Many computational algorithms applied to geometry operate on discrete representations of shape. It is sometimes necessary to first simplify, or coarsen, representations found in modern datasets for practicable or expedited processing. The utility of a coarsening algorithm depends on both, the choice of representation as well as the specific processing algorithm or operator. e.g. simulation using the Finite Element Method, calculating Betti numbers, etc. We propose a novel method that can coarsen triangle meshes, tetrahedral meshes and simplicial complexes. Our method allows controllable preservation of salient features from the high-resolution geometry and can therefore be customized to different applications. Salient properties are typically captured by local shape descriptors via linear differential operators -- variants of Laplacians. Eigenvectors of their discretized matrices yield a useful spectral domain for geometry processing (akin to the famous Fourier spectrum which uses eigenfunctions of the derivative operator). Existing methods for spectrum-preserving coarsening use zero-dimensional discretizations of Laplacian operators (defined on vertices). We propose a generalized spectral coarsening method that considers multiple Laplacian operators defined in different dimensionalities in tandem. Our simple algorithm greedily decides the order of contractions of simplices based on a quality function per simplex. The quality function quantifies the error due to removal of that simplex on a chosen band within the spectrum of the coarsened geometry.

翻译：许多应用于几何的计算算法处理形状的离散表示。有时有必要先简化或粗化现代数据集中的表示，以实现可行或加速处理。粗化算法的效用既取决于表示的选取，也取决于具体的处理算法或算子，例如使用有限元法进行模拟、计算贝蒂数等。我们提出了一种新颖方法，能够粗化三角网格、四面体网格和单纯复形。该方法可控制性地保留高分辨率几何中的显著特征，因此可针对不同应用进行定制。显著属性通常通过线性微分算子（拉普拉斯算子的变体）的局部形状描述符捕获。其离散化矩阵的特征向量为几何处理提供了一个有用的谱域（类似于著名的傅里叶谱，后者利用导数算子的特征函数）。现有的保谱粗化方法使用拉普拉斯算子的零维离散化（定义在顶点上）。我们提出了一种广义谱粗化方法，该方法同时考虑在不同维度上定义的多重拉普拉斯算子。我们的简单算法基于每个单纯形的质量函数贪婪地决定单纯形收缩的顺序。该质量函数量化了在粗化几何的选定谱带内去除该单纯形所导致的误差。