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 distances between them using the Fourier transform on the ambient space. In particular, the inhomogeneous Sobolev norms of negative order are approximated from data in the frequency space, which amounts to comparing measures after appropriate smoothing. Such Fourier-based distances allow several advantages including high accuracy due to fast-converging numerical quadrature rules, acceleration by the nonuniform fast Fourier transform, parallelization on massively parallel architectures. 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 truncations.

翻译：提出一种新算法，用于定量比较嵌入三维欧氏空间中的紧致曲面。核心思想是将这些对象与相关的曲面测度等同，并利用环境空间上的傅里叶变换计算它们之间的距离。具体而言，从频域数据中近似负阶非齐次索伯列夫范数，这相当于在适当平滑后比较测度。此类基于傅里叶的距离具有多项优势，包括因快速收敛的数值求积规则而实现的高精度、通过非均匀快速傅里叶变换加速，以及在大规模并行架构上的并行化。在数值实验中，将傅里叶变换显式已知的范例——二维球面，与其二十面体离散化进行比较，观察到分段线性近似以二次速率收敛至光滑对象，直至较小的截断误差。