L. Younes. Computable elastic distances between shapes. SIAM Journal of Applied Mathematics, 58:565--586, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....clear distinction between motion and deformation. Another line of work uses variational methods and the solution of partial di#erential equations (PDEs) to model shape and to compute distances and similarity. In this framework, not only can the notion of alignment or distance be made precise [5, 43, 31, 20, 35], but quite sophisticated theories that encompass perceptually relevant aspects, can be formalized in terms of the properties of the evolution of PDEs (e.g. 21] The work of Kimia et al. 18] describes a scale space that corresponds to various stages of evolution of a di#using PDE, and a ....

L. Younes. Computable elastic distances between shapes. SIAM J. of Appl. Math., 1998.

....(x u) dx. 16) We want to compute the derivative of G along a test function v G(u zv) G(u) lim (17) We first introduce the following notations 1 : c(el (x u) H = H( x u) c( x u) By expanding the numerator in (17) up to first order, we get the following three terms DA 21ve(x u)lE ve Ix u .v]dx (18) ID.fSH2 V(p(x u)V[ aV(p Ix u oF] dx (19) Iv x u l Now we app integration by pas on (19) and assume Neum bod conditions, we get (19) ldiv(D, rSH VO (x u) Ive(x (20 Thus, term (18) is canceled the numerator of (17) by negative (18) from the expansion of (19) ....

....the derivative of G along a test function v G(u zv) G(u) lim (17) We first introduce the following notations 1 : c(el (x u) H = H( x u) c( x u) By expanding the numerator in (17) up to first order, we get the following three terms DA 21ve(x u)lE ve Ix u . v]dx (18) ID.fSH2 V(p(x u)V[ aV(p Ix u oF] dx (19) Iv x u l Now we app integration by pas on (19) and assume Neum bod conditions, we get (19) ldiv(D, rSH VO (x u) Ive(x (20 Thus, term (18) is canceled the numerator of (17) by negative (18) from the expansion of (19) Let us her work on the tegrd of the ....

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L. Younes, "Computable elastic distances between shapes," Siam Journal on Applied Mathematics, vol. 58, pp. 565-586, 1998.

....refers to the use of warping techniques and free form deformations that are quite popular in graphics, animation and rendering [4] The essence of traditional FFD is to deform an object by manipulating a regular control lattice P overlaid on the volumetric embedding space. Thin plate splines [6, 19] is a popular non rigid transformation technique that requires finding two sets of corresponding landmark points, a difficult and ill posed problem. Cubic B spline based Free Form Deformations (FFD) is an alternative to TPS. It can model local transformations in a computationally efficient manner. ....

L. Younes. Computable elastic distances between shapes. SIAM Journal of Applied Mathematics, 58:565--586, 1998.

....manifold, and the variations between the shapes are modeled by the action of Lie groups (deformations) on this manifold. Low dimensional groups, such as rotation, translation and scaling, keep the shapes unchanged, while high dimensional di#eomorphism groups smoothly change the object shapes ([30, 33, 32, 20, 8]) This idea forms a mathematical basis of the deformable template theory outlined in [7] One limitation of this approach is the need to embed shapes in bigger Euclidean spaces (R ) such that di#eomorphism groups can be applied. Since the cost of finding di#eomorphisms is high, these methods ....

L. Younes. Computable elastic distance between shapes. SIAM Journal of Applied Mathematics, 58:565--586, 1998.

.... shape fixed (see for example [8] while the high dimensional groups, such as di#eomorphism groups, alter the shapes ( 7] This representation forms a mathematical basis of deformable templates as treated in [6, 1] Use of di#eomorphisms in matching images (or shapes) has also been described in [17, 19, 18, 12]. Such tools have found remarkable success in analysis of anatomical shapes. However, one limitation is their need for embedding shapes in larger Euclidean spaces, e.g. curves in R surfaces in R . After embedding, one derives di#eomorphic transformations between two Euclidean spaces such ....

....between two Euclidean spaces such that the embedded shapes transform from one to another. Computational cost of di#eomorphic solutions is rather large, and this paper suggests more e#cient algorithms based on intrinsic transformations amongst the shapes without requiring Euclidean embeddings. In [19, 18], Younes quantifies di#erences of planar (open and closed) shapes using metrics derived from the energy needed to deform a shape into another via elastic deformations that allow both stretching and bending. The consideration of both forms of elastic energy has the advantage of (infinitesimally) ....

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L. Younes. Computable elastic distance between shapes. SIAM Journal of Applied Mathematics, 58:565--586, 1998.

....of shape context based on a point cloud representation [4] gives an alignment and nally a distance. Since at the matching stage only distances are used, this method falls into the metric space class. Also, approaches where an outline curve is elastically deformed into correspondence with another [3, 12, 33], and those that rely on matching medial axis representations [34, 18, 26, 28, 21, 30] fall into this class. Unfortunately, unlike the rst class of representations where the structure of the underlying Euclidean space is used to organize and query the database, it is not clear how the metric ....

