# Total Variation of the Normal Vector Field as Shape Prior

## TL;DR

This paper introduces a total variation prior for the normal vector field on 3D shape boundaries, analyzing its properties using differential geometry and proposing an extension of the split Bregman method for non-differentiable cases.

## Contribution

It develops a novel total variation prior for normal vector fields on 3D shapes and extends optimization methods to handle non-differentiability on flat regions.

## Key findings

- Spheres are stationary points under area constraints.
- The total variation functional is non-differentiable on flat boundary regions.
- An extended split Bregman method for manifold-valued functions is proposed.

## Abstract

An analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. The analysis of the total variation of the normal vector field is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is shown that spheres are stationary points when the total variation of the normal is minimized under an area constraint. Shape calculus is used to characterize the relevant derivatives. Since the total variation functional is non-differentiable whenever the boundary contains flat regions, an extension of the split Bregman method to manifold valued functions is proposed.