Computing weak distance between the 2-sphere and its nonsmooth approximations

TL;DR

The paper addresses how to quantify differences between compact surfaces in $\mathbb{R}^3$ by identifying each surface with its surface measure and measuring their distance in a negative-order Sobolev space via the Fourier transform on the ambient space. It develops a computational pipeline that (i) approximates surface measures using affine simplices, (ii) computes Sobolev norms from frequency-space data using a bounded 3D trapezoidal rule, and (iii) accelerates the frequency-domain summations with nonuniform FFTs and GPUParallelization. Numerical results show that the 2-sphere and its icosahedral discretizations converge in the $H_s(\mathbb{R}^3)$ norm with quadratic rate up to truncation (notably for $s=-10$), and that the GPU-accelerated NUFFT implementation dramatically speeds up computations compared to CPU and direct-sum approaches. The method offers a robust, scalable tool for geometry processing and mesh quality assessment, enabling precise, measure-based comparisons that are less sensitive to sampling artifacts than point-cloud distances.

Abstract

A novel algorithm is proposed for quantitative comparisons between compact surfaces embedded in the three-dimensional Euclidian space. The key idea is to identify those objects with the associated surface measures and compute a weak distance between them using the Fourier transform on the ambient space. In particular, the inhomogeneous Sobolev norm of negative order for a difference between two surface measures is evaluated via the Plancherel theorem, which amounts to approximating a weighted integral norm of smooth data on the frequency space. This approach allows several advantages including high accuracy due to fast-converging numerical quadrature rules, acceleration by the nonuniform fast Fourier transform, and parallelization on many-core processors. In numerical experiments, the 2-sphere, which is an example whose Fourier transform is explicitly known, is compared with its icosahedral discretization, and it is observed that the piecewise linear approximations converge to the smooth object at the quadratic rate up to small truncation.

Figures (4)

• Figure 1: \newlabelfig:plot_bessel_trap0 Convergence of trapezoidal rule (\ref{['eq:trap_rule_R3']}) on $D=[-\xi_\text{max},\xi_\text{max}]^3$ : (a) exact $|\widehat{\sigma}|^2$ and weighted function (\ref{['eq:ft_2sphere_quad1D']}) for $s=-10$, and (b) plots of relative errors for $s=-5,-10$ and $\xi_\text{max}=5$.
• Figure 2: \newlabelfig:plot_isocad0 Illustrations of icosahedral discretization: Initial icosahedron $\mathcal{T}_0$ (left), and polyhedron $\mathcal{T}_1$ after first subdivision (right).
• Figure 3: \newlabelfig:conv_isocad0 Convergence of icosahedral discretization : (a) plots for exact $\widehat{\sigma}$ and $\widehat{\mathcal{T}_n}$ with $N=0,1,2$, and (b) quadratic convergence in $\|\,\cdot\,\|_{H_{-10}(\mathbb{R}^3)}$ and theoretical $\mathcal{O}(4^{-N})$.
• Figure 4: \newlabelfig:time_cpu_gpu0 Computation time with parallelization on OpenMP and GPU in single precision : (a) variable $N$ with $(M,\xi_\text{max}) =(128,5)$, and (b) variable $M$ with $(N,\xi_\text{max}) =(5,5)$.