In a first part I'll will broadly cover the topic of what the Laplace- Beltrami is, its different characterization and its many uses in computer graphics.
Then I'll cover our works on the topic of building this operator (along with a wider digital calculus framework) on digital surfaces (boundary of voxels). These surfaces cannot benefit directly from the classical mesh calculus frameworks. In our recent work, we propose two new calculus frameworks dedicated to digital surfaces, which exploit a corrected normal field. 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. 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.