T i;j;k +1 , w e can rewrite equation (3 : 5) as ( T i;j;k T i 1 ;j;k ) 2 + ( T i;j;k T i;j 1 ;k ) 2 + ( T i;j;k T i;j;k 1 ) 2 = F 2 (3.8) In the case of t w o solutions, the larger v alue is accepted as the solution since w e are assuming that the unkno wn v alue is larger than the three kno wn v alues. PAGE 35 21 Figure 3.2: Stencil for the fast marc hing neigh b orho o d 3.2.2 T agging Algorithm The pro cess of the fast marc hing metho d giv es the unsigned distance image for a giv en p oin t set. This leads to the task of iden tifying the v o xels inside and outside the surface with the surface as the zero set, th us giving a signed distance function. This can b e done b y using the tagging algorithm under the assumption that the ob ject under consideration is w ell inside the lattice structure, has b een mark ed as the zero-set and is connected. Starting with one corner of the 3D lattice whic h is kno wn to b e outside the surface and hence can b e mark ed as p ositiv e (assuming outside is p ositiv e and inside is negativ e) the algorithm pro ceeds as follo ws: 1. Mark the starting v o xel as p ositiv e and iden tify its 26 neigh b ors. 2. Chec k if these neigh b ors are not on the zero-set, push them in to a stac k. 3. Extract the top elemen t of the stac k, mark it as p ositiv e and c hec k its neigh b ors no w. 4. Rep eat step 2 and 3 till the stac k is empt y . The v o xels left unmark ed can then b e tagged as negativ e. Th us the en tire 3D lattice with a signed distance v alue w.r.t to the surface. PAGE 36 22 Figure 3.3: Up date pro cedure for F ast Marc hing Metho d PAGE 37 23 3.3 Non-Rigid Registration The problem of fnding the non-rigid estimation can b e form ulated as a motion estimation task, in particular, estimation of the optical ro w b et w een the t w o giv en shap es. V em uri et al. [54] ha v e prop osed a lev el set form ulation to estimate the optical ro w, giv en b y )Tj-0.0106 0 Td(! V t = [ I 2 ( X ) )Tj/T1_3 11.9552 Tf11.9894 0 Td(I 1 ( )Tj-0.0106 0 Td(! V ( X ))] r I 1 ( )Tj-0.0106 0 Td(! V ( X )) k r I 1 ( )Tj-0.0106 0 Td(! V ( X )) k (3.9) w ith )Tj-0.0106 0 Td(! V ( X ; 0) = )Tj-0.0106 0 Td(! 0 where )Tj-0.0106 0 Td(! V = ( u; v ; w ) T is the displacemen t v ector at X and the op eration )Tj-0.0106 0 Td(! V ( X ) = ( x )Tj/T1_3 11.9552 Tf12.5894 0 Td(u; y )Tj/T1_3 11.9552 Tf12.5894 0 Td(v ; z )Tj/T1_3 11.9552 Tf12.5894 0 Td(w ). Since the gradien t is highly sensitiv e to the noise, I ( X , t ) is con v olv ed with the Gaussian k ernel b efore taking the gradien t. This leads to the follo wing mo difcation in the previous equation )Tj-0.0106 0 Td(! V t = [ I 2 ( X ) )Tj/T1_3 11.9552 Tf11.9894 0 Td(I 1 ( )Tj-0.0106 0 Td(! V ( X ))] r ( G I 1 ( )Tj-0.0106 0 Td(! V ( X )))) k r ( G I 1 ( )Tj-0.0106 0 Td(! V ( X ))) k (3.10) w ith )Tj-0.0106 0 Td(! V ( X ; 0) = )Tj-0.0106 0 Td(! 0 T o mak e the computation of the lev el-set motion stable in nature when, ( G I 1 ( )Tj-0.0106 0 Td(! V ( X ))) is near to zero, a stabilizing factor is added to previous equation )Tj-0.0106 0 Td(! V t = [ I 2 ( X ) )Tj/T1_3 11.9552 Tf11.9894 0 Td(I 1 ( )Tj-0.0106 0 Td(! V ( X ))] r ( G I 1 ( )Tj-0.0106 0 Td(! V ( X ))) k r ( G I 1 ( )Tj-0.0106 0 Td(! V ( X ))) k + (3.11) w ith )Tj-0.0106 0 Td(! V ( X ; 0) = )Tj-0.0106 0 Td(! 0 where is a small p ositiv e n um b er. F urther, a smo othness constrain t is added to the lev el-set motion mo del whic h mo difes the equation (3 : 11) as follo ws )Tj-0.0106 0 Td(! V t = [ I 2 ( X ) )Tj/T1_3 11.9552 Tf11.9894 0 Td(I 1 ( )Tj-0.0106 0 Td(! V ( X ))] r ( G I 1 ( )Tj-0.0106 0 Td(! V ( X ))) k r ( G I 1 ( )Tj-0.0106 0 Td(! V ( X ))) k + + 0 @ u v w 1 A (3.12) w ith )Tj-0.0106 0 Td(! V ( X ; 0) = )Tj-0.0106 0 Td(! 0 where denotes the Laplacian op erator. In the implemen tation, the smo othness constrain t is added externally b y con v olving the motion feld with a Gaussian flter after eac h iteration. PAGE 38 24 3.3.1 Numerical Implemen tation T o solv e (3.12), V em uri et al. [54] ha v e suggested the use of the minmo d fnite dierence sc heme b y Kimmel et al. [26 ] to ac hiev e the lev el-set based optical ro w. The minmo d function is defned as follo ws m ( x; y ) = sig n ( x )min ( j x j ; j y j ) if xy > 0 0 if xy 0 (3.13) Defning C ij k = G ( I 1 ) i )Tj/T1_5 7.9701 Tf6.5992 0 Td(u;j )Tj/T1_5 7.9701 Tf6.5992 0 Td(v ;k )Tj/T1_5 7.9701 Tf6.5992 0 Td(w , w e get ( C x ) ij k = m ( D + x C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -7.8 Td(x C ij k ) ( C y ) ij k = m ( D + y C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -7.92 Td(y C ij k ) (3.14) ( C z ) ij k = m ( D + z C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -7.92 Td(z C ij k ) where D + x , D )Tj/T1_5 7.9701 Tf-0.36 -7.32 Td(x , D + y , D )Tj/T1_5 7.9701 Tf-0.36 -7.32 Td(y , D + z and D )Tj/T1_5 7.9701 Tf-0.36 -7.32 Td(z are defned as follo ws D + x E ij k = E i +1 ;j;k )Tj/T1_2 11.9552 Tf11.8694 0 Td(E i;j;k x ; D )Tj/T1_5 7.9701 Tf-0.36 -7.8 Td(x E ij k = E i;j;k )Tj/T1_2 11.9552 Tf11.9894 0 Td(E i )Tj/T1_6 7.9701 Tf6.5992 0 Td(1 ;j;k x D + y E ij k = E i;j +1 ;k )Tj/T1_2 11.9552 Tf11.8694 0 Td(E i;j;k y ; D )Tj/T1_5 7.9701 Tf-0.36 -7.8 Td(y E ij k = E i;j;k )Tj/T1_2 11.9552 Tf11.9894 0 Td(E i;j )Tj/T1_6 7.9701 Tf6.5992 0 Td(1 ;k y (3.15) D + z E ij k = E i;j;k +1 )Tj/T1_2 11.9552 Tf11.8694 0 Td(E i;j;k z ; D )Tj/T1_5 7.9701 Tf-0.36 -7.92 Td(z E ij k = E i;j;k )Tj/T1_2 11.9552 Tf11.9894 0 Td(E i;j;k )Tj/T1_6 7.9701 Tf6.5992 0 Td(1 z Using these defnitions and a forw ard dierence in time, (3.12) can b e rewritten in a discretized form: 0 @ u s +1 ij k v s +1 ij k w s +1 ij k 1 A = 0 @ u s ij k v s ij k w s ij k 1 A + 0 @ u s ij k v s ij k w s ij k 1 A + (3.16) t ( I 2 ) ij k )Tj/T1_0 11.9552 Tf11.9894 0 Td(( I 1 ) i )Tj/T1_5 7.9701 Tf6.5992 0 Td(u s ij k ;j )Tj/T1_5 7.9701 Tf6.5992 0 Td(v s ij k ;k )Tj/T1_5 7.9701 Tf6.5992 0 Td(w s ij k q m 2 ( D + x C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -6.36 Td(x C ij k ) + m 2 ( D + y C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -6.36 Td(y C ij k ) + m 2 ( D + z C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -6.36 Td(z C ij k ) + 0 @ m ( D )Tj/T1_5 7.9701 Tf-0.36 -7.2 Td(x C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -7.2 Td(x C ij k ) m ( D + y C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -7.32 Td(y C ij k ) m ( D + z C ij k ; D )Tj/T1_5 7.9701 Tf-0.36 -7.32 Td(z C ij k ) 1 A w ith 0 @ u 0 ij k v 0 ij k w 0 ij k 1 A = )Tj-0.0106 0 Td(! 