Elastic comparison of curves using distances built on stichastic and deterministic models.

Summary This thesis deals with curve comparison in image processing. In the first part, we build a distance from a stochastic deformation model of a curve closed by random similarities. We study the displacement between the initial curve and the deformed curve. Starting from a discrete curve model, we establish the continuous limit model, which is identified with a stochastic diffusion, solution of a stochastic differential equation. In the demonstration we highlight the relevant types of similarity variances. We give a method for simulating the random displacement, and then argue for a modification of the displacement to have displacements with zero barycenter. These modified displacements are no longer diffusions. We then use a large deviation inequality to deduce from this type of displacement a distance between curves. We then show that by choosing a particular variance function, we find a distance between curves already used in image processing, although found according to other premises, thus bridging the gap between purely deterministic methods and a stochastic method. In the second part, we start from an already established distance and show the need for a multiscale algorithm for the computation of optimal parameters. We describe methods to speed up the minimization process, and provide a criterion for deciding whether or not to use a multi-resolution analysis. Application examples are given.