# Bilateral Operators for Functional Maps

## TL;DR

This paper introduces bilateral operators within the functional maps framework for shape correspondence, leveraging pointwise descriptors and intrinsic geometry to improve local smoothing and correspondence accuracy.

## Contribution

It develops a novel bilateral operator that combines pointwise descriptors and heat kernel information to enhance shape matching in the spectral domain.

## Key findings

- Bilateral operators improve shape correspondence accuracy.
- The method effectively exploits descriptor and geometry information.
- Enhanced local smoothing leads to better matching results.

## Abstract

A majority of shape correspondence frameworks are based on devising pointwise and pairwise constraints on the correspondence map. The functional maps framework allows for formulating these constraints in the spectral domain. In this paper, we develop a functional map framework for the shape correspondence problem by constructing pairwise constraints using point-wise descriptors. Our core observation is that, every point-wise descriptor allows for the construction a pairwise kernel operator whose low frequency eigenfunctions depict regions of similar descriptor values at various scales of frequency. By aggregating the pairwise information from the descriptor and the intrinsic geometry of the surface encoded in the heat kernel, we construct a hybrid kernel and call it the bilateral operator. Analogous to the edge preserving bilateral filter in image processing, the action of the bilateral operator on a function defined over the manifold yields a descriptor dependent local smoothing of that function. By forcing the correspondence map to commute with the Bilateral operator, we show that we can maximally exploit the information from a given set of pointwise descriptors in a functional map framework.