On the optimality of shape and data representation in the spectral domain

TL;DR

This paper proves the optimality of Laplace-Beltrami eigenfunctions for representing smooth surface functions and explores their applications in shape analysis, including spectral shape processing, geodesic distance approximation, and enhanced PCA methods.

Contribution

It provides a theoretical proof of eigenfunction optimality and demonstrates practical applications in shape processing and data analysis using spectral methods.

Findings

Eigenfunctions of the Laplace-Beltrami operator are optimal for smooth surface function representation.

A scale-invariant pseudo-metric enables better geometric structure capturing.

Spectral methods improve efficiency in geodesic distance approximation and PCA.

Abstract

A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant-Fischer min-max principle adapted to our case. % The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilisation of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. % We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudo-metric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface, and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can handle cases that go beyond the scope defined by the observation set that is handled by regular PCA.