Structured Regularization of Functional Map Computations

TL;DR

This paper introduces a new regularization method for functional map shape matching based on the Laplacian resolvent, addressing theoretical limitations of previous approaches and improving correspondence accuracy.

Contribution

We propose a theoretically sound regularization approach using the Laplacian resolvent, enhancing shape matching accuracy over existing methods.

Findings

Improved shape correspondence quality across various scenarios.

The new regularizer outperforms traditional commutativity-based regularizers.

Enhanced pointwise correspondence accuracy before refinement.

Abstract

We consider the problem of non-rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace-Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well-justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one-parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly-used refinement techniques.