We introduce a novel computational framework for digital geometry processing, based upon the derivation of a nonlinear operator associated to the total variation functional. Such operator admits a generalized notion of spectral decomposition, yielding a sparse multiscale representation akin to Laplacian-based methods, while at the same time avoiding undesirable over-smoothing effects typical of such techniques. Our approach entails accurate, detail-preserving decomposition and manipulation of 3D shape geometry while taking an especially intuitive form: non-local semantic details are well separated into different bands, which can then be filtered and re-synthesized with a straightforward linear step. Our computational framework is flexible, can be applied to a variety of signals, and is easily adapted to different geometry representations, including triangle meshes and point clouds. We showcase our method throughout multiple applications in graphics, ranging from surface and signal denoising to detail transfer and cubic stylization.

中文翻译：

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通过 TV 变换进行非线性光谱几何处理

我们为数字几何处理引入了一种新颖的计算框架，基于与总变分函数相关的非线性算子的推导。这种算子承认频谱分解的广义概念，产生类似于基于拉普拉斯的方法的稀疏多尺度表示，同时避免此类技术典型的不良过度平滑效应。我们的方法需要对 3D 形状几何进行准确、保留细节的分解和操作，同时采用一种特别直观的形式：非局部语义细节被很好地分成不同的波段，然后可以通过简单的线性步骤对其进行过滤和重新合成。我们的计算框架灵活，可以应用于各种信号，并且很容易适应不同的几何表示，包括三角形网格和点云。我们在图形中的多个应用程序中展示了我们的方法，从表面和信号去噪到细节转移和立方体风格化。