The spectrum of the Laplace‐Beltrami operator is instrumental for a number of geometric modeling applications, from processing to analysis. Recently, multiple methods were developed to retrieve an approximation of a shape that preserves its eigenvectors as much as possible, but these techniques output a subset of input points with no connectivity, which limits their potential applications. Furthermore, the obtained Laplacian results from an optimization procedure, implying its storage alongside the selected points. Focusing on keeping a mesh instead of an operator would allow to retrieve the latter using the standard cotangent formulation, enabling easier processing afterwards. Instead, we propose to simplify the input mesh using a spectrum‐preserving mesh decimation scheme, so that the Laplacian computed on the simplified mesh is spectrally close to the one of the input mesh. We illustrate the benefit of our approach for quickly approximating spectral distances and functional maps on low resolution proxies of potentially high resolution input meshes.

中文翻译：

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光谱网格简化

Laplace-Beltrami 算子的频谱有助于许多几何建模应用程序，从处理到分析。最近，开发了多种方法来检索尽可能保留其特征向量的形状的近似值，但是这些技术输出了没有连通性的输入点子集，这限制了它们的潜在应用。此外，从优化过程中获得的拉普拉斯算子结果，意味着它与所选点一起存储。专注于保持网格而不是操作员将允许使用标准余切公式检索后者，从而使之后的处理更容易。相反，我们建议使用频谱保留网格抽取方案来简化输入网格，这样在简化网格上计算的拉普拉斯算子在光谱上接近输入网格之一。我们说明了我们的方法在潜在高分辨率输入网格的低分辨率代理上快速逼近光谱距离和功能图的好处。