Table Of Content

Modeling and Simulation in Science, Engineering and Technology Series Editor Nicola Bellomo Politecnico di Torino Italy Advisory Editorial Board M. Avellaneda (Modeling in Economics) H.G. Othmer (Mathematical Biology) Courant Institute of Mathematical Sciences Department of Mathematics New York University University of Minnesota 251 Mercer Street 270A Vincent Hall New York, NY 10012, USA Minneapolis, MN 55455, USA [email protected] [email protected] K.J. Bathe (Solid Mechanics) L. Preziosi (Industrial Mathematics) Department of Mechanical Engineering Dipartimento di Matematica Massachusetts Institute of Technology Politecnico di Torino Cambridge, MA 02139, USA Corso Duca degli Abruzzi 24 [email protected] 10129 Torino, Italy [email protected] P. Degond (Semiconductor & Transport Modeling) V. Protopopescu (Competitive Systems, Mathématiques pour l'Industrie et la Physique Epidemiology) Université P. Sabatier Toulouse 3 CSMD 118 Route de Narbonne Oak Ridge National Laboratory 31062 Toulouse Cedex, France Oak Ridge, TN 37831-6363, USA [email protected] [email protected] M.A. Herrero Garcia (Mathematical Methods) K.R. Rajagopal (Multiphase Flows) Departamento de Matematica Aplicada Department of Mechanical Engineering Universidad Complutense de Madrid Texas A&M University Avenida Complutense s/n College Station, TX 77843, USA 28040 Madrid, Spain [email protected] [email protected] Y. Sone (Fluid Dynamics in Engineering W. Kliemann (Stochastic Modeling) Sciences) Department of Mathematics Professor Emeritus Iowa State University Kyoto University 400 Carver Hall 230-133 Iwakura-Nagatani-cho Ames, IA 50011, USA Sakyo-ku Kyoto 606-0026, Japan [email protected] [email protected] Statistics and Analysis of Shapes Hamid Krim Anthony Yezzi, Jr. Editors Birkhaüser Boston • Basel • Berlin HamidKrim AnthonyYezzi,Jr. NorthCarolinaStateUniversity GeorgiaInstituteofTechnology DepartmentofElectrical SchoolofElectrical andComputerEngineering andComputerEngineering Raleigh,NC27606-7914 Atlanta,GA30332-0250 USA USA Mathematics Subject Classification: 05C10, 05C12, 05C35, 05C62, 05C65, 14J17, 14J29, 14J70, 14J80,14J81,14R05,14R10,14R15,14R20,26B12,26E15,26E20,26E25,26E30,28A25,28A33, 28A35,35A15,35A17,35A18,35A20,35A21,35F20,35F25,35F30,35F99,35Gxx,35G05,35G10, 35G15,35G20,35G25,35G30,37C05,37C10,37C15,37C20,37D15,37C20,37D15,37G10,46C05, 46N30,49Q10,55P55,55Q07,57N25,62H11,68T10,90C90,92C55 LibraryofCongressControlNumber:2005938892 ISBN-100-8176-4376-1 e-ISBN0-8176-4481-4 ISBN-13978-0-8176-4376-8 Printedonacid-freepaper. (cid:1)c2006BirkhaüserBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit- tenpermissionofthepublisher(BirkhaüserBoston,c/oSpringerScience+BusinessMediaLLC,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. (KeS/SB) 987654321 www.birkhauser.com Contents Preface vii Medial Axis Computation and Evolution Sylvain Bouix, Kaleem Siddiqi, Allen Tannenbaum and Steven W. Zucker.............................................. 1 Shape Variation of Medial Axis Representations via Principal Geodesic Analysis on Symmetric Spaces P. Thomas Fletcher, Stephen M. Pizer and Sarang C. Joshi.... 29 2D Shape Modeling using Skeletal Graphs in a Morse Theoretic Framework Sajjad Hussain Baloch and Hamid Krim........................ 61 Matching with Shape Contexts Serge Belongie, Greg Mori and Jitendra Malik.................. 81 Shape Recognition Based on an a Contrario Methodology Pablo Musé, Frédéric Sur, Frédéric Cao, Yann Gousseau and Jean-Michel Morel.............................................. 107 Integral Invariants and Shape Matching Siddharth Manay, Daniel Cremers, Byung-Woo Hong, Anthony Yezzi, Jr. and Stefano Soatto.......................... 137 On the Representation of Shapes Using Implicit Functions N. Paragios, M. Taron, X. Huang, M. Rousson and D. Metaxas 167 Computing with Point Cloud Data Facundo Mémoli and Guillermo Sapiro......................... 201 vi Contents Determining Intrinsic Dimension and Entropy of High-Dimensional Shape Spaces Jose A. Costa and Alfred O. Hero III........................... 231 Object-Image Metrics for Generalized Weak Perspective Projection Gregory Arnold, Peter F. Stiller and Kirk Sturtz................ 253 WulffShapesatZeroTemperatureforSomeModelsUsedinImage Processing Xavier Descombes and Eugène Pechersky....................... 281 Curve Shortening and Interacting Particle Systems Sigurd Angenent, Allen Tannenbaum, Anthony Yezzi, Jr. and Ofer Zeitouni................................................... 303 Riemannian Structures on Shape Spaces: A Framework for Statistical Inferences Shantanu Joshi, David Kaziska, Anuj Srivastava and Washington Mio................................................ 313 Modeling Planar Shape Variation via Hamiltonian Flows of Curves Joan Glaunès, Alain Trouvé and Laurent Younes............... 335 Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics Guillaume Charpiat, Olivier Faugeras, Renaud Keriven and Pierre Maurel.................................................. 