0 This fnite dierence sc heme is kno wn to preserv e lo cal min/max for the equation, ensuring stable solutions for the lev el set motion mo del. PAGE 39 25 3.3.2 Adaptiv e Time Step t T o stabilize the n umerical in tegration pro cess, Y e[56]proposetheuseof adaptiv e time step sc heme. Let I 2 ( X ) b e the target image, I ( X ; t ) b e the source image at time t and E ( X ) = G I ( X ). The time step t can b e calculated as follo ws t = 1 max fj H u j + j H v j + j H w jg if max ( j H u j + j H v j + j H w j ) 6= 0 k otherwise (3.17) where u = E x ; v = E y ; w = E z H u = u p u 2 + v 2 + w 2 + ( I 2 ( X ) )Tj/T1_9 11.9552 Tf11.9894 0 Td(I ( X ; t )) H v = v p u 2 + v 2 + w 2 + ( I 2 ( X ) )Tj/T1_9 11.9552 Tf11.9894 0 Td(I ( X ; t )) (3.18) H w = w p u 2 + v 2 + w 2 + ( I 2 ( X ) )Tj/T1_9 11.9552 Tf11.9894 0 Td(I ( X ; t )) k = small p ositiv e n um b er (3.19) and max( j H u j + j H v j + j H w j ) is computed in eac h iteration using the maxim um absolute v alue of H u ; H v and H w defned on the 3D lattice ( H u ; H v and H w is computed for eac h grid p oin t). PAGE 40 CHAPTER 4 SHAPE CLASSIFICA TION Shap e classication deals with the problem of assigning a class to eac h of the considered shap e. This in v olv es an imp ortan t step of selection of appropriate features to represen t the shap es under consideration. The c haracterization should b e suc h that that v alues of the features should b e similar for the ob jects in one class while v ery dieren t for the ob ject in dieren t categories. Their c hoice is problem and domain dep enden t and in an ideal case should b e easy to extract, in v arian t under certain geometrical transformations suc h as scaling and rotation and robust to noise. Giv en these features, the classication can then b e approac hed in t w o w a ys, namely sup ervise d and unsup ervise d . In sup ervised classication, an input ob ject is assigned to one of the predened protot yp es based on some distance measures (suc h as the Mahalanobis distance). This pro cess in v olv es t w o stages: learning, corresp onding to the stage where criterion and metho ds are tried on the protot yp es and recognition, corresp onding to classify new en tities using the trained system. On the other hand, unsup ervised classication in v olv es dening or iden tifying the in v olv ed classes in a p opulation of previously unclassied shap es. This sc heme is commonly kno wn as clustering . F urther, the discriminan t function in these classication sc hemes ma y b e linear or non-linear in nature. The classical approac h b egins with the optimal Ba y es classier b y assuming the normal distribution for the classes. In this c hapter w e will discuss Fisc her-based linear discriminan t analysis follo w ed b y non-linear discriminan t metho ds based on Mercer k ernels, namely Kernel Fisher [33 ] and Supp ort V ector Mac hines [50 , 41]. Finally , a discussion comparing the c haracteristics of k ernel Fisher 26 PAGE 41 27 and Supp ort V ector Mac hine is giv en along with the justication of using k ernel Fisher as the non-linear classier in our w ork. 4.1 Fisc her Linear Discriminan t Linear discriminan t functions can b e mo deled in the follo wing form g i ( x ) = w Ti x + w i 0 (4.1) where w i is the v ector of w eigh ts for the class i , w i 0 is the bias and g i ( x ) denotes the pro jection of the input v ector x on to the w eigh t v ector w i . The Fisher approac h is based on pro jecting d-dimensional data on to a line with the hop e that the pro jections are w ell separated b y class. Th us, the line is orien ted to maximize this class separation. F or the t w o class problem, let 1 = f x 11 ; x 12 ; : : : ; x 1n 1 g and 2 = f x 21 ; x 22 ; : : : ; x 2n 2 g b e the samples for the training set suc h that = f 1 ; 2 g and n 1 + n 2 = n . The feature v ector pro jections are giv en b y y i = w T x i = < w ; x i > i = 1 ; 2 ; : : : ; n (4.2) By constraining k w k = 1, eac h y i is the pro jection of x i on to a line in the direction of w . 4.1.1 Measures of Pro jected Data Class Separation One measure of separation of the pro jections is the square of the dierence of the means of the pro jections giv en b y j Y 1 Y 2 j 2 where Y i = E f y i j x i 2 w i g = E f w T x i j x i 2 w i g (4.3) This measure can b e sho wn to relate the 1 and 2 samples means through w : m i = 1 n i X x i 2 i x i (4.4) The mean of the pro jected data is giv en b y: m i = 1 n i X x i 2 i w T x i = 1 n i X y i 2 Y i y i PAGE 42 28 = w T 1 n i X x i 2 x i = w T m i (4.5) where m i is the sample mean of the v ectors in i . Therefore, the dierence of the pro jected means using sample data can b e giv en b y: j m i m 2 j = j w T ( m 1 m 2 ) j (4.6) T o ac hiev e w ell-separated class pro jections, v ariances of y i relativ e to the means m ust also b e considered. Th us the class separation measure can b e giv en b y: J ( w ) = ( Y 1 Y 2 ) 2 2 Y 1 + 2 Y 2 (4.7) In the case of sample data, it can b e expressed as J ( w ) = ( m 1 m 2 ) 2 s 21 + s 22 (4.8) where s 2i = X y 2 Y i ( y m i ) 2 is dened as the within-class scatter of the pro jected data. Dening a sc atter matrix S i as S i = X x 2 i ( x m i )( x m i ) T i = 1 ; 2 (4.9) and S W = S 1 + S 2 (4.10) the denominator of (4 : 8) can b e form ulated as s 21 + s 22 = w T S W w (4.11) Similarly , the n umerator of (4 : 8) can b e expressed in terms of the sample means as follo ws ( m 1 m 2 ) 2 = w T ( m 1 m 2 )( m 1 m 2 ) T w = w T S B w (4.12) where S B is the b et w een-class scatter matrix. Therefore, equation (4 : 8) b ecomes J ( w ) = w T S B w w T S W w (4.13) PAGE 43 29 F orming @ J =@ w = 0 leads to S W ^ w ( ^ w T S B ^ w )( ^ w T S W ^ w ) )Tj/T1_93 7.9701 Tf6.5992 0 Td(1 = S B ^ w (4.14) whic h is a generalized eigen v ector problem with ( ^ w T S B ^ w )( ^ w T S W ^ w ) )Tj/T1_93 7.9701 Tf6.5993 0 Td(1 = . Th us, w e seek solution to S W ^ w = S B ^ w (4.15) If S )Tj/T1_93 7.9701 Tf6.5992 0 Td(1 W exists, then the direction of ^ w is giv en b y ^ w = ( S )Tj/T1_93 7.9701 Tf6.5992 0 Td(1 W S B ) ^ w (4.16) whic h can b e found b y solving for an e-v ector of ( S )Tj/T1_93 7.9701 Tf6.5992 0 Td(1 W S B ). An alternate solution can b e found based on the fact that S B ^ w in (4 : 14) has direction m 1 )Tj/T1_90 11.9552 Tf12.9494 0 Td(m 2 , since ( m 1 )Tj/T1_90 11.9552 Tf11.8694 0 Td(m 2 )( m 1 )Tj/T1_90 11.9552 Tf11.9894 0 Td(m 2 ) T ^ w = ( m 1 )Tj/T1_90 11.9552 Tf11.8694 0 Td(m 2 ) k . Therefore, ^ w = S )Tj/T1_93 7.9701 Tf6.5992 0 Td(1 W ( m 1 )Tj/T1_90 11.9552 Tf11.8694 0 Td(m 2 ) (4.17) The decision function for a new pattern x can b e defned as follo ws g ( x ) = x 0 ^ w ; f ( x ) = sig n ( g ( x ) )Tj/T1_90 11.9552 Tf11.9894 0 Td(b ) (4.18) where b is a threshold whic h can b e determined exp erimen tally suc h that the training set error is minim um. 4.1.2 Connection to theOptimalLinearBayesClassifer The optimal Ba y es classifer assigns a class to an input pattern b y comparing the a p osteriori probabilities of all classes. Since a-p osteriori probabilities are usually unkno wn and need to b e estimated from a fnite sample, the task to fnd a closed form solution b ecomes dicult in nature. By assuming normal distributions for all classes, quadratic discriminan t analysis can b e ac hiev ed. F urther, b y assuming equal co v ariance structure for all classes, quadratic discriminan t analysis b ecomes PAGE 44 30 linear. Although the assumptions are not true in man y applications, Fisher linear discriminan t has pro v en v ery useful. Mik a et al.[33]mentionthatthisispossible b ecause the linear mo del is rather robust against noise and most lik ely will not o v erft. Ho w ev er, they also sa y that estimation of the scatter matrices is v ery crucial as they can b e highly biased. Th us, linear metho ds b eing to o limited motiv ate to deriv e more general class separabilit y . 