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. (in: Teillaud (ed) Proceedings of the 24th ACM symposium on computational geometry, College Park, MD, USA, pp 278–287, 2008, https://doi.org/10.1145/1377676.1377725) for triangular meshes to topological border of subsets of $$\mathbb {Z}^n$$Zn. 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.