Computable Elastic Distances Between Shapes

We dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally dened from a left invariant Riemannian distance on an innite dimensional group acting on the curves, which can be explicitly computed. The obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed. An analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem, which can eciently be implemented, as illustrated by experiments. 1.1. Generalities. The problem of matching two objects together is very impor- tant in computer vision and shape recognition. In many cases, recognition is based on shapes (outlines), with the help of some suitably designed distance. A general principle is to associate with any pair (O1;O 2) of objects to be compared a measure of discrepancy d(O1;O 2 ):The recognition of an observed object O may be done by nding, from a dictionary of \templates," the previously recorded object Otemp, for which d(O;Otemp) is minimal. Clearly, the denition of the distance is the crucial step of the method, and much research has been done in this direction. We shall not try here to provide a review of the huge literature existing on the subject (see, for example, (17)) but rather focus on methods related to deformable templates, with which we are directly concerned. Instead of basing recognition on a nite collection of points of interest (primitives) taken from the outline of an object (corners, inflexion points, etc.), which is a popular way of handling the problem, our purpose is to base the comparison on the whole outline, considered as a plane curve. The distance we shall dene incorporates some deformation energy between the curves. The approach, as we will see, turns out to be intrinsic and robust to usual Euclidean transformations. The method is related to the wide literature on \snakes" (14), (7), (21) etc. in the way that our distance corresponds to some continuous process of deformation of one curve into another. It is also related to papers on elastic matching, such as (8); however, we provide an elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes, taking into account possible invariance to scaling or Euclidean transformations in the case they are required. From this point of view, our results are indebted to the seminal work of Grenander on group theory applied to pattern recognition (cf. (10) and (11), in particular; see also (1), (2), 12). Another source of inspiration may come from mathematical physics, since we are going to look to the path (process) of lowest energy which deforms one object into