4.2 Supp ort V ector Mac hine (SVM) SVM is a non-linear classifcation sc heme based on the idea of constructing a linear decision b oundary for the patterns in a high-dimensional space, th us implicitly giving a non-linear discriminan t in the input feature space. The mapping to the high dimensional space from the input space is ac hiev ed using Mercer Kernels [40]. The k ey feature of the metho d is to maximize the margin b et w een the nearest p ositiv e and negativ e samples (in k ernel feature space), whic h has sho wn go o d generalization p erformance in man y domains [22 ]. Giv en a t w o-class recognition problem, with l training set samples x 2 < d assigned a class +1/-1, the learning pro cess b egins with mapping the data set in to a v ery high or infnite dimensional space F using a mapping . W e need to fnd a h yp erplane in this new space suc h that w ( x ) + b = 0 (4.19) and the patterns can b e separated as follo ws y i (( w ( x ) + b ) (4.20) for some > 0. Also, w e can rescale ( w , b) suc h that min i =1 ;:::;l j ( w : ( x i )) + b j = 1 (4.21) PAGE 45 31 i.e. the p oin ts closest to the h yp erplane ha v e a distance of 1/ k w k . Then (4.20) b ecomes y i (( w ( x ) + b ) 1 (4.22) The ab o v e equation can b e solv ed b y minimizing k w k whic h will implicitly maximize the margin. In practice, a separating h yp erplane ma y not exit. This limitation can b e o v ercome b y in tro ducing slac k parameters [11], i 0 ; i = 1 ; : : : ; l (4.23) suc h that y i (( w ( x ) + b ) 1 i ; i = 1 ; : : : ; l (4.24) Using Lagrange m ultipliers i , and the Kuhn-T uc k er theorem of optimization theory [42], the solution can b e found to b e w = l X i =1 y i i ( x i ) (4.25) with the nonzero co ecien ts i corresp onding to examples ( ( x i ) ; y i ) meeting the constrain t (4.22). These ( x i ) are called the supp ort v ectors. The remaining examples ha v e i = 0 and th us automatically satisfy the constrain t (4.22). The co ecien ts i can b e solv ed b y maximizing W ( ) = l X i =1 i 1 2 l X i;j =1 i j y i y j ( ( x i ) ( x j )) (4.26) sub ject to 0 i C ; i = 1 ; : : : ; l ; and l X i =1 i y i = 0 (4.27) where C is the cost for the errors. The h yp erplane decision function can th us b e written as [42] f ( ( x )) = sgn l X i =1 y i i ( ( x ) ( x i ) + b ) ! (4.28) PAGE 46 32 As can b e seen w e need the mapping function to solv e the equation (4.26) and (4.28) whic h ma y not b e easy to dene. This limitation can b e o v ercome b y using the Mercer k ernels prop osed in [50, 41]. These k ernels allo w to compute the dot-pro ducts required in (4.26) and (4.28) b y using the k ernel function dened in the input space i.e. k ( x ; y ) = ( ( x ) ( y )). P opular c hoices of k ernels include Gaussian RBF, k ( x ; y ) = exp ( k x y k 2 = c ) or p olynomial k ernels, k ( x ; y ) = ( x y ) d , for some p ositiv e constan ts c and d resp ectiv ely . Th us using these k ernels, equation (4.26) can b e written as W ( ) = l X i =1 i 1 2 l X i;j =1 i j y i y j K ( x i ; x j ) (4.29) and the decision function can b e rewritten as f ( ( x )) = sgn l X i =1 y i i K ( x; x j ) + b ) ! (4.30) 4.3 Fisher Discriminan t Analysis with Kernels In [33] Mik a et al. prop ose a non-linear classication tec hnique based on the Fisher discriminan t using the k ernel idea originally in tro duced in Supp ort V ector Mac hines [50, 41]. The metho d rst maps the data in to some feature space F and then computes Fisher linear discriminan t there, th us implicitly yielding a non-linear discriminan t in input space. Let b e a non-linear mapping to some feature space F . Th us (4 : 8) can w e written as follo ws: J ( w ) = w T S B w w T S W w (4.31) where w 2 F , S B and S W are dened as follo ws S B = ( m 1 m 2 )( m 1 m 2 ) T (4.32) S W = X i =1 ; 2 X x 2 i ( ( x ) m i )( ( x ) m i ) T (4.33) with m i = 1 l i l i X j =1 ( x ij ) and the optim um w can b e found b y setting @ J ( w ) =@ w = 0 . PAGE 47 33 With F as a v ery high dimensional space, maximizing (4 : 31) directly is not p ossible. The limitation can b e o v ercome b y using the k ernel idea prop osed in [42 , 50 , 41 ] b y form ulating an algorithm whic h uses only dot-pro ducts ( ( x ) ( y )) of the training patterns. The dot-pro duct form can then b e ev aluated using Mercer k ernels [40] without ev er explicitly mapping to F as explained in the previous section ( k ( x ; y ) = ( ( x ) ( y ))). As sho wn in [33], to nd the Fisher discriminan t in the feature space F , equation (4 : 31) is form ulated in terms of dot pro ducts of input patterns in the follo wing w a y . The theory of repro ducing k ernels sa ys that an y solution w 2 F m ust lie in span of all training samples in F . Th us, w can b e expressed as follo ws w = l X i =1 i ( x i ) (4.34) Using the denition of m i , (4 : 34) can b e written as w T m i = 1 l i l X j =1 l i X k =1 j k ( x j ; x ik ) = T M i (4.35) where ( M i ) j = 1 l i l i X k =1 k ( x j ; x ik ). Using the denition of S B and (4 : 35), the n umerator of (4 : 31) can b e expressed as w T S B w = T M (4.36) where M = ( M 1 M 2 )( M 1 M 2 ) T . Similarly the denominator can b e expressed as w T S W w = T N (4.37) where N = K K 0 , K is the l l k ernel matrix with K ij = k ( x i ; x j ) [13 ]. Com bining (4 : 36) and (4 : 37), (4 : 31) can b e written as J ( ) = T M T N (4.38) PAGE 48 34 The solution can b e found b y taking the leading eigen v ector of N )Tj/T1_3 7.9701 Tf6.5992 0 Td(1 M . This approac h is called as the Kernel Fisher Discriminant (KFD) [33]. The pro jections of a new v ector x on to w can b e obtained b y ( w ( x )) = l X i =1 i k ( x i ; x ) (4.39) and the decision function in this case can b e giv en as F ( x ) = sig n ( w ( x ) )Tj/T1_1 11.9552 Tf11.8694 0 Td(b ) (4.40) where b is a threshold whic h can b e determined similarly as in linear Fisher. Mik a et al. [33] men tion that the prop osed setting is ill-p osed due to the estimation of l -dimensional co v ariance structures from l samples. The n umerical problems can cause the matrix N not to b e p ositiv e defnite. They prop ose a solution to the problem b y adding a m ultiple iden tit y matrix to N suc h that N = N + I (4.41) As stated in Mik a et al. [33 ], this solution mak es the problem n umerically more stable. 4.4 Kernel Fisher vs SVM Use of Mercer k ernels in k ernel Fisher and SVM has giv en a new approac h to the problem of pattern recognition. The wide v ariet y of k ernels allo ws v arious non-linear classifers. Also the ease of implemen ting these k ernel function mak es the en tire pro cess easy . Ho w ev er, a crucial step is the c hoice of the k ernel function whic h is en tirely data dep enden t. It can b e seen that for solving the Lagrangian in SVM is a quadratic problem with constrain ts and is kno wn to b e NP-complete. On the other hand, the training algorithm used in Kernel Fisher is m uc h simpler to implemen t and do es not require non-linear optimization. The data set considered in this thesis is small and th us using the SVM as the classifer will lead to a large n um b er PAGE 49 35 of training patterns to b e supp ort v ectors. Th us, the adv an tage of constructing a decision function using a few samples (ab out 5%-10%) of the training set will not b e ac hiev ed. Keeping in view the adv an tages of computational eciency with resp ect to the data set in hand, w e are using k ernel Fisher as the non-linear classifer. PAGE 50 CHAPTER 5 IMPLEMENT A TION RESUL TS In this c hapter, w e presen t the exp erimen tal results obtained b y testing the p erformance of Linear Fisher and Kernel Fisher for the statistical analysis of shap e deformations that migh t indicate the hemispheric lo cation of an epileptic fo cus. The scans of t w o classes of patien ts of epilepsy , those with a righ t and those with a left medial temp oral lob e fo cus (RA TL and LA TL), as v alidated b y clinical consensus and subsequen t surgery , are compared to a set of age and sex matc hed health y v olun teers using b oth v olume and shap e based features. The data set consists of 24 con trols, 11 LA TL and 14 RA TL patien ts. F or eac h health y v olun teer/patien t, w e are giv en the p oin t sets placed b y a neuroscien tist on the MRI scan of the left and righ t hipp o campii. These p oin t sets then undergo three stages, namely mo del tting using the Snak e P edal Mo del [51], shap e registration (b oth rigid and non-rigid) and nally classication to recognize the group. Shap e based features are deriv ed from the lo cal displacemen t eld b et w een the left and righ t hipp o campii of a health y v olun teer/patien t. The results sho w a signican t impro v emen t in distinguishing b et w een the con trols and the rest (RA TL and LA TL) using only the shap e as opp osed to v olume based features. W e also ac hiev e a reasonable impro v emen t in the eciency to distinguish b et w een RA TL and LA TL based on shap e in comparison to v olume information. A discussion of the results is presen ted in the next c hapter. 