Laplace–Beltrami Operator on Digital Surfaces


This article presents a novel discretization of the Laplace–Beltrami operator on digital surfaces.We adapt an existing convolution technique proposed by Belkin et al. [5] for triangular meshes to topological border of subsets of $ℤ^n$. The core of the method relies on first-order estimation of measures associated with our discrete elements (such as length, area etc.). We show strong consistency (i.e. pointwise convergence) of the operator and compare it against various other discretizations.

Journal of Mathematical Imaging and Vision

Caption: Comparison of various Laplace-Beltrami operator on a digital sphere with decreasing grid-step.

      author = {Thomas Caissard and David Coeurjolly and Jacques-Olivier Lachaud and Tristan Roussillon},
      doi = {10.1007/s10851-018-0839-4},
      journal = {Journal of Mathematical Imaging and Vision},
      number = {3},
      pages = {359--379},
      title = {Laplace–Beltrami Operator on Digital Surfaces},
      volume = {61},
      year = {2019}