Shape Partitioning via L\(_p\) Compressed Modes

• Published in: arXiv.org
• Main Authors: Huska, Martin, Lazzaro, Damiana, Morigi, Serena
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 20.04.2018
• Subjects: Chemical partition; Eigenvectors; Iterative algorithms; Operators (mathematics); Optimization; Orthogonality; Partitioning; Patching; Segmentation
• ISSN: 2331-8422

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.