5.1 P art I: Mo del Fitting In this section, w e presen t a set of three exp erimen ts whic h demonstrate the shap e reconstruction with the snak e p edal for the hipp o campus in health y v olun teers, 36 PAGE 51 37 patien ts with righ t medical temp oral lob e fo cus (RA TL) and patien ts with left medial temp oral lob e fo cus (LA TL). The input consists of a p oin t set placed b y a neuroscien tist on the MRI scans of brain to iden tify the hipp o campus. This is follo w ed b y selecting a region of in terest on the MRI scan with the R OI dimension as 64 x 64 in the x and y directions and n um b er of slices for the z direction, to initialize the snak e p edal. Setting the parameters (suc h as b ending, scaling) and using 3D force, desired ttings are ac hiev ed. All the ttings are done at in teractiv e rates on an SGI-O2. The results are organized as follo ws: in Fig. (5 : 1), left to righ t, the images sho w a slice of an MR brain scan in whic h the shap e of in terest,the hipp o campus, has b een iden tied b y a neuroscien tist via sparsely placed p oin ts on the shap e b oundary (in red). The image also sho ws initialized snak e p edal mo del (in green). Next image sho ws the tted mo del follo w ed b y the particular slice with the tting. As can b e seen from the gures, a visually accurate t is ac hiev ed in all the three classes. Using the discretization of the mo del as explained in Section 2.2.2, w e get a new p oin t set with 3D p oin ts placed on a mesh of 21 X 40, th us giving 840 3D p oin ts for eac h hipp o campus considered. 5.2 P art I I: Shap e Registration In this section, w e presen t the rigid and non-rigid registration results obtained on the p oin t sets obtained b y mo del tting of corresp onding left and righ t hipp o campii of a health y v olun teer/patien t. The pro cess of rigid registration b egins with taking a sub ject as the reference and registering all the left and righ t hipp o campii to the left and righ t hipp o campii of the reference using the ICP algorithm prop osed b y Besl and McKa y [7]. This is done b ecause the data for dieren t sub jects (con trols/patien ts) migh t ha v e b een tak en initially in a dieren t co ordinate system. Th us in order to compare all the sub jects, it is necessary to main tain a common co ordinate frame. This is follo w ed b y estimating the rotation and translation b et w een the corresp onding left and righ t PAGE 52 38 Figure 5.1: Mo del Fitting using Snak e P edal. First ro w: MRI scan of a health y v olun teer; p oin ts placed b y a neuroscien tist on a slice (in red) and initialized snak e (left), tted mo del (cen ter), the particular slice sho wing tted mo del. Second ro w: MRI scan of a patien t with left medial temp oral lob e fo cus. Third ro w: MRI scan of a patien t with righ t medial temp oral lob e. PAGE 53 39 hipp o campii using the ICP algorithm [7]. The next step is to remo v e the global scaling factor among a pair of hipp o campii using the metho d describ ed in Section 3.1.2. The results of this en tire pro cess are sho wn in Fig. (5.2): from left to righ t, the rst image sho ws the initial p osition of the left hipp o campus with resp ect to the righ t hipp o campus b efore the rigid registration and the next image sho ws the t w o after the registration using rotation, translation and scaling parameters. The ab o v e pro cedure leads us to the next step of nding the non-rigid registration using the lev el-set form ulation [51 ]. This requires the signed distance images whic h are formed using the F ast Marc hing Metho d and the tagging algorithm as explained in Section 3.2. The lattice for whic h the signed-distance is dened with resp ect to the giv en shap e, is of dimension 128 x 128 x 128 with v o xel distance as 0.2 (this v alue w as selected empirically). T o ensure that the giv en shap e is connected, w e are in terp olating the p oin t set using Spline in terp olation. Fig. (5.3) sho ws the zero set obtained from the signed distance image with sup erp osition of the original p oin t set. Using the signed distance images so obtained, the lo cal deformation is found b y using the equation (3 : 12). The v arious parameters used can b e summarized as follo ws: = 1.2 for smo othing the source image after applying displacemen t eld, = 1 for smo othing the displacemen t eld at eac h iteration, as the stabilizing factor has v alue 0.01 and = 5. The results obtained are sho wn in gures Fig.(5.4), Fig.(5.5) and Fig.(5.6) for health y v olun teer, patien t with left medial temp oral lob e fo cus (LA TL) and patien t with righ t medial temp oral lob e fo cus (RA TL) resp ectiv ely . The organization is as follo ws: rst (upp er) image sho ws the left hipp o campus sup erimp osed on the righ t after the rigid registration and second (lo w er) image sho ws the left hipp o campus sup erimp osed on the righ t after the non-rigid registration. As can b e seen, the dierence b et w een the source and target image after the rigid registration is eliminated nearly p erfectly using the lo cal deformation eld. PAGE 54 40 -20 -10 0 10 20 -10 -5 0 5 10 -10 -5 0 5 10 15 20 10 0 10 20 10 5 0 5 10 6 4 2 0 2 4 6 8 10 20 10 0 10 20 10 5 0 5 10 10 5 0 5 10 15 20 10 0 10 20 10 5 0 5 10 10 5 0 5 10 15 20 10 0 10 20 10 5 0 5 10 6 4 2 0 2 4 6 8 10 20 15 10 5 0 5 10 15 10 5 0 5 10 6 4 2 0 2 4 6 8 Figure 5.2: Rigid registration of left(in red) and righ t(in blue) hipp o campii using ICP algorithm and scaling. First ro w: P oin t sets for a health y v olun teer; b efore registration (left), after registration (righ t). Second ro w: P oin t sets for a patien t with left medial temp oral lob e fo cus. Third ro w: patien t with righ t medial temp oral lob e fo cus. PAGE 55 41 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8 10 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 8 10 -6 -4 -2 0 2 4 6 8 10 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 8 10 -6 -4 -2 0 2 4 6 8 10 -15 -10 -5 0 5 10 15 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8 -15 -10 -5 0 5 10 15 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8 -10 -5 0 5 10 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8 -10 -8 -6 -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8 Figure 5.3: P oin t set o v erla y ed on the zero-set obtained from the corresp onding signed distance image. First ro w: Health y v olun teer; left hipp o campus (left), righ t hipp o campus (righ t). Second Ro w: P atien t with left medial temp oral lob e fo cus. Third Ro w: P atien t with righ t medial temp oral lob e fo cus. PAGE 56 42 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 Figure 5.4: Non-rigid registration of the left and righ t hipp o campii for a health y v olun teer using lev el set form ulation. First ro w: After rigid registration. Second ro w: After non-rigid registration. PAGE 57 43 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 Figure 5.5: Non-rigid registration of the left and righ t hipp o campii for a patien t with left medial temp oral lob e fo cus using lev el set form ulation. First ro w: After rigid registration. Second ro w: After non-rigid registration. PAGE 58 44 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 Figure 5.6: Non-rigid registration of the left and righ t hipp o campii for a patien t with righ t medial temp oral lob e fo cus using lev el set form ulation. First ro w: After rigid registration. Second ro w: After non-rigid registration. PAGE 59 45 T o v alidate the accuracy of the registration, w e calculate a parameter dened as the ratio of the n um b er of mismatc hes in the sign of the distance images of corresp onding left and righ t hipp o campii and the total n um b er of v o xels in the image. The mean v alue b efore the non-rigid registration is found to b e 2.57% 1.63%, while it decreases to 1.01% 0.37%. Th us, the non-rigid registration is able to capture the lo cal deformations with high eciency . 