Shape Partitioning via Lp Compressed Modes

• Published in: Journal of mathematical imaging and vision Vol. 60; no. 7; pp. 1111 - 1131
• Main Authors: Huska, Martin, Lazzaro, Damiana, Morigi, Serena
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2018
• Subjects: Applications of Mathematics; Computer Science; Image Processing and Computer Vision; Mathematical Methods in Physics; Signal,Image and Speech Processing; Patch-based partitioning; Geometry processing; Sparsity; Alternating directions method of multipliers; Compressed modes; norm penalty; Mesh segmentation
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-018-0799-8

The eigenfunctions of the Laplace–Beltrami operator (manifold harmonics) define a function basis that can be used in spectral analysis on manifolds. In Ozoli et al. (Proc Nat Acad Sci 110(46):18368–18373, 2013 ) 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.