Shape Partitioning via $${L}_{p}$$ L p Compressed Modes

Abstract

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.