5.3 P art I I I: Shap e Classication After the pro cess of rigid and non-rigid registration, the nal task is to select appropriate features and classier to capture the shap e dierences in the classes considered. With v olume based study b eing p opular in this particular application, our objectiv e is to compare it with the shap e based features in learning the shap e dierences b et w een the three groups. Hence, w e are using b oth v olume and shap e features for our purp ose. The feature v ector obtained from v olume information is of length t w o with L/R and (L-R)/(L+R) as the comp onen ts, where L and R are the v olume of the left and righ t hipp o campii resp ectiv ely . Shap e based features are b eing deriv ed from the displacemen t eld obtained b y non-rigid registration and can b e group ed in to t w o categories. The rst t yp e is called sign of displacemen t and the other is called direction of the displacemen t v ector. The sign of displacemen t is dened as follo ws. Giv en the displacemen t v ector for a p oin t on the zero-set of the source image, determine the cub e in whic h the displaced p oin t falls in the source image. Using the sign information of the v ertices (since eac h v ertex w as assigned +/sign while forming the distance image) of the enclosing cub e, assign a sign to the magnitude of the displacemen t. The direction v ector is obtained b y nding the unit v ector corresp onding to the displacemen t v ector at eac h p oin t on the zero-set of source image. The feature v ector for sign of displacemen t is of length 762 while displacemen t v ector is of length 762 x 3 (since eac h p oin t has x,y ,z comp onen t of displacemen t). This PAGE 60 46 comes from the fact that there are 840 p oin t on the zero set and the frst and last ro w of 21 x 40 mesh represen ts the north and south p ole as describ ed in [51]. Since our data set is small, w e are restricting our analysis to linear Fisher and Kernel Fisher with b oth v olume and shap e features for the three classes. The classifcation is done in t w o stages, namely frst distinguish b et w een con trols and patien ts with epilepsy and then determine the hemispheric lo cation of an epileptic fo cus in the patien ts. 5.3.1 Health y V olun teers vs. P atien ts withEpilepsy W e b egin our analysis with distinguishing b et w een health y v olun teers and the rest (RA TL and LA TL). Fig. (5.7) sho ws the classifcation obtained b et w een the t w o classes using v olume based information for linear and non-linear classifer. The plots on the left sho w the con tours for dieren t discriminan ts and plots on the righ t sho w the pro jection of the data sets on the separating line (either in input space or pro jected space). As can b e seen from the plots, the t w o classes cannot b e separated using only v olume based study . Ev en the non-linear classifer is not able to separate them out as their c haracteristics are mixed in nature. The results are further v alidated using lea v e-one-out test and ha v e b een summarized in T able (5.1). The a v erage training accuracy ac hiev ed is nearly 64% while the maxim um test set accuracy is 61.2%. The next stage of the analysis is based on shap e features for distinguishing the t w o classes. W e b egin with the linear classifer and sign of displacemen t as the feature v ector. As can b e seen in Fig. (5.8), the classifcation impro v es considerably , though not p erfect. Analysis using lea v e-one-out as sho wn in T able (5.2) also sho ws an impro v emen t o v er using only v olume information. The next step is to use k ernel Fisher as the non-linear classifer since linear classifer is not giving a p erfect classifcation. W e are using b oth p olynomial (with degree = 2) and radial basis (with radius = 1000), b oth parameters b eing determined exp erimen tally based on the data PAGE 61 47 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.58 0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0 50 100 150 200 250 300 350 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 2 1.5 1 0.5 0 0.5 1 x 10 4 0 1 2 3 4 5 6 7 8 x 10 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 2 1.5 1 0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 Figure 5.7: Classication results for con trols (red) vs rest (blue) using v olume based feature. First ro w: scatter plot of feature v ector (left), classication using linear sher (righ t). Second ro w: con tour plot with p olynomial basis(left), classication using p olynomial basis of degree 2 (righ t). Third ro w: con tour plot using RBF (radius = 0.05) (left), classication using RBF (righ t). PAGE 62 48 15 10 5 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 15 10 5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0 2 4 6 8 10 12 14 Figure 5.8: Classication results for con trols (red) vs rest (blue) using sign of displacemen t as feature: linear Fisher (left), k ernel Fisher with P oly . basis (degree=2) (cen ter), k ernel Fisher with radial basis (r = 1000) (righ t). 10 5 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 15 10 5 0 5 10 15 0 0.5 1 1.5 2 2.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0 2 4 6 8 10 12 Figure 5.9: Classication results for con trols (red) vs rest (blue) using direction v ector as feature: linear Fisher (left), k ernel Fisher with P oly . basis (degree=2) (cen ter), k ernel Fisher with radial basis (r = 1000) (righ t). PAGE 63 49 T able 5.1: Con trols vs Rest, Fv ec: V olume (Con trols=24, Rest=25) Linear Fisher KF with P oly . basis (d=2) KF with RBF (r=0.05) T raining 64.43% 55.2% 64.93% T esting(Lea v e-1-out) 61.22% 55% 61.22% T able 5.2: Con trols vs Rest, Fv ec: Sign of Displacemen t (Con trols=24, Rest=25) Linear Fisher KF with P oly . basis (d=2) KF with RBF (r=1000) T raining 95.92% 100% 100% T esting(Lea v e-1-out) 87.76% 91.84% 89.98% v alue, as our k ernel functions. Fig.