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.

翻译：提出了一种新算法，用于定量比较嵌入三维欧几里得空间中的紧致曲面。其核心思想是将这些对象与相应的曲面测度进行识别，并利用环境空间上的傅里叶变换计算它们之间的弱距离。具体而言，通过普兰歇尔定理评估两个曲面测度之差的负阶非齐次索博列夫范数，这相当于在频率空间中逼近光滑数据的加权积分范数。该方法具有多项优势，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实现的加速，以及在多核处理器上的并行化能力。在数值实验中，将傅里叶变换显式已知的示例——二维球面与其二十面体离散化进行比较，观察到分段线性逼近以二次速率收敛至光滑对象，直至达到小截断误差。