We present an analysis of total-variation (TV) on non-Euclidean parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work explains recent experimental findings in shape spectral TV [Fumero et al., 2020] and adaptive anisotropic spectral TV [Biton and Gilboa, 2022]. A new way to generalize set convexity from the plane to surfaces is derived by characterizing the TV eigenfunctions on surfaces. Relationships between TV, area, eigenvalue, eigenfunctions and their discontinuities are discovered. Further, we expand the shape spectral TV toolkit to include versatile zero-homogeneous flows demonstrated through smoothing and exaggerating filters. Last but not least, we propose the first TV-based method for shape deformation, characterized by deformations along geometrical bottlenecks. We show these bottlenecks to be aligned with eigenfunction discontinuities. This research advances the field of spectral TV on surfaces and its application in 3D graphics, offering new perspectives for shape filtering and deformation.

翻译：本文针对三维图形学中形状的自然表示——非欧几里得参数化曲面，开展了全变分（TV）分析。我们的工作解释了近期关于形状频谱全变分[Fumero等人，2020]和自适应各向异性频谱全变分[Biton和Gilboa，2022]的实验发现。通过刻画曲面上的TV本征函数，推导出一种将集合凸性从平面推广至曲面的新方法。我们发现了TV、面积、本征值、本征函数及其不连续性之间的内在关联。此外，我们扩展了形状频谱全变分工具包，纳入了通过平滑和夸张滤波器展示的通用零齐次流。最后但同样重要的是，我们提出了首个基于TV的形状形变方法，其特征在于沿几何瓶颈的形变。研究表明这些瓶颈与本征函数的不连续性相一致。本研究推进了曲面上频谱全变分领域及其在三维图形学中的应用，为形状滤波与形变提供了新视角。