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：提出了一种新的算法，用于比较嵌入三维欧几里德空间中的紧致曲面之间的定量差异。关键思想是在关联的曲面度量下识别这些对象，并使用傅里叶变换计算它们之间的距离。特别地，负阶不均匀嵌入式 Sobolev 范数是从频率空间中的数据逼近的，这相当于在适当平滑之后比较度量。这些基于傅里叶的距离具有几个优势，包括由于快速收敛数值积分规则而高精度，由于非均匀快速傅里叶变换而加速，可以在大规模并行体系结构上并行化。在数值实验中，将2-球与其 20 面体离散化进行比较，观察到分段线性逼近以二次速率收敛到平滑对象，直到小截断。