Digital Calculus Frameworks and Comparative Evaluation of Their Laplace-Beltrami Operators

Abstract

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 two new calculus frameworks dedicated to digital surfaces, which exploit a corrected normal field, in a manner similar to the recent digital calculus of [3]. 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.

This work was partly funded by StableProxies ANR-22-CE46-0006 research grant.

A Details on the Interpolated Corrected Calculus

Let $\sigma$ be a surfel aligned with x and y and with normal aligned with z. The flat operator has the following expression:

The metric matrix for 0-forms is defined as the matrix such that, for any bilinearly interpolated functions $\phi ,\psi$, we obtain on surfel $\sigma$ the scalar:

Let us now define weighted sums for components of $\textbf{u}$ over the quad. We number the edges when turning along the boundary of the surfel $\sigma$ from 0 to 3, such that edges 0, 1, 2, 3 connect vertex pairs $(\textbf{x}_{00},\textbf{x}_{10})$, $(\textbf{x}_{10},\textbf{x}_{11})$,$(\textbf{x}_{01},\textbf{x}_{11})$, $(\textbf{x}_{01},\textbf{x}_{00})$, respectively. We define

By integration of left-hand side, we obtain for a surfel with normal z:

$$\begin{aligned}\textbf{M}_0 = \frac{1}{144} \begin{bmatrix} \bar{\textbf{u}}^z_{00} &{} \bar{\textbf{u}}^z_{00,10} &{} 4 \bar{\textbf{u}}^z &{} \bar{\textbf{u}}^z_{01,00} \\ \bar{\textbf{u}}^z_{00,10} &{} \bar{\textbf{u}}^z_{10} &{} \bar{\textbf{u}}^z_{10,11} &{} 4 \bar{\textbf{u}}^z \\ 4 \bar{\textbf{u}}^z &{} \bar{\textbf{u}}^z_{10,11} &{} \bar{\textbf{u}}^z_{11} &{} \bar{\textbf{u}}^z_{11,01} \\ \bar{\textbf{u}}^z_{11,01} &{} 4 \bar{\textbf{u}}^z &{} \bar{\textbf{u}}^z_{11,01} &{} \bar{\textbf{u}}^z_{01} \end{bmatrix}\,. \end{aligned}$$