363 Preface Shapes have been among man’s fascinations from the stone age to the space age. The scientific study of shapes may indeed be traced back to D’Arcy Thompson in his pioneering book On Growth and Form where shape was shown to be dependent on functionality [6]. Numerous definitions of a notion of a shape have been proposed in the past, each and every one highlighting aspects relevant to a particular application of interest. The advent of digital imagery, together with the ubiquitous exploitation of its characteristics in a variety of applications, have triggered a renewed and keen interest in further refining and possibly unifying the notion of shape. The present contributed book is, to a large extent, motivated by this upsurge in activity and by the need for an update on recent accomplishments and trends. Theresearchactivityinshapeanalysisisdistinguishedbytwomainschools of thought: — Thefirstapproximatesshapesbyafinite-dimensionalrepresentation(aset of landmarks), which is then subjected to various transformations to account for variability and to subsequently derive models. — The second, on the other hand, interprets shapes as closed contours in an infinite-dimensional space, which, when subjected to transformations, morph into other shapes, thereby yielding a notion of similarity in the space of shapes. 1 Landmark-Based Shape Representation Shapeisaboutscale,orientation,andrelationshipamongtheso-calledcharac- teristic points/landmarks of an object-delineating contour. Such information about a data set better defines a shape. Equivalently, when such information istakenoutoftwodatasets,theresultingshapesmaybecompared.Aplanar shape commonly coincides with a closed curve enclosedin a regionof a plane Ω ∈ R2, bearing landmarks given by a vector τ = {(x1i,x2i)}i=1,...,N}. With viii Preface additional constraints on these coordinates (e.g., centered and normalized), theyrepresentaconstrainedsubsetofR2 alsoreferredtoasapreshapespace. If we subject a preshape τ (or rather the plane it lies in) to all rotations, we obtain orbits O(τ) of a preshape. Equivalence classesof shapes τi are a space of such orbits and form what is referred to as a shape space Σn, which was 2 shown to form a Riemannian manifold by Kendall1 [3] (see Chapters 13 and 15). A metric (or a geodesic) on this manifold, which affords a comparison of shapes, is induced by a metric on the preshape space (or a sphere of pre- shapes), and may be written as d[O(τ1),O(τ2)]=inf{d[θ1(τ1),θ2(τ2)]:0≤θ1,θ2 <2π}. (1) Related to landmark-based shapes, but independently proposed were graph- based representations of shapes (See Chapters 1, 2, 3, 4), with more recent extensions to 3D (or higher) shapes (See Chapters 6, 7, 8, 9, and 10). The so-called Shock Medial and Topological graphs may in fact be thought of as a collapse of a set of equivalent landmarks (an equivalence class) to a graph edge. The nodes of the graph depict transitions among different classes. Another twist on landmark-oriented shapes is the pioneering work of Grenander[2]ondeformabletemplates,whichinsteadsimplifiesashapebyus- ing a polygonal approximation to a shape (i.e., using a linear spline between two landmarks). The variability is addressed by rotation, translation, and scaling of linear segments in tight coordination with their neighbors so as to preserveacoherenceofashapeinthecourseofitsevolution.Thesedeformable models, together with those described above, have been extensively used in shape classification and recognition, and more efficient and novel techniques are continually being proposed. Alsorelatedareparticle-basedmodelsinspiredbypatternformationinsta- tistical physics. The particles may, for instance, be distributed over a region and interact as a system of spin particles to yield a shape (see Chapter 11). They may alsomodel a limiting caseofa landmark-basedshape where a particle diffuses along a trajectory describing the shape in question (see Chap- ter 12). Non-probabilistic versions of these limiting cases (infinite number of pointsonashape)formwhatisreferredtoasactivecontours,whichisfurther discussed below. 2 Infinite-Dimensional Shape Representation Analternativetothelandmarkapproachtoshaperepresentationandanalysis is the infinite-dimensional approachin which a shape is represented and ana- 1Similar ideas were independentlyproposed by Bookstein [1]. Preface ix lyzedinitsentirety2asthelocusofaninfinitenumberofpoints(asopposedto afinitenumberoflandmarkpoints).Astandardwaytorepresentsuchalocus istoassociatetoeachpointavalueofaparameterpdefinedoverarealinterval I (whichisoftenchosencanonicallytobetheinterval[0,1])andto(cid:1)encodeth(cid:2)e coordinatesofeachsuchpointbyamappingC :I →R2, C(p)= x(p),y(p) . This parametric representation was used in capturing object boundaries in images by way of so-called snakes or active contours first proposed by Kass, Witkin, and Terzopoulos [4]. While