L. Younes. Computable elastic distance between shapes. SIAM J. Appl. Math., 58:565-586, 1998.

....robust to visual transformations like articulation and deformation of parts, viewpoint variation, occlusion. Thus, the shape representation has to effectively capture the variations in shape due to these transformations. In previous recognition applications, shapes have been represented as curves [1, 2], point sets or feature sets, and by medial axis [3, 4, 5, 6, 7] among others. This paper compares two techniques for matching shapes, one based on matching their outline curves [8] and the second based on matching their shock graphs [9] In many object recognition and content based image ....

....recognition and content based image indexing applications, the objects are represented by their outline curves and matched. Outline curves typically do not represent a notion of the interior of the shapes. Despite this well known drawback, it has been effectively used in certain applications [1, 10, 2]. Matching typically involves finding a mapping from one curve to the other that minimizes an elastic performance functional, which penalizes stretching and bending [2] The minimization problem in the The support of NSF grants IRI 9700497 and BCS 9980091 is gratefully acknowledged. ....

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Laurent Younes, "Computable elastic distance between shapes," SIAM J. Appl. Math., vol. 58, pp. 565-- 586, 1998.

....of the outline based representation by examining the e ectiveness of the recognition system on a variety of examples. 1 Introduction The representation of the shape of objects can have a signi cant impact on the effectiveness of a recognition strategy. Shapes have been represented as curves [12, 22, 15, 2, 7, 23], point sets [1, 16, 21] feature sets [3, 8] and by medial axis [24, 18, 19, 14, 13, 11, 10] among others. This paper develops an approach to object recognition based on a curve based representation of shape outline using the proposed concept of an alignment curve, and identi es the strengths ....

....and content based image indexing applications, the object outlines are represented as curves and matched. The matching relies on either aligning feature points using an optimal similarity transformation [1, 16, 21] or on a deformation based approach to aligning the properties of the two curves [12, 22, 15, 2, 7, 6, 23]. Deformation based methods typically involve nding a mapping from one curve to the other that minimizes an elastic performance functional, which penalizes stretching and bending [4, 20, 2, 23] The minimization problem in the discrete domain is transformed into one of matching shape ....

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L. Younes. Computable elastic distance between shapes. SIAM J. Appl. Math., 1996.

....2 Z 1 0 (AE ff ff Gamma AE fi OE fi OE ) 2 dt 2 Z 1 0 (AE ff ff Gamma AE fi OE fi OE ) 2 dt The problem of matching curves across different brains is defined as one of finding the particular diffeomorphism that minimizes the above distance metric. This is akin to the work of Younes [4]. Problem Statement: Given two curves ff(t) fi(t) t 2 [0; 1] and the set of diffeomorphisms Phi = fOE : OE(ff(t) fi(t)g, matching the two curves is equivalent to finding the diffeomorphism that minimizes the above distance metric with a constraint on the velocity field: OE(ff; fi) j ....

L. Younes, "Computable elastic distances between shapes," To appear in SIAM J. Applied Math., 1998. 9

....premise of this approach is that the goodness of the optimal match is the sum of goodness of the optimal matches be tween two corresponding subsegments. This allows an energy functional to convey the goodness of a match as a function of the correspondence or alignment of the two curves [5, 16, 7]. Let a mapping g, g : 0; L] 0; L] g(s) s; which represents an alignment of the two curves. Cohen et al. 5] uses bending and stretching energies in a physical analogy similar to the one used in formulating active contours or snakes [10] Specifically they compare the displacement ....

.... active contours or snakes [10] Specifically they compare the displacement velocities and bending energies in the form of [g] R C fi fi s ( C(s) Gamma C(s) fi fi 2 ds R R C ( C (s) Gamma C (s) 2 ds where is the curvature along the curves and s = g(s) Younes [16] uses a similar functional which measures the variation in the displacement vector ( Gamma C C) and can be written as [g] R C (g s Gamma 1) 2 ds R C ( s) ffi g(s) Gamma (s) 2 ds where and are the angles that the curves C and C make with the horizontal axis, ....

L. Younes. Computable elastic distance between shapes. SIAM Journal of Applied Mathematics, 1996. (a) (b) (c)

.... with respect to and 0 (matching to 0 or 0 to are distinct operations) As an example of symmetric matching functional, let us quote [1] with (among other proposals) L ; 0 (OE) Z I j OE(x) Gamma 1jdx Z I j (x) Gamma OE(x) 0 ffi OE(x)jdx (2) or [13], with L ; 0 (OE) length(I) Gamma Z I q OE(x) fi fi fi fi cos (x) Gamma 0 ffi OE(x) 2 fi fi fi fi dx : 3) The last two examples provide, after minimization over OE, a distance d( 0 ) between the functions and 0 . In this paper, all the matching ....

