# Laplacian Spectral Basis Functions

arXiv: https://arxiv.org/abs/1906.03856 · HTML: https://mlsanity.com/paper/1906.03856/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2019.

## Built on by (most cited first, 2 of 2)

- [Non-Isometric Shape Matching via Functional Maps on Landmark-Adapted Bases](https://mlsanity.com/paper/2205.04800/) (2022)
- [A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domains](https://mlsanity.com/paper/2409.05508/) (2024)

## Abstract (arXiv)

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.