algorithmically convenient, this approach with its nonunique parameterization scheme presented a fundamental difficulty in developing a systematic machinery for basic analysis, such as computation of averages, distances, just to name a few. An alternative implicit representation addresses the issue of nonunique parameterizationandmaybefoundintheseminalworkofOsherandSethian [5],namelythatoflevel-set methods(seeChapters5,6,7).Here,areal-valued function ψ : Ω → R is defined over a domain Ω ∈ R2 where all the contour points reside.Agivenpoint (x,y)in this domainis then determinedeither to belong to or be excluded from the contour based upon the value of ψ(x,y). However, the problem of nonunique parameterizations is replaced here by nonunique choices of ψ(x,y) for points that do not belong to the contour. Both representations, therefore, suffer from an infinite-dimensional ambi- guity if our goal is to do shape analysis in their respectively corresponding functional spaces. The arbitrary parameterization of parametric approaches or arbitrary choice of level-set functions make shape analysis difficult: its evaluation for two different curves, for instance, may not necessarily be of analytical utility. Attempts to “geometrize” the parameter space for explicit representations (e.g., by using the arclength parameter) or the level-setfunc- tion for implicit representations (e.g., by using the popular signed distance function) are of little help since such geometric representations live in highly nonlinear spaces: the arithmetic mean of two equal length curves parameterized by arclength rarely yields a new curve parameterized by arclength, nor does the arithmetic meanofthe signeddistancefunctions oftwogivencurves yield a new signed distance function. 2.1 Analysis in Infinite-Dimensional Space While the ambiguities of implicit/explicit representations of shape may be overcomebyadoptingageometricformulation(signeddistancefunctionorarc length parameterization), one’s inability to operate in convenient functional 2It is inevitable that any representation or calculation implemented on a computer will ultimately be discretized and therefore be finite dimensional. We differ- entiate between finite- and infinite-dimensional shape representations, therefore, in terms of how they are mathematically modelled prior to final implementation on a computer. x Preface spacesstillpersists.Towardsmitigatingsuchlimitations,somerecentresearch has embarked on exploiting the machinery of Riemannian geometry. The fundamental problem in collective shape analysis is to derive a mea- sureofdistancebetweendifferentshapes.Suchametricinhandexpeditesthe derivation from first principles of various other statistics of shapes. To arrive at a distance measure in the framework of Riemannian geometry, one uses a differential approach. Failing to be a vector space, the space of shapes is a manifold M where two curves,C0 andC1, wouldlie as individual points.For thesakeofbriefandsimpleexposition,letMbethespaceofallclosed,simply connected,smooth3 planarcurves.Next,consideratrajectoryγ :[0,1]→M of smoothly varying curves (in M) between the two points C0 and C1 in M, starting from γ(0)=C0 and ending at γ(1)=C1. The next step is to assign a length to any such trajectory. The standard way to do this is to imagine dividing the trajectory γ into a large number of small incremental segments whoseindividuallengthsaresummedtogethertoobtainthetotallengthofγ. Since we consider smoothly varying trajectories, we may adopt the limiting process by integrating the differential increment of γ from 0 to 1. The differential increment dγ corresponds to an infinitesimal deformation of a curve (recallthateachpointalongthe trajectoryγ representsanentire curvetaken from the smooth morph from C0 to C1). We may represent this infinitesimal deformation by a vector field along the curve itself, where each vector indi- catesthedirectionandspeedwithwhichthecorrespondingpointonthecurve will evolve as we progress along the trajectory γ. We will denote this entire vector field along the curve by dγ, where t ∈ [0,1] represents the parameter dt forthetrajectoryγ.Thelengthofthetrajectoryisnowgivenbythefollowing integral. (cid:4) (cid:3) (cid:5) (cid:6) 1 dγ dγ Length(γ)= , dt (2) dt dt 0 Different adoptions of norms yield different algorithmic techniques (Chap- ters 12, 13, 14, 15). The final step is to consider all possible trajectories γ connecting two curvesC0andC1andtodefinethedistancebetweenC0andC1astheinfimum of the lengths of all such trajectories. 3 Goal of this Book While the history of shape analysis is long, the topic remains wide open and exciting. Fundamental problems in both schools of thought remain. This is 3For most of the contributed chapters in this volume, a sufficient notion of smoothness is that parametric representations of the curve are twice differentiable, therebygivingthecurveawell-definedunittangentandnormal,aswellascurvature at every point.