# Laplacian Spectral Basis Functions

## TL;DR

This paper explores various Laplacian-based spectral basis functions for 3D shape analysis, comparing their properties and computational aspects to improve signal approximation and shape processing techniques.

## Contribution

It introduces diffusion and Laplacian spectral basis functions, analyzing their properties and providing a comprehensive comparison with existing basis functions for shape analysis.

## Key findings

- Laplacian spectral basis functions exhibit desirable invariance and locality properties.
- Diffusion basis functions offer improved stability over traditional eigenfunctions.
- Geometric metrics effectively characterize basis function properties.

## Abstract

Representing a signal as a linear combination of a set of basis functions is central in a wide range of applications, such as approximation, de-noising, compression, shape correspondence and comparison. In this context, our paper addresses the main aspects of signal approximation, such as the definition, computation, and comparison of basis functions on arbitrary 3D shapes. Focusing on the class of basis functions induced by the Laplace-Beltrami operator and its spectrum, we introduce the diffusion and Laplacian spectral basis functions, which are then compared with the harmonic and Laplacian eigenfunctions. As main properties of these basis functions, which are commonly used for numerical geometry processing and shape analysis, we discuss the partition of the unity and non-negativity; the intrinsic definition and invariance with respect to shape transformations (e.g., translation, rotation, uniform scaling); the locality, smoothness, and orthogonality; the numerical stability with respect to the domain discretisation; the computational cost and storage overhead. Finally, we consider geometric metrics, such as the area, conformal, and kernel-based norms, for the comparison and characterisation of the main properties of the Laplacian basis functions.