A PDE-based Method for Shape Registration

TL;DR

This paper introduces a PDE-based numerical method for shape registration that efficiently solves a non-convex variational problem, providing global convergence and improved computational performance.

Contribution

It presents a novel PDE approach using Hamilton-Jacobi-Bellman equations for shape registration, with a backtracking scheme for geodesic computation.

Findings

Quadratic complexity and global convergence of the method

Linear numerical convergence and improved efficiency

Ability to compute shape space geodesics

Abstract

In the square root velocity framework, the computation of shape space distances and the registration of curves requires solution of a non-convex variational problem. In this paper, we present a new PDE-based method for solving this problem numerically. The method is constructed from numerical approximation of the Hamilton-Jacobi-Bellman equation for the variational problem, and has quadratic complexity and global convergence for the distance estimate. In conjunction, we propose a backtracking scheme for approximating solutions of the registration problem, which additionally can be used to compute shape space geodesics. The methods have linear numerical convergence, and improved efficiency compared previous global solvers.