Laplace–Beltrami Operator on Digital Surfaces

Abstract

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 $\mathbb {Z}^n$. 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.

Appendices

Appendices

Lemma 4

Let $a,b,c\in \mathbb {R}$, if $a \le 0$ and $c \ge 0$, then

$$\begin{aligned} a \le b \le c \implies |b| \le \max \{|a|,c\}\,. \end{aligned}$$

Proof

We split the proof into two cases depending on the sign of b. If $b \le 0$, then $|b| \le |a|$ and therefore $|b| \le \max \{|a|,c\}$. If $b > 0$, then $|b| = b \le |c| = c$ as $c \ge 0$ and therefore $|b| \le \max \{|a|,c\}$ which concludes the proof. $\square$

Lemma 5

Let $x\in \mathbb {R}$,

$$\begin{aligned}&e^{\frac{x}{2}} -1 \le x \iff 0 \le x \\&\quad \le -\frac{1}{2}\left[ 2\, W_{-1}\left( - \frac{1}{2 \sqrt{e}} \right) + 1 \right] \approx 2.51286 \end{aligned}$$

where $W_{-1}$ is the lower branch of Lambert W-function (also called omega function or product logarithms).

Fig. 12

Plot of the two principal branches of the Lambert W-function. $W_0$ is in orange, and $W_{-1}$ in green. The two branches join at the point $(-1/e,-1)$ (Color figure online)

Proof

We use the Lambert W-function to prove this lemma. In-depth study of this object can be found in the book of Corless, Gonnet, Hare and Knuth [12]. The function is defined as the multivalued function W that satisfies

$$\begin{aligned} z = W(z)e^{W(z)} \end{aligned}$$

for $z\in \mathbb {C}$. It is equivalently the inverse function of $f(w) = we^w$. The graph of the Lambert W-function in the real numbers is drawn in Fig. 12. The function has two real branches $W_0$ and $W_{-1}$ in the interval $-1/e< x < 0$ which join at $x=-1/e$. This means that the equation $x = we^w$ has two solutions in this interval (one per branch). We will use both branches: $W_{-1}$ which is decreasing in its interval, and $W_0$ which is increasing in this interval. We also use the identity $W(x e^x) = x$. We do a proof by equivalence of inequalities:

$$\begin{aligned}&e^{\frac{x}{2}} - 1 \le x \\&\quad \iff e^{\frac{x}{2}} \le x + 1 \\&\quad \iff -(x+1) e^{-\frac{x}{2}} \le -1 \\&\quad \iff -\left( \frac{x}{2} + \frac{1}{2} \right) e^{- \left( \frac{x}{2} + \frac{1}{2}\right) } \\&\qquad \le - \frac{1}{2 \sqrt{e}}. \,\,\hbox { by multiplying by } \frac{1}{2\sqrt{e}} \end{aligned}$$

Putting $X = - \frac{1}{2}(x + 1)$ we have

$$\begin{aligned} X \ge W_{-1}\left( - \frac{1}{2 \sqrt{e}}\right) \text { and } X \le W_{0}\left( - \frac{1}{2 \sqrt{e}}\right) \end{aligned}$$

which leads to

$$\begin{aligned}&- \left( 2 W_{0}\left( - \frac{1}{2\sqrt{e}}\right) + 1 \right) = 0 \le x \\&\quad \le - \left( 2 W_{-1}\left( - \frac{1}{2\sqrt{e}}\right) + 1 \right) \approx 2.51286 \end{aligned}$$

as $W_{0}( - \frac{1}{2\sqrt{e}}) = \frac{1}{2}$. $\square$