Shape Partitioning via L$_p$ Compressed Modes

TL;DR

This paper introduces a novel spectral basis for shape partitioning on 3D manifolds using an $L_p$ penalization to achieve localized solutions, improving shape analysis and mesh segmentation.

Contribution

It proposes an $L_p$ penalization framework with an ADMM algorithm to obtain compactly supported eigenfunctions for shape partitioning, advancing spectral methods in geometry processing.

Findings

Effective shape partitioning demonstrated on 3D meshes

Improved localization of spectral basis functions

Enhanced mesh segmentation performance

Abstract

The eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis that can be used in spectral analysis on manifolds. In [21] the authors recast the problem as an orthogonality constrained optimization problem and pioneer the use of an $L_1$ penalty term to obtain sparse (localized) solutions. In this context, the notion corresponding to sparsity is compact support which entails spatially localized solutions. We propose to enforce such a compact support structure by a variational optimization formulation with an $L_p$ penalization term, with $0<p<1$. The challenging solution of the orthogonality constrained non-convex minimization problem is obtained by applying splitting strategies and an ADMM-based iterative algorithm. The effectiveness of the novel compact support basis is demonstrated in the solution of the 2-manifold decomposition problem which plays an important role in shape geometry processing where the boundary of a 3D object is well represented by a polygonal mesh. We propose an algorithm for mesh segmentation and patch-based partitioning (where a genus-0 surface patching is required). Experiments on shape partitioning are conducted to validate the performance of the proposed compact support basis.