Mumford-Shah Mesh Processing using the Ambrosio-Tortorelli Functional

TL;DR

This paper introduces a novel discrete exterior calculus approach to applying the Ambrosio-Tortorelli approximation of the Mumford-Shah functional for various mesh processing tasks, including denoising, inpainting, and segmentation.

Contribution

It presents a new discretization method and shape optimization routine that extend Mumford-Shah functional applications to a broad range of mesh processing problems.

Findings

Effective mesh denoising demonstrated

Successful mesh inpainting and segmentation

Unified framework for multiple mesh processing tasks

Abstract

The Mumford-Shah functional approximates a function by a piecewise smooth function. Its versatility makes it ideal for tasks such as image segmentation or restoration, and it is now a widespread tool of image processing. Recent work has started to investigate its use for mesh segmentation and feature lines detection, but we take the stance that the power of this functional could reach far beyond these tasks and integrate the everyday mesh processing toolbox. In this paper, we discretize an Ambrosio-Tortorelli approximation via a Discrete Exterior Calculus formulation. We show that, combined with a new shape optimization routine, several mesh processing problems can be readily tackled within the same framework. In particular, we illustrate applications in mesh denoising, normal map embossing, mesh inpainting and mesh segmentation.