Model Order Reduction for Efficient Descriptor-Based Shape Analysis

TL;DR

This paper introduces a model order reduction framework for PDE-based shape descriptors, significantly improving efficiency and accuracy in 3D shape analysis, and provides a comprehensive comparison of spectral and integration methods.

Contribution

It presents a novel MOR-based computational framework for shape analysis, including technical innovations and a detailed comparison of existing PDE solution techniques.

Findings

MOR significantly improves PDE integration efficiency.

The framework yields more accurate shape signatures.

Soft correspondences enhance the MOR approach.

Abstract

In order to investigate correspondences between 3D shapes, many methods rely on a feature descriptor which is invariant under almost isometric transformations. An interesting class of models for such descriptors relies on partial differential equations (PDEs) based on the Laplace-Beltrami operator for constructing intrinsic shape signatures. In order to conduct the construction, not only a variety of PDEs but also several ways to solve them have been considered in previous works. In particular, spectral methods have been used derived from the series expansion of analytic solutions of the PDEs, and alternatively numerical integration schemes have been proposed. In this paper we show how to define a computational framework by model order reduction (MOR) that yields efficient PDE integration and much more accurate shape signatures as in previous works. Within the construction of our framework we introduce some technical novelties that contribute to these advances, and in doing this we present some improvements for virtually all considered methods. As part of the main contributions, we show for the first time an extensive and detailed comparison between the spectral and integration techniques, which is possible by the advances documented in this paper. We also propose here to employ soft correspondences in the context of the MOR methods which turns out to be highly beneficial with this approach.