Laplacian Spectral Basis Functions

TL;DR

The main aspects of signal approximation are addressed, such as the definition, computation, and comparison of basis functions on arbitrary 3D shapes, and the main properties of the Laplacian basis functions are discussed.

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.

Figures (17)

• Figure 1: Overview of the proposed approach for the definition of Laplacian spectral basis functions based on the solution to the harmonic equation, the Laplacian eigenproblem, and the diffusion equation, and on the filtering of the Laplacian spectrum.
• Figure 2: Distribution and shape of the level-sets of harmonic basis functions, centred at seed points.
• Figure 3: Level-sets of three Laplacian eigenfunctions.
• Figure 4: (b,c) Level-sets of the diffusion basis functions at different scales and centered at different seed points in (a). The example confirms the locality, smoothness, and shape-awareness of the diffusion basis functions, computed with the spectrum-free method (Sect. \ref{['sec:SPEC-DIST-COMP']}).
• Figure 5: Level-sets of diffusion basis functions at three scales and centred at the same seed point.