(5.8) sho ws a considerable impro v emen t in the classication with the non-linear classier. V alidation results using lea v e-one-out in T able (5.2) sho w a p erfect training set accuracy and an impro v emen t in the test set accuracy o v er the linear classier. Fig. (5.8) and T able (5.3) summarize the results obtained with direction v ector as the feature v ector. It can b e seen that the results impro v e with this feature v ector whic h can b e explained using the fact that the direction v ector has more information than the sign of displacemen t (whic h uses only the magnitude information of the displacemen t eld instead of the individual comp onen ts). Eviden tly , b oth the shap e based features sho w a considerable impro v emen t in ac hieving the task of distinguishing b et w een con trols and patien ts with epilepsy as opp osed to v olume information. T able 5.3: Con trols vs Rest, Fv ec: direction v ector (Con trols=24, Rest=25) Linear Fisher KF with P oly . basis (d=2) KF with RBF (r=1000) T raining 96.5% 100% 100% T esting(Lea v e-1-out) 85.71% 95.9% 93.8% PAGE 64 50 5.3.2 LA TL vs RA TL Giv en the classication b et w een the health y v olun teers and patien ts with epilepsy , the next task is to iden tify the hemispheric lo cation of an epileptic fo cus in RA TL and LA TL. The pro cess b egins with v olume information only . As can b e seen in the Fig. (5.10), v olume is not able to distinguish b et w een RA TL and LA TL using either linear Fisher or k ernel Fisher. Lea v e-one-out p erformance, as sho wn in T able (5.4), also sho ws lo w training and test set accuracy and do es not impro v e using non-linear classier. Ho w ev er, it can b e seen that v olume information is able to distinguish among the patien ts of epilepsy with a little higher accuracy as compared to distinguishing b et w een con trols and patien ts of epilepsy . The analysis is follo w ed b y using shap e based information for distinguishing b et w een the t w o classes. Both linear Fisher and k ernel Fisher are tested for the shap e based features. Fig. (5.11) sho ws the classication obtained using sign of displacemen t as the feature v ector. It can b e seen that though the classes are w ell separated, test set accuracy impro v es marginally o v er the v olume based study . T able (5.5) sho ws the quan titativ e results obtained b y lea v e-one-out test. Results obtained using direction v ector as the feature v ector are sho wn is Fig. (5.12) and T able (5.6). It can b e seen that the p erformance of linear Fisher impro v es with this feature v ector particularly for the training test. Ho w ev er, it can b e seen that the p erformance of this feature v ector is at par with the other shap e feature. Ho w ev er, using the non-linear classier, though the training set accuracy b ecomes p erfect, test set accuracy is still not impro v ed m uc h. This can b e b ecause the patterns are to o close to the decision b oundary and an y c hange in it is unable to impro v e the eciency . PAGE 65 51 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 2 1.5 1 0.5 0 0.5 1 x 10 4 0 0.5 1 1.5 2 2.5 3 x 10 4 Figure 5.10: Classication results for LA TL (red) vs RA TL (blue) using v olume based feature. First ro w: (left) scatter plot of feature v ector, (righ t) classication using linear sher. Second ro w: (left) con tour plot with p olynomial basis, (righ t) classication using p olynomial basis of degree 2. Third ro w: (left) con tour plot using RBF (radius = 0.5), (righ t) classication using RBF. PAGE 66 52 T able 5.4: LA TL vs RA TL, Fv ec: V olume (LA TL=11, RA TL=14) Linear Fisher KF with P oly . basis (d=2) KF with RBF (r=0.5) T raining 66.88% 66.88% 66.88% T esting(Lea v e-1-out) 64% 64% 64% 15 10 5 0 5 10 15 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 10 5 0 5 10 15 0 1 2 3 4 5 6 7 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 60 70 Figure 5.11: Classication results for LA TL (red) vs RA TL (blue) using sign of displacemen t feature; (left) classication using linear sher, (cen ter) classication using p olynomial basis of degree 2, (righ t) classication using RBF (r=900). T able 5.5: LA TL vs RA TL, Fv ec: Sign of displacemen t (LA TL=11, RA TL=14) Linear Fisher KF with P oly . basis (d=2) KF with RBF (r=900) T raining 74.88% 100% 100% T esting(Lea v e-1-out) 64% 72% 72% 15 10 5 0 5 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 15 10 5 0 5 10 15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0 5 10 15 20 25 30 35 40 Figure 5.12: Classication results for LA TL (red) vs RA TL (blue) using direction v ector as feature v ector; (left) classication using linear sher, classication using P oly . basis (d=2) (cen ter), (righ t) classication using RBF (r=900). PAGE 67 53 T able 5.6: LA TL vs RA TL, Fv ec: Direction v ector (LA TL=11, RA TL=14) Linear Fisher KF with P oly . basis (d=2) KF with RBF (r=900) T raining 88% 100% 100% T esting(Lea v e-1-out) 68% 72% 68% 5.4InterpretationoftheResults F rom the exp erimen ts, it can b e seen that w e are able to distinguish b et w een the con trols and patien ts with epilepsy with a high accuracy using only shap e information. W e also observ e a reasonable impro v emen t in distinguishing b et w een RA TL and LA TL using shap e based features in comparison to v olume features. In this section, w e will presen t the plots for the shap e based features whic h can help understand the results obtained and can also explain the dierence in the recognition accuracy as seen when w e classify con trol and patien ts and then among patien ts. W e b egin our analysis with sign of displacemen t as the feature v ector. The feature v ector is defned for the 21 x 40 mesh obtained from mo del ftting. Th us, for eac h lo cation on this mesh, w e ha v e defned a +1/-1 sign as explained b efore. This can b e visualized using a binary image. Fig.(5.13), Fig.(5.14) and Fig.(5.15) (with blue as negativ e displacemen t and bro wn as p ositiv e displacemen t)sho w the examples of some of the patterns b elonging to the class of health y v olun teers, patien ts with left medial temp oral lob e fo cus and patien ts with righ t medial temp oral lob e fo cus. As can b e seen from the fgures, con trols ha v e p ositiv e displacemen t near the p oles, while the patien ts with epilepsy ha v e b oth p ositiv e and negativ e displacemen t near the p oles. This can explain the high accuracy in distinguishing the t w o classes. Ho w ev er, the pattern is mixed for LA TL and RA TL and th us ma y ha v e lead to a lo w er eciency . PAGE 68 54 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 Figure 5.13: Examples sho wing the colormap corresp onding to the sign of displacemen t as the feature v ec for the 21 x 40 mesh for health y v olun teers 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 Figure 5.14: Examples sho wing the colormap corresp onding to the sign of displacemen t as the feature v ec for the 21 x 40 mesh for patien ts with left medial temp oral lob e fo cus PAGE 69 55 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 Figure 5.15: Examples sho wing the colormap corresp onding to the sign of displacemen t as the feature v ec for the 21 x 40 mesh for patien ts with righ t medial temp oral lob e fo cus F urther, w e p erform principal comp onen t analysis on this feature v ector and nd the eigen v ectors corresp onding to non-zero eigen v alues. W e then follo w a stepwise pro cedure of forming a new basis based on these eigen v ectors (starting with the eigen v ector corresp onding to largest eigen v alue) and p erform classication at eac h step to understand the signicance of the feature comp onen ts in classication b et w een LA TL and RA TL. Fig.