Article Dans Une Revue Journal of Mathematical Imaging and Vision Année : 2025

Corrected Laplace–Beltrami Operators for Digital Surfaces

David Coeurjolly

Résumé

Defining consistent calculus frameworks on discrete meshes is useful for processing the geometry of meshes or model numerical simulations and variational problems onto them. However, digital surfaces (boundary of voxels) cannot benefit directly from the classical mesh calculus frameworks, since their vertex and face geometry is too poor to capture the geometry of the underlying smooth Euclidean surface well enough. This paper proposes three new calculus frameworks dedicated to digital surfaces, which exploit a corrected normal field, in a manner similar to the recent digital calculus of Coeurjolly and Lachaud (IAPR second international conference on discrete geometry and mathematical morphology, Springer, Berlin 2022). First, we build a corrected interpolated calculus by defining inner products with position and normal interpolation in the Grassmannian. Second, we present a corrected finite element method which adapts the standard Finite Element Method with a corrected metric per element. Third, we present a corrected virtual refinement method adapting the method of Bunge et al. (Comput Graph Forum 39(2):303-313, 2020, https://doi.org/10.1111/cgf.13931). Experiments show that these digital calculus frameworks seem to converge toward the continuous calculus, offer a valid alternative to classical mesh calculus, and induce effective tools for digital surface processing tasks. We then use these corrected Laplace-Beltrami operators in order to build a regularization method for digital surface, using geometric information given by discrete normal and curvature estimators.
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hal-04913546 , version 1 (03-02-2025)

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Colin Weill-Duflos, David Coeurjolly, Jacques-Olivier Lachaud. Corrected Laplace–Beltrami Operators for Digital Surfaces. Journal of Mathematical Imaging and Vision, 2025, 67 (2), pp.11. ⟨10.1007/s10851-024-01226-6⟩. ⟨hal-04913546⟩
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