Shape Analysis via Functional Map Construction and Bases Pursuit

TL;DR

This paper introduces a novel method for shape analysis that constructs functional maps and basis functions optimized for feature matching, improving accuracy over traditional Laplace--Beltrami-based approaches.

Contribution

It presents a new spectrum selection approach for functional maps, leveraging high-dimensional degrees of freedom and regularized optimization for better shape correspondence.

Findings

Improved shape matching accuracy demonstrated on multiple datasets.

Effective optimization scheme with good convergence properties.

Versatile application to quadrangulation and function transfer.

Abstract

We propose a method to simultaneously compute scalar basis functions with an associated functional map for a given pair of triangle meshes. Unlike previous techniques that put emphasis on smoothness with respect to the Laplace--Beltrami operator and thus favor low-frequency eigenfunctions, we aim for a spectrum that allows for better feature matching. This change of perspective introduces many degrees of freedom into the problem which we exploit to improve the accuracy of our computed correspondences. To effectively search in this high dimensional space of solutions, we incorporate into our minimization state-of-the-art regularizers. We solve the resulting highly non-linear and non-convex problem using an iterative scheme via the Alternating Direction Method of Multipliers. At each step, our optimization involves simple to solve linear or Sylvester-type equations. In practice, our method performs well in terms of convergence, and we additionally show that it is similar to a provably convergent problem. We show the advantages of our approach by extensively testing it on multiple datasets in a few applications including shape matching, consistent quadrangulation and scalar function transfer.