Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

Abstract

Abstract. This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the characteristic functions of the subsets of R 2 and their distance functions. The L 2 norm of the difference of characteristic functions, the L ∞ and the W 1,2 norms of the difference of distance functions define interesting topologies, in particular the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with image shapes defined on finite grids of pixels, we restrict our attention to subsets of R 2 of positive reach in the sense of Federer [16], with smooth boundaries of bounded curvature. For this particular set of shapes we show that the three previous topologies are equivalent. The next problem we consider is that of warping a shape onto another by infinitesimal gradient descent, minimizing the corresponding distance. Because the distance function involves an inf, it is not differentiable with respect to the shape. We propose a family of smooth approximations of the distance function which are continuous with respect to the Hausdorff topology, and hence with respect to the other two topologies. We compute the corresponding Gâteaux derivatives. They define deformation flows that can be used to warp a shape onto another by solving an initial value problem. We show several examples of this warping and prove properties of our approximations that relate to the existence of local minima. We then use this tool to produce computational definitions of the empirical mean and covariance of a set of shape examples. They yield an analog of the notion of principal modes of variation. We illustrate them on a variety of examples.

Generalized Gradients: Priors on Minimization Flows

“…In this context, the energy functional to be minimized is a measure of dissimilarity between the evolving contour and a target contour. The study of shape metrics is still an active research area [34,33,7,32], and there are many candidates for the dissimilarity measure. In this paper, we use a differentiable approximation of the well-known Hausdorff distance, as proposed in [7], to warp the contours of two different hands.…”

“…The study of shape metrics is still an active research area [34,33,7,32], and there are many candidates for the dissimilarity measure. In this paper, we use a differentiable approximation of the well-known Hausdorff distance, as proposed in [7], to warp the contours of two different hands. Figure 1 compares the evolution of the contour when using the L 2 gradient descent (top row) and a modified gradient descent favoring rigid plus scaling motions (bottom row) as in paragraph 4.1.1.…”

“…We use here the W 1,2 -norm of the distance functions as proposed in [7]. Despite an initial rigid registration, the L 2 gradient descent is unable to give satisfying results.…”

An SL(2) Invariant Shape Median

“…The notion of a shape space was already introduced by Kendall [23] in 1984. Charpiat et al [9,10] discuss shape averaging and shape statistics based on the notion of the Hausdorff distance and on the H 1 norm of the difference of the signed distance functions of shapes. They study gradient flows for energies defined as functions over these distances for the warping between two shapes.…”

A New Prior Shape Model for Level Set Segmentation

“…The localization along the level set shall enhance the control of the edge based stopping functional. To formulate the shape energy, we adopt the symmetric dissimilarity measure betweentwoshapesproposedbyChanandZhu [3],Riklin-Raviv [11],Charpiat [4]:…”