We present a comprehensive analysis of total variation (TV) on non-Euclidean domains and its eigenfunctions. We specifically address parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work sheds new light on the celebrated Beltrami and Anisotropic TV flows and explains experimental findings from recent years on shape spectral TV [Fumero et al. 2020] and adaptive anisotropic spectral TV [Biton and Gilboa 2022]. A new notion of convexity on surfaces is derived by characterizing structures that are stable throughout the TV flow, performed on surfaces. We establish and numerically demonstrate quantitative relationships between TV, area, eigenvalue, and eigenfunctions of the TV operator on surfaces. Moreover, we expand the shape spectral TV toolkit to include zero-homogeneous flows, leading to efficient and versatile shape processing methods. These methods are exemplified through applications in smoothing, enhancement, and exaggeration filters. We introduce a novel method that, for the first time, addresses the shape deformation task using TV. This deformation technique is characterized by the concentration of deformation along geometrical bottlenecks, shown to coincide with the discontinuities of eigenfunctions. Overall, our findings elucidate recent experimental observations in spectral TV, provide a diverse framework for shape filtering, and present the first TV-based approach to shape deformation.

我们对非欧几里得域及其特征函数的总变分（TV）进行了全面分析。我们专门解决参数化表面，即 3D 图形中使用的形状的自然表示。我们的工作为著名的贝尔特拉米和各向异性电视流提供了新的视角，并解释了近年来形状光谱电视的实验结果 [Fumero 等人，2017]。 2020] 和自适应各向异性光谱电视 [Biton 和 Gilboa 2022]。表面凸度的新概念是通过表征在表面上执行的整个电视流中稳定的结构而得出的。我们建立并数值证明了电视、面积、特征值和表面上电视算子的特征函数之间的定量关系。此外，我们扩展了形状光谱电视工具包以包括零均匀流，从而实现高效且通用的形状处理方法。这些方法通过在平滑、增强和夸张滤波器中的应用进行了举例说明。我们引入了一种新颖的方法，首次使用电视解决形状变形任务。这种变形技术的特点是变形集中在几何瓶颈上，这与本征函数的不连续性一致。总的来说，我们的研究结果阐明了光谱电视的最新实验观察结果，为形状过滤提供了多样化的框架，并提出了第一个基于电视的形状变形方法。