Section: New Results

Steepest Descent in Banach Spaces with Application to Piecewise-Rigid Evolution of Curves

Participant : Guillaume Charpiat.

This is joint work with Gabriel Peyré (CNRS, Ceremade, Université Paris-Dauphine). We intend to favor piecewise-rigid motions, i.e. articulated movements, during shape evolutions, especially when computing morphings or image segmentation with shape prior. To do this, we first need a dissimilarity measure between shapes, whose gradient is meaningful. We formulate one using kernels and bistochastization.

The parameters of these kernels are automatically estimated in a fixed-point scheme that guarantees physical relevance, and the notion of bistochastization is extended to continuous distributions. Finally, piecewise rigidity is ensured during gradient descents by a change of the norm from which the gradient is derived. This norm is formulated so as to favor sparse second derivatives, which produces articulated movements without knowing by advance the location of the articulations.

The formula of the norm is actually elegantly simple, involving simple geometric quantities, derivatives, and the $L_1$ norm. Note that this norm does not derive from an inner product but defines a gradient in the sense of [5] as the minimizer of an energy. It turns out that in our case the energy defining the gradient is actually convex, and efficient minimization follows.