(5.18) sho ws the scatter plot of the pro jections of the t w o classes in the t w o dimension and sup erimp osed is the con tour plot obtained in the follo wing w a y . Dep ending on the range of the pro jections of the t w o classes, w e dene a mesh of test v ectors in t w o dimensions. Using the k ernel Fisher as the classier, w e estimate the pro jection of the test v ector in the k ernel space and using MA TLAB plot these con tours. The analysis is done for p olynomial basis with d=1 (whic h will b e a linear classier) and d=2 and then with radial basis (r=900). As can b e seen the gure, the pro jections of the t w o classes sho w a mixed nature. Th us, the h yp erplane separating the t w o classes is highly sensitiv e to the the data, whic h can PAGE 70 56 also b e seen from the results obtained from lea v e-one-out tests. W e also observ e that using 17 eigen v ectors corresp onding to 17 largest eigen v alues, giv es the b est training (100%) and test-set (80%) accuracy using linear Fisher. Our next set of exp erimen ts is for direction v ector as the feature v ector. Again w e consider the 21 x 40 mesh. Here for eac h co ordinate lo cation w e ha v e a length three v ector represen ting the displacemen t in x, y and z directions. This can b e used to represen t a R GB (dx = red, dy = green and dz = blue) v alue at that lo cation. Fig.(5.17), Fig.(5.18) and Fig.(5.19) sho w the colormaps obtained for the health y v olun teers, patien ts with left medial temp oral lob e fo cus and patien ts with righ t medial temp oral lob e fo cus. The gures sho w that the con trols ha v e more displacemen t in the z direction to w ards the middle while patien ts with epilepsy ha v e a com bination of displacemen t in x and z direction in the same lo cation. Th us again w e can explain the high accuracy in con trols vs. rest but the mixed nature for the patien ts, limits the test set accuracy among the patien ts. The PCA analysis with direction v ector as the feature and using t w o principal comp onen ts as the new basis is sho wn in Fig.(5.20). The gures again sho w a mixed nature and the h yp erplane b eing to o sensitiv e to the data. Again, w e observ e that using 16 most signican t eigen v ectors giv es the b est training (100%) and test-set (88%) accuracy . Th us it can b e concluded that the shap e dierence among the pathologies ma y b e highly correlated, hence making it dicult to separate them out. Also the n um b er of data samples for the patien ts with epilepsy is quite small hindering a sucien t represen tation of the p opulation whic h migh t ha v e led to high training accuracy but lo w test accuracy . W e also observ e an impro v emen t in the training and test set accuracy using lo cal shap e features with PCA analysis. This implies that the high frequency comp onen ts of the feature v ector ma y corresp ond to the noise in the data, th us leading to lo w er eciency with the en tire feature. In case of PCA analysis, PAGE 71 57 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 Figure 5.16: PCA on sign of displacemen t: k ernel Fisher with d=1 (left), k ernel Fisher with d=2 (cen ter) and k ernel Fisher with RBF (righ t) PAGE 72 58 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 Figure 5.17: Examples sho wing the colormap corresp onding to the direction v ector as the feature v ec for the 21 x 40 mesh for health y v olun teers PAGE 73 59 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 Figure 5.18: Examples sho wing the colormap corresp onding to the direction v ector as the feature v ec for the 21 x 40 mesh for patien ts with left medial temp oral lob e fo cus PAGE 74 60 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 10 20 30 40 5 10 15 20 Figure 5.19: Examples sho wing the colormap corresp onding to the direction v ector as the feature v ec for the 21 x 40 mesh for patien ts with righ t medial temp oral lob e fo cus PAGE 75 61 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 15 10 5 0 5 10 15 Figure 5.20: PCA on direction v ector: k ernel Fisher with d=1 (left), k ernel Fisher with d=2 (cen ter) and k ernel Fisher with RBF (righ t) PAGE 76 62 w e also see b est classication accuracy with linear Fisher whic h indicates that the features are strong enough to separate the classes using a linear b oundary . PAGE 77 CHAPTER 6 CONCLUSION In this thesis, w e presen t an automatic tec hnique to distinguish b et w een health y v olun teers and patien ts of epilepsy and also indicates the hemispheric location of an epileptic fo cus in the patien ts. The k ey feature is the use of shap e based features as opp osed to v olume information to ac hiev e the task. Since the data set considered is small, the analysis is done using linear and non-linear Fisher (based on k ernel metho ds) whic h mak es the pro cess computationally fast and meaningful. Also w e hop e that shap e can pro vide enough information to distinguish b et w een the dieren t classes and hence they can b e linearly separated either in the input space or pro jected space. Selecting the optimal feature v ector pla ys a crucial role in the analysis. Use of sign of displacemen t and the direction v ector ha v e sho wn to capture the shap e differences in the classes considered. The results sho w a high accuracy in distinguishing b et w een the hipp o campal shap es of health y v olun teers and patien ts with pathology using shap e in comparison to v olume features. W e also ac hiev e a reasonable impro v emen t in the eciency to distinguish among pathology . Ho w ev er, the success rate in this case migh t b e impro v ed with an increase in the size of the data set. 63 PAGE 78 REFERENCES [1]J.A.Baerentzen,OnTheImplementationofFastMarchingMethodsfor3D L attic es , T ec hnical Rep ort IMM, 2001. [2] C. Baillard, P . Hellier and C. Barillot, Se gmentation of 3D Br ain Structur es using L evel Sets and Dense R e gistr ation , IEEE W orkshop on Mathematical Metho ds in Biomedical Image Analysis, Hilton Head Island, 2000, pp. 94-101. [3] R. Ba jscy and S. Ko v acic, Multir esolution Elastic Matching , Computer Vision, Graphics and Image Pro cessing 46, 1989, pp. 1-21. [4] E. Bardinet, N. Ay ac he and L.D. Cohen, Fitting of Iso-Surfac e Using Sup er quadrics and F r e e-F orm Deformations , Pro c. IEEE W orkshop Biomedical Image Analysis WBIA 1994, pp. 184-193. [5] A.H. Barr Glob al and L o c al Deformations of Solid Primitives , Computer Graphics (A CM) 18(3), 1984, pp. 21-30. [6] J.L. Barron, D.J. Fleet and S.S. Beauc hemin, Performanc e of Optic al Flow T e chniques , In tern. J. Comput. Vision 1(12), 1994, pp. 43-77. [7] P .J. Besl and N.D. McKa y , A Metho d of R e gistr ation of 3-D Shap es , IEEE T rans. of P attern Analysis and Mac hine In telligence 14(2), 1992, 239-255. [8] M.J. Blac k and P . Anandan, A F r amework for R obust Estimation of Optic al Flow , Pro c. In tl. Conf. on Computer Vision, Berlin, German y , 1993, pp. 231236. [9] G.E. Christensen, M.I. Miller and M. V annier, Individualizing Neur o anatomic al A tlases Using a Massively Par al lel Computer , IEEE Computer 1(29), 1996, pp. 32-38. [10]A.Collignon,F.Maes,D.Delaere,D.Vandermeulen,P.SuetensandG.Marchal, A utomate d Multimo dality Image R e gistr ation using Information The ory , Pro c. Information Pro cessing in Medical Imaging (Brest) Klu w er, Dordrec h t, 1995, pp. 263-274. [11] C. Cortes and V. V apnik, Supp ort V e ctor Networks , Mac hine Learning 20, 1995, pp. 273-297. [12] J.G. Csernansky , S. Joshi, L. W ang, J.W. Haller, M. Gado, J.P . Miller, U. Grenander and M.I. Miller, Hipp o c amp al Morphometry in Schizophr enia by High Dimensional Br ain Mapping , Neurobiology 95(10), 1998, pp. 11406-11411. [13] D. DeCoste, M.C. Burl, A. Hopkins and N.S. Lewis, Supp ort V e ctor Machines and Kernel Fisher Discriminants: A Case Study using Ele ctr onic Nose Data , F ourth W orkshop on Mining Scien tifc Datasets, 2001. 