Functional Maps Representation on Product Manifolds

TL;DR

This paper introduces a novel framework for representing and manipulating shape maps using densities on product manifolds, leveraging spectral analysis and discretization for efficient processing across 2D and 3D shapes.

Contribution

It proposes a new geometric and spectral approach to shape map representation using product manifolds, including discretization and localized spectral analysis techniques.

Findings

Framework applies to 2D and 3D shape maps without adjustments

Efficient implementation with sparse matrix operations

Provides new tools for map analysis and manipulation

Abstract

We consider the tasks of representing, analyzing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace--Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.