Finsler Steepest Descent with Applications to Piecewise-regular Curve Evolution

Guillaume Charpiat (1) , Giacomo Nardi (2) , Gabriel Peyré (2) , François-Xavier Vialard (2)

1 STARS - Spatio-Temporal Activity Recognition Systems
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Curve evolution; Finsler space; gradient flow; shape registration

Domaines: Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Guillaume Charpiat, Giacomo Nardi, Gabriel Peyré, François-Xavier Vialard. Finsler Steepest Descent with Applications to Piecewise-regular Curve Evolution. 2013. ⟨hal-00849885v1⟩