Spectral Total-Variation Processing of Shapes: Theory and Applications

TL;DR

This paper extends total variation (TV) based spectral processing from Euclidean domains to smooth parametric surfaces $M \subset \mathbb{R}^3$, enabling nonlinear spectral filtering and deformation of 3D shapes. It develops a surface NETV framework with eigenfunction theory, introduces a non-Euclidean notion of convexity via eigensets, and derives exact relations such as $NETV(C)=per(C)$ and $ \lambda|C| = NETV(C)$ that link eigenfunctions to geometric quantities. It then builds a flow-based, zero-homogeneous spectral framework and proposes three shape-processing methods (M1–M3) plus a TV-based deformation approach that concentrates on bottlenecks. Numerical experiments validate linear decay of eigenfunctions, piecewise-constant deformation along low-perimeter boundaries, and enhanced geometric detail, demonstrating a practical toolkit for shape filtering and deformation on surfaces.

Abstract

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.

Figures (21)

• Figure 1: Spectral TV filtering of a shape, demonstrating the application of different filters to isolate TV eigenfunctions and reveal geometric details. Left: The original shape with an all-pass filter, displaying the complete spectrum. Middle: A low-pass filter uncovers the shape’s fundamental structure. From a theoretical point of view - it isolates eigenfunctions as lower-frequency components. Right: A high-pass filter brings forth the geometrical details on-top the foundational structure. The geometry is partitioned into three distinct subsets based on eigenfunction discontinuities (details in Fig. \ref{['sinc']}). This delineates the original shape by its bottlenecks, which are also observed in the boundaries of each of the three nested, concentric surfaces of the right panel. These bottleneck boundaries are a direct manifestation of the eigenfunction properties, which our manuscript investigates through novel theoretical and empirical analysis.
• Figure 2: "Bust of Queen Nefertiti": Nonlinear spectral filtering applied using one of our proposed methods, presented in Sec. \ref{['resmoothing method']}
• Figure 3: $M$ is a surface-revolution of a part of a translated sinc curve, inducing a non-Euclidean metric. $f$ is initialized as an indicator function of a "sleeve set". Upper row: linear diffusion. Bottom row: $NETV$ minimizing flow. Unlike linear diffusion, the function remains piecewise constant throughout the flow - dividing the surface $M$ to subsets. Iter 50: New boundaries, of small perimeters emerge; iter 100: Initial boundaries subside; iter 150: The sets merge so that only the minimal perimeter remains - which is numerically shown to be an eigenfunction (by Thm. 2.3 in bungert2020asymptotic). When an eigenfunction indicates a set (as is the case here), we call it an eigenset. In the Euclidean case - the eigensets' shape and behaviour have well studied properties andreu2001minimizingbellettini2002total. In contrast, eigensets of the non-Euclidean case are less understood . We introduce novel theoretical properties of the eigensets in non-Euclidean domains.
• Figure 4: $NETV$ minimizing flow on a spherical manifold $M$. The initial function $f$ assumes two options: An indicator function of a geodesically convex spherical cap $C$ (lower row), or of $M \backslash C$, which is not geodesically-convex (upper row). From right to left: Values of $q_1 \in C$ throughout the flow, Values of $q_2 \in M \backslash C$ throughout the flow, and the flow portrayed as color on $M$. $C$ and $M \backslash C$ remain intact throughout the flow, and their values change linearly with time $t$, until they decay completely. Such a behaviour implies that both $f$s are eigenfunctions, i.e. $C$ and $M \backslash C$ are eigensets. Noteably $M \backslash C$ is not geodesically convex. However, in the Euclidean case, eigensets must be convex sets. This raises the question: Is there an alternative notion of convexity on manifolds, other than geodesical convexity, which characterizes eigensets similarly to the Euclidean case? In the following we define and prove such a notion.
• Figure 5: Weighted indicator functions on Torus and their $NETV$ flow. First row: Two geodesical disks. The small disk is defined in "Euclidean position", i.e. results for Euclidean case carry over. Namely - the small disk is both geodesically convex, and a (locally) minimal perimeter set. The large disk is not in Euclidean position, and Euclidean laws do not carry over: It is not geodesically convex, nor is it of locally minimal perimeter. Throughout the flow it becomes a sleeve-like set, which is a minimal perimeter set, after which it decays completely. Bottom row: Indicator function of two sleeve sets $C_1, C_2$, weighted by the ratio of their areas: $f = \tilde{\chi}^{C_1} + \frac{|C_2|}{|C_1|}\tilde{\chi}^{C_2}$. The sleeve sets are eigensets, as shown in the appendix. By Eq. \ref{['eq: eigenvalue perimeter area']}, and since $per(C_1)=per(C_2)$, the eigenvalue ratio should be inverse the area ratio i.e. $\frac{\lambda_2}{\lambda_1}=\frac{|C_1|}{|C_2|}$. Thus we can reformulate as $f = \tilde{\chi}^{C_1} + \frac{\lambda_1}{\lambda_2}\tilde{\chi}^{C_2}$. If this is true, then by Eq. \ref{['eq: linear decay']} we expect the sets to completely decay at the same time - and indeed they do.

Theorems & Definitions (16)

• definition 1
• definition 2
• Claim 1
• proof
• Claim 2
• proof
• Corollary
• definition 3
• definition 4
• theorem 1