Piecewise rigid curve deformation via a Finsler steepest descent
Guillaume CharpiatINRIA Saclay, Université Paris Sud, Orsay, France
Giacomo NardiUniversité Paris IX - Paris Dauphine, France
Gabriel PeyréUniversité Paris IX - Paris Dauphine, France
François-Xavier VialardUniversité Paris IX - Paris Dauphine, France
This paper introduces a novel steepest descent flow in Banach spaces. This extends previous works on generalized gradient descent, notably the work of Charpiat et al. , to the setting of Finsler metrics. Such a generalized gradient allows one to take into account a prior on deformations (e.g., piecewise rigid) in order to favor some specific evolutions. We define a Finsler gradient descent method to minimize a functional defined on a Banach space and we prove a convergence theorem for such a method. In particular, we show that the use of non-Hilbertian norms on Banach spaces is useful to study non-convex optimization problems where the geometry of the space might play a crucial role to avoid poor local minima.We show some applications to the curve matching problem. In particular, we characterize piecewise rigid deformations on the space of curves and we study several models to perform piecewise rigid evolution of curves.
Cite this article
Guillaume Charpiat, Giacomo Nardi, Gabriel Peyré, François-Xavier Vialard, Piecewise rigid curve deformation via a Finsler steepest descent. Interfaces Free Bound. 18 (2016), no. 1, pp. 1–44DOI 10.4171/IFB/355