Piecewise rigid curve deformation via a Finsler steepest descent

TL;DR

This paper develops a Finsler steepest descent method in Banach spaces to better handle non-convex optimization problems, especially for curve matching with piecewise rigid deformations, by incorporating prior deformation constraints.

Contribution

It introduces a Finsler gradient descent framework in Banach spaces that extends previous generalized gradient methods to include non-Hilbertian norms for improved non-convex optimization.

Findings

Proves convergence of the Finsler gradient descent method.

Characterizes piecewise rigid deformations of curves.

Demonstrates applications to curve matching problems.

Abstract

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.