J Math Imaging Vis 25 341–352, 2006 c ○ 2006 Springer Science + Business Media, LLC. Manu.pdf
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1、J Math Imaging Vis25: 341352, 2006c 2006 Springer Science + Business Media, LLC. Manufactured in The Netherlands.DOI: 10.1007/s10851-006-7249-8HarmonicHarmonic EmbeddingsEmbeddings forfor LinearLinear ShapeShape AnalysisAnalysisALESSANDRO DUCIComputer Science Department, University of California at
2、Los Angeles, Los Angeles - CA 90095alessandro.ducisns.itANTHONY YEZZIElectrical and Computer Engineering, Georgia Institute of Technology,Atlanta - 30332ayezziece.gatech.eduSTEFANOSOATTOComputer Science Department, University of California at Los Angeles, Los Angeles - CA 90095soattoucla.eduKELVINRO
3、CHAElectrical and Computer Engineering, Georgia Institute of Technology,Atlanta - 30332gtg185jmail.gatech.eduPublishedPublished online:online: 9 9 OctoberOctober 20062006Abstract.Abstract.We present a novel representation of shape for closed contours in R2or for compact surfaces in R3explicitlydesig
4、nedtopossessalinearstructure.Thisgreatlysimplifieslinearoperationssuchasaveraging,principalcomponent analysis or differentiation in the space of shapes when compared to more common embedding choicessuch as the signed distance representation linked to the nonlinear Eikonal equation. The specific choi
5、ce of implicitlinear representation explored in this article is the class of harmonic functions over an annulus containing thecontour. The idea is to represent the contour as closely as possible by the zero level set of a harmonic function,thereby linking our representation to the linear Laplace equ
6、ation. Wenote that this is a local represenation withinthe space of closed curves as such harmonic functions can generally be defined only over a neighborhood of theembedded curve. We also make no claim that this is the only choice or even the optimal choice within the classof possible linear implic
7、it representations. Instead, our intent is to show how linear analysis of shape is greatlysimplified (and sensible) when such a linear representation is employedin hopes to inspire newideas and additionalresearch into this type of linear implicit representations for curves. We conclude by showing an
8、 application forwhich our particular choice of harmonic representation is ideally suited.1. 1.IntroductionIntroductionThe analysis and representation of shape is at thebasis of many visual perception tasks, from classifi-cation and recognition to visual servoing.This is avastand complex problem, whi
9、ch we have no intention ofaddressing in its full generality here. Instead, we con-centrate on a specific issue that relates to the represen-tation of closed, planar contours or compact surfacesin 3D space. Even this issue has receivedconsiderableattention in the literature. In particular, in their w
10、orkon statistical shape influence in segmentation 20,342Duci et al.Leventon et al. have proposed representing a closedplanar contour as the zero level set of a function inorder to perform linear operations such as averag-ing or principal component analysis. The contour isrepresented by the embedding
11、 function, and all op-erations are then performed on the embedded repre-sentation. They choose as their embedding functionthe signed distance from the contour (whose differen-tial structure is described by the non-linear EikonalEquation) and implement its evolution in the numer-ical framework of lev
12、el sets pioneered by Osher andSethian 27.While this general program has proven effectivein various applications, the particular choice of em-bedding function presents several difficulties, becausesigned distance functions are not a closed set underlinear operations: the sum or difference of two sign
13、eddistance functions is not a signed distance function(an immediate consequence of their nonlinear differ-ential structures). Consequently, the space cannot beendowed with a probabilistic structure in a straight-forward manner, and repeated linear operations, in-cludingincrementsanddifferentiation,e
14、ventuallyleadto computational difficulties that are not easily ad-dressed within this representation. Alternative meth-ods that possess a linear structure rely on parametricrepresentations. For instance various forms of splines6, 7, cannot guarantee that topology (or even theembeddedness) of the sha
15、pe is preserved under sig-nificant variations of the control points. Furthermore,such representations are not geometric as they dependupon an arbitrary choice of parameterization for thecontour.Geometricizedparametricrepresentationsuti-lizing the arclength parameter suffer from the samenonlinearity
16、problem as implicit representations uti-lizing the signed distance to the curve. While it ispossible to generalize the notions of mean shapes andPCAtononlinearrepresentations,especiallywhentheycan be endowed with a Riemannian structure, (seefor example 17), it is decidely more complicatedand often i
17、nvolves rather expensive computationalalgorithms.In this paper, we present a novel implicit represen-tation of shape for closed planar contours and compactsurfacesinR3thatisgeometricand explicitlydesignedto possess a (locally) linear structure. This allows lin-ear operations such as principal compon
18、ent analysis ordifferentiation to be naturally defined and easily car-ried out. The basic idea consistsof, again,representingthecontourorsurfaceasthezerolevelsetofafunction,but this time the function belongs to a linear (or quasi-linear1) space. While previous methods relied on the(non-linear) Eikon
19、al equation, ours relies on Laplaceequation,whichislinear.Ourrepresentationallowsex-ploring the neighborhood of a givenshape while guar-anteeingthatthetopologyandtheembeddednessoftheoriginalshapeispreserved,evenunderlargevariationsof the parameters in our representation (the boundaryvalues of the ha
20、rmonic function).We should point out that the primary goal here isto show through an example how a linear implicit rep-resentation of contours or surfaces can simplify (andjustify) linear operations such as averaging and prin-ciple component analysis. It is not our claim that theuse of Laplaces equa
21、tion is the optimal choice. It is,however,a well studied PDE whose known propertiesallowustoconcludevariousanalyticalandtopologicalproperties of the curves we seek to represent. Furthersome recent attention to shape analysis via Rieman-nain structures based on nonlinear representations ofcurves usin
22、g harmonic functions is presented in 29(The authors wonder, in fact, if the linear representa-tionpresentedheremaybetiedtooflocallinearizationof this representation in the neighborhood of a givenshape).WealsoshowinSection5anapplicationthatisideallysuitedforourparticularchoiceofharmonicem-bedding. Be
23、yond these considerations, however,mostof what we are illustrating in this work could carrythrough for other classes of linear or quasilinear em-bedding functions.Weintroducethesimplestformofharmonicembed-ding in Section 2, where we point out some of its diffi-culties. Wethen extend the representati
24、on to a relatedanisotropicoperator inSection3,anddiscussitsfinite-dimensional implementation in Section 4. In Section 5we show an application to measuring tissue thicknessonsegmentedmedicalimagedata.Finally,weillustratesome of the properties of this representation in Section6. While the detailed dis
25、cussion is restricted to the 2Dcase, the extension to 3D is straight-forward and obvi-ous.Wethereforewillmakenotrepeatthesamedetailsin 3D, but we will however show 3D examples in theresults section.1.1.Relation to PreviousWorkThe literature on shape modeling and representation istoo vast to review i
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