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Mesh-Free Discrete Laplace–Beltrami Operator

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Abstract

In this work we propose a new discretization method for the Laplace–Beltrami operator defined on point-based surfaces. In contrast to the existing point-based discretization techniques, our approach does not rely on any triangle mesh structure, turning out truly mesh-free. Based on a combination of Smoothed Particle Hydrodynamics and an optimization procedure to estimate area elements, our discretization method results in accurate solutions while still being robust when facing abrupt changes in the density of points. Moreover, the proposed scheme results in numerically stable discrete operators. The effectiveness of the proposed technique is brought to bear in many practical applications. In particular, we use the eigenstructure of the discrete operator for filtering and shape segmentation. Point-based surface deformation is another application that can be easily carried out from the proposed discretization method.

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