Composition Operators, Matrix Representation, and the Finite Section Method: A Theoretical Framework for Maps between Shapes

TL;DR

This paper develops a theoretical framework for the functional maps approach in shape analysis, connecting it to composition operators, matrix representations, and the Finite Section Method, and discusses numerical solutions with convergence guarantees.

Contribution

It introduces a functional analytic foundation for functional maps, linking them to composition operators and matrix representations, and analyzes numerical methods for solving related equations.

Findings

Functional maps are modeled as composition operators between manifolds.

Two variants of the Finite Section Method are analyzed for solving operator equations.

Convergence results are established for the proposed numerical methods.

Abstract

This paper intends to lay the theoretical foundation for the method of functional maps, first presented in 2012 by Ovsjanikov, Ben-Chen, Solomon, Butscher and Guibas in the field of the theory and numerics of maps between shapes. We show how to analyze this method by looking at it as an application of the theories of composition operators, of matrix representa- tion of operators on separable Hilbert spaces, and of the theory of the Finite Section Method. These are three well known fruitful topics in functional analysis. When applied to the task of modelling of correspondences of shapes in three-dimensional space, these concepts lead directly to functional maps and its associated functional matrices. Mathematically spoken, functional maps are composition operators between two-dimensional manifolds, and functional matrices are infinite matrix representations of such maps. We present an introduction into the notion and theoretical foundation of the functional analytic framework of the theory of matrix repre- sentation, especially of composition operators. We will also discuss two numerical methods for solving equations with such operators, namely, two variants of the Rectangular Finite Section Method. While one of these, which is well known, leads to an overdetermined system of linear equations, in the second one the minimum-norm solution of an underdetermined system has to be computed. We will present the main convergence results related to these methods.