Nonlinear Spectral Geometry Processing via the TV Transform

TL;DR

This paper presents a new nonlinear spectral framework for geometry processing that enables detail-preserving, multiscale manipulation of 3D shapes, avoiding over-smoothing and applicable to various geometry representations.

Contribution

It introduces a nonlinear operator based on total variation for spectral decomposition, offering a flexible, detail-preserving alternative to Laplacian methods in geometry processing.

Findings

Effective detail-preserving decomposition demonstrated on 3D shapes

Applicable to meshes and point clouds with flexible signal processing

Successful applications in denoising, detail transfer, and stylization

Abstract

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.