....interpreted from a geometrical point of view, as rescaling (a portion of) a plane curve so that it has, let s say, length 1. But applying such a scale change may have some impact not only on the variable x (which here represents the length) but also on the values of the geometric features . In [13], for example, the geometric features were the orientations of the tangents, which are not affected by scale change, so that focus invariance is in this case equivalent to geometric scale invariance. Letting be the curvature computed along the curve, the same invariance would be true if we had ....

L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math, 58 (1998), pp. 565--586.

....set of the measurable functions from a finite dimensional Riemannian compact manifold M without boundary to a finite dimensional manifold X. This definition matches numerous examples. For instance, the closed curves in R 2 correspond to the case M = R=Z (1 dimensional torus) and X = R 2 (see [12, 11]) The case of the periodic grey level images correspond to M = R 2 =Z 2 (2 dimensional torus) and X = R. In both last case, X is a vector space. However, more general situations arise if we work with images with bounded greylevel values (X = R or X = a; b] or with images where f(m) is ....

L. Younes. Computable elastic distance between shapes. Preprint, 1995. LMENS/DIAM, URA 762, Ecole Normale Sup' erieure, 45, rue d'Ulm, 75230 Paris cedex 05, France E-mail address: trouve@ens.ens.fr

....object rather than the other, and symmetrical matching seems appealing in many contexts. This symmetry constraint will be one of the features we will discuss in the design of a matching functional. As an example of symmetric matching functional, let us quote the one which has been introduced in [11], which is L ; 0 (OE) Z I q OE(x) fi fi fi fi cos (x) Gamma 0 ffi OE(x) 2 fi fi fi fi dx (2) which in this case has to be maximized in OE. In fact, it has been shown that d( 0 ) arccos(sup(L ; 0 ( is a distance (satisfying, in particular, the triangle ....

....a geometrical point of view, as rescaling (a portion of) a plane curve so that it has, let s say, length 1. But, in this case, applying such a scale change may have some impact not only on the variable x (which here represents the length) but also on the values of the geometric features . In [11], for example, the geometric feature was the orientation of the tangents, which is not affected by scale change, so that focus invariance is in this case equivalent to geometric scale invariance. Letting be the curvature computed along the curve, the same invariance would be true if we had taken ....

L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math, 58 (1998), pp. 565--586.

....is defined, and a probability model describing the shapes of small organelles is devised from the deformation of a circle. In [15] an infinite dimensional group action is introduced to model deformations of plane curves, in polar representation (distance to the origin vs. rotating angle) In [22], a similar group action is introduced, acting on the representation s 7 (s) of a plane curve, s being the arc length and (s) the unit tangent vector, with [3] containing transformations of curves in three dimensions. In [21] a theoretical study of a large class of infinite Date: December 8, ....

....dx fi fi fi fi 2 dxdt Z 1 0 Z 1 0 L fi fi fi fi T t T x v fi fi fi fi 2 dtdx : 22) where we have made the usual change of variables v = dg dt ffi g Gamma 1 and T = T 0 ffi g Gamma1 . For this particular context, the distance D on A can be explicitely computed (cf. [22]) Letting a 0 = g 0 ; L 0 ; T 0 ) and a 1 = g 1 ; L 1 ; T 1 ) be two registered objects, we have D(a 0 ; a 1 ) L 0 L 1 Gamma p 2L 0 L 1 Z 1 0 p g 0 (x) g 1 (x) 1 hT 0 (x) T 1 (x)i)dx : The associated optimal deformation paths being also explicitely known. 22 M. I. MILLER AND ....

L Younes. Computable elastic distances between shapes. SIAM J. Appl. Math, 58(2):565--586, 1998.

.... to match plane curves on the basis of their curvatures; for example, one of them takes the form U(g) Z 1 0 j g Gamma 1jds Z 1 0 ( g 1)jf( I ffi g) Gamma f( J )jds where f( c Gamma sign( e Gammaff , or U(g) Z 1 0 j g Gamma 1jds Z 1 0 j g I ffi g Gamma J jds In [71], matching was performed using U(g) arccos p 2 2 Z 1 0 p g (1 h I ffi g ; J i)ds To match 3D curves, the authors in [11] have used a characterization through the Fr enet s frame instantaneous rotation, which is a 3 by 3 matrix depending in the curvature and the torsion of the curve. The ....

L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math, 58 (1998), pp. 565--586.

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L. Younes. Computable elastic distances between shapes. SIAM Journal of Applied Mathematics, 58:565--586, 1998.

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L. Younes. Computable elastic distances between shapes. SIAM Journal of Applied Mathematics, 58:565--586, 1998.

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L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math. 58(2) (1998), 565--586.

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L. Younes. Computable elastic distance between shapes. SIAM J. Appl. Math., 58:565--586, 1998.

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L. Younes. Computable elastic distances between shapes. SIAM J. Appl. Math., 58(2):565--586, 1998.

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Younes, L.: Computable Elastic Distance Between Shapes. SIAM Journal of Applied Mathematics 58 (1998) 565--586. 6

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Younes, L. [1998], Computable elastic distances between shapes, SIAM J. Appl. Math. 58, 565--586.

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L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), pp. 565--586.

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