64 PAGE 79 65 [14] J.H. Duncon and T.C. Chou, On the Dete ction of Motion and the Computations of Optic al Flow IEEE T rans. P att. Anal. mac h. In tell. P AMI-14(3) pp. 346-352. [15] N. Duta, M. Sonk a and A.K. Jain, L e arning Shap e Mo dels fr om Examples using A utomatic Shap e Clustering Pr o crustes A nalysis , IPMI, 1999, pp. 370-375. [16] W.R. Dyksen, T ensor Pr o duct Gener alize d ADI Metho ds for Sep ar able El liptic Pr oblems , Siam J. Numerical Analysis 24(1), 1987, pp. 59-76. [17] G. Gerig, M. St yner, M.E. Shen ton and J.A. Lieb erman, Shap e vs Size: Impr ove d Understanding of the Morpholo gy of Br ain Structur es , MICCAI, 2001, pp. 24-32. [18] P . Golland, W. Eric, L. Grimson and R. Kikinis, Statistic al Shap e A nalysis Using Fixe d T op olo gy Skeletons: Corpus Cal losum Study , Pro c. IPMI LNCS 1613 Springer-V erlag, 1999, pp. 382-387. [19] P . Golland, W. Eric, M.E. Shen ton and R. Kikinis, Smal l Sample Size L e arning for Shap e A nalysis of A natomic al Structur es , Pro c. MICCAI LNCS 1935, 2000, pp. 72-82. [20] P . Golland, W. Eric, L. Grimson, M.E. Shen ton and R. Kikinis, Deformation A nalysis for Shap e Base d Classifc ation , Pro c. IPMI LCNS 2082, 2001, pp. 517530. [21] Y. Guo, Snake Pe dals: A ctive Ge ometric Mo dels for Shap e Mo deling and R e c overy , PhD Dissertation, Univ ersit y of Florida, 1998. [22] I. Guyon, Online SVM Application List, www.clopinet.com/isabelle/Projects/SVM/applist.html,2000. [23] M. Heb ert, K. Ik euc hi and H. Delingette, A Spheric al R epr esentation for R e c o gnition of F r e e-F orm Surfac es , IEEE P attern Analysis and Mac hine In telligence 17(7), 1995, pp. 681-690. [24] B.P .K. Horn and B.G. Sc h unk, Determining Optic al Flow , Artifcial In telligence 17, 1981, pp. 185-203. [25] M. Kass, A. Witkin and D. T erzop oulos, Snakes: A ctive Contour Mo dels , In t. Journal of Computer Vision 1, 1987, pp. 321-331. [26] R. Kimmel, A. Amir and A.M. Bruc kstein, Finding Shortest Path on Surfac es using L evel Set Pr op agation , IEEE T rans. on P attern Analysis and Mac hine In telligence 17(6), 1995, pp. 635-640. [27] S.H. Lai and B.C. V em uri, R obust and Ecient Computation of Optic al Flow , IEEE Symp osium on CVS, Miamii, Florida, 1995, pp. 455-460. [28] R.J. LeV eque, Numeric al Metho ds for Conservation L aws , Birkhauser, Boston, MA, 1992. [29] A. Lu and E.L. W ac hspress, Solution of Lyapunov Equations by A lternating Dir e ction Implicit Iter ation , Computers Math. Applications 21(9), 1991, pp. 43-58. [30] R.E. Lync h, J.R. Ric h and D.H. Thomas, T ensor Pr o duct A nalysis of A lternating Dir e ction Implicit Metho ds , J. So c. Industrial and applied Math. 13(4), 1965, pp. 995-1007. PAGE 80 66 [31] D. Metaxas, J. P ark and L. Axel, A nalysis of L eft V entricular Wal l Motion Base d on V olumetric Def ormable Mo dels and MRI-Sp amm , Medical Image Analysis 1(1), 1996, pp. 53-72. [32] C.R. Mey er, J.L. Bo es, B. Kim, P .H. Bland, K.R. Zasadn y , P .V. Kison, K. Koral, K.A. F rey and R.L. W ahl, Demonstr ating the A c cur acy and Clinic al V ersatility of Mutual Information for A utomatic Mult imo dality Image F usion using Ane and Thin-Plate Spline warpr e d Ge ometric Deformations , Medical Image Analysis 1(3), 1997, pp. 195-206. [33] S. Mik a, G. R atsc h and J. W eston, Fisher Discriminant A nalysis with Kernels , Neural Net w orks for Signal Pro cessing IX, IEEE, 1999, 41-48. [34] S. Mik a, A. Smola and B. Sc h olk opf, A n impr ove d T r aining A lgorithm for Kernel Fisher Discriminant , Pro ceedings AIST A TS, 2001, 98-104. [35] T. O'Donnel, T. Boult and A. Gupta, Glob al Mo dels with Par ametric Osets as applie d to Car diac , IEEE Pro c. Conf. Computer Vision and P attern Recognition, 1996 pp. 293-299. [36] M. Okuomi and T. Kanade, A L o c al ly A daptive Window for Signal Matching , In ternational Journal of Computer Vision 7(2), 1992, pp. 143-162. [37] D.V. Ouellette, Schur Complements and Statistics , Linear Algebra and its Applications 36, 1981, pp. 187-295. [38] W.H. Press, S.S. T euk olsky , W.T. V etterling and B.P . Flannery , Numeric al R e cipies in C , Cam bridge Univ. Press, Cam bridge, 1992. [39] E. Rouy and A. T ourin, A Visc osity Solutions Appr o ach to Shap e-F r om-Shading , SIAM J. Num. Anal 29(3), 1992, pp. 867-884. [40] S. Saitoh, The ory of R epr o ducing Kernels and its Applic ation , Longman Scientifc & T ec hnical, Harlo w, England, 1988. [41] B. Sc h olk opf, C. Burges and A. Smola, Adv ances in Kernel Metho ds-Supp ort V ector Learning, MIT Press, Cam bridge MA, 1999. [42] B. Sc h olk opf, A. Smola and K,-R. M uller, Nonline ar Comp onent A nalysis as a Kernel Eigenvalue Pr oblem , Neural Computation 10, 1998, pp. 1299-1319. [43] T.W. Sederb erg and S.R. P arry , F r e e-F orm Deformation of Solid Ge ometric Mo dels , A CM SIGGRAPH, 1986, pp. 151-160. [44] J.A. Sethain, L evel Set Metho ds and F ast Mar ching Metho ds: Evolving Interfac es in Computational Ge ometry, Fluid Me chanics, Computer Vision and Material Scienc e , Cam bridge Univ ersit y Press, Cam bridge, 1999. [45] F. Solina and R. Ba jscy , R e c overy of Par ametric Mo dels fr om R ange Images: The Case for Sup er quadrics with Glob al Deformation , IEEE T rans. on P attern Analysis and Mac hine in telligence 12, 1990, pp. 131-146. [46] C. Studholme, D.L.G. Hill and D.J. Ha wk es, A utomate d 3D R e gistr ation of MR and CT Images in the He ad , Medical Image Analysis 1(2), 1996, pp. 163-175. [47] R. Szeliski and J. Coughlan, Hier ar chic al Spline-Base d Image R e gistr ation , IEEE Conf. Comput. Vision P att. Recog., Seattle, W A, 1994, 194-201. PAGE 81 67 [48] D. T erzop oulos and D. Metaxas, Dynamic 3D Mo dels with L o c al and Glob al Deformations: Deformable Sup er quadrics , IEEE T rans. P attern Analysis and Mac hine In telligence 13(7), 1991, pp. 703-714. [49] J.P . Thirion, Extr emal Points: Defnition and Applic ation to 3D R e gistr ation , Pro c. of IEEE CVPR, Seattle, W A, 1994, pp. 587-592. [50] V. V apnik, The Natur e of Statistic al L e arning The ory , Springer V erlag, New Y ork, 1995. [51] B.C. V em uri and Y. Guo, Comp act and V ersatile Ge ometric Mo dels with PhysicsBase d c ontr ol , IEEE T rans. on P attern Analysis and Mac hine In telligence 22(5) , 2000, pp. 445-459. [52] B.C. V em uri, S. Huang, S. Sahni, C.M. Leonard, C. Mohr, R. Gilmore and J. Fitzsimmons, A n Ecient Motion Estimator with Applic ation to Me dic al Image R e gistr ation , Medical Image Analysis, Oxford Univ ersit y Press 2(1), 1998, pp. 79-98. [53] B.C. V em uri and A. Radisa vljevic, Multir esolution Sto chastic Hybrid Shap e Mo dels with F r actal Priors , A CM T rans. Graphics, 1994, pp. 177-207. [54] B.C. V em uri, J. Y e, Y. Chen and C. Leonard, A L evel-Set b ase d Appr o ach to Image R e gistr ation , W orkshop on Mathematical Metho ds in Biomedical Image Analysis June 11-12, 2000, pp. 86-93. [55] P . Viola and W.M. W ells I I I, A lignment by Maximization of Mutual Information , In tl. Journal of Computer Vision 24(2), 1997, pp. 137-154. [56] J. Y e, A L evel-Set Base d Appr o ach to Image R e gistr ation , M.S. Thesis, Univ ersit y of Florida, 2000. [57] W.M. W ells I I I, P . Viola and H. A tsumi, Multi-mo dal V olume R e gistr ation by Maximization of Mutual Information , Medical Image Analysis 1(1), 1996, pp. 35-91 PAGE 82 BIOGRAPHICAL SKETCH Neeti V ohra w as b orn in New Delhi, India. She receiv ed her Bac helor of T ec hnology degree in electrical engineering from the Indian Institute of T ec hnology , Delhi, India, inAugust2000.Shewillreceiveher theMasterofSciencedegreein electrical and computer engineering from the Univ ersit y of Florida, Gainesville in August 2002. Her researc h in terests include signal and image pro cessing, medical imaging, pattern recognition, computer vision, m ultimedia and comm unications. 68 |