Multiscale Anisotropic Harmonic Filters on non Euclidean domains

TL;DR

This paper presents Multiscale Anisotropic Harmonic Filters (MAHFs) for analyzing signal variations on non-Euclidean domains like meshes, point clouds, and graphs, combining heat diffusion and directional information for multi-scale filtering.

Contribution

The introduction of MAHFs that integrate heat kernel smoothing with local directional data for effective pattern analysis on complex non-Euclidean structures.

Findings

Effective multi-scale filtering demonstrated on 3D meshes.

Successful detection of curvature and signal variations.

Application to surface normals and luminance signals.

Abstract

This paper introduces Multiscale Anisotropic Harmonic Filters (MAHFs) aimed at extracting signal variations over non-Euclidean domains, namely 2D-Manifolds and their discrete representations, such as meshes and 3D Point Clouds as well as graphs. The topic of pattern analysis is central in image processing and, considered the growing interest in new domains for information representation, the extension of analogous practices on volumetric data is highly demanded. To accomplish this purpose, we define MAHFs as the product of two components, respectively related to a suitable smoothing function, namely the heat kernel derived from the heat diffusion equations, and to local directional information. We analyse the effectiveness of our approach in multi-scale filtering and variation extraction. Finally, we present an application to the surface normal field and to a luminance signal textured to a mesh, aiming to spot, in a separate fashion, relevant curvature changes (support variations) and signal variations.