Self Functional Maps

TL;DR

Self Functional Maps provide a novel algebraic surface representation that is invariant under isometries, enabling surface classification by comparing matrices derived from eigenfunctions of different surface Laplacians.

Contribution

The paper introduces Self Functional Maps, a new surface representation method that encodes geometric information into matrices, facilitating shape comparison and classification.

Findings

The method produces a universal isometry invariant matrix for each surface.

Surface similarity can be effectively measured through algebraic distances between matrices.

The approach leverages eigenfunctions of regular and scale-invariant Laplacians for robust shape analysis.

Abstract

A classical approach for surface classification is to find a compact algebraic representation for each surface that would be similar for objects within the same class and preserve dissimilarities between classes. We introduce Self Functional Maps as a novel surface representation that satisfies these properties, translating the geometric problem of surface classification into an algebraic form of classifying matrices. The proposed map transforms a given surface into a universal isometry invariant form defined by a unique matrix. The suggested representation is realized by applying the functional maps framework to map the surface into itself. The key idea is to use two different metric spaces of the same surface for which the functional map serves as a signature. Specifically, in this paper, we use the regular and the scale invariant surface laplacian operators to construct two families of eigenfunctions. The result is a matrix that encodes the interaction between the eigenfunctions resulted from two different Riemannian manifolds of the same surface. Using this representation, geometric shape similarity is converted into algebraic distances between matrices.