Total Roto-Translational Variation

TL;DR

This paper introduces a convex representation of curvature-dependent variational models in roto-translation space, enabling improved shape and image regularization with a novel numerical solution approach.

Contribution

It presents a tight convex formulation for curvature models in roto-translation space and develops a discretization and optimization method for practical image processing applications.

Findings

Convex representation is tight for smooth shapes.

Discretization using Raviart-Thomas elements is effective.

Numerical results demonstrate improved shape regularization.

Abstract

We consider curvature depending variational models for image regularization, such as Euler's elastica. These models are known to provide strong priors for the continuity of edges and hence have important applications in shape-and image processing. We consider a lifted convex representation of these models in the roto-translation space: In this space, curvature depending variational energies are represented by means of a convex functional defined on divergence free vector fields. The line energies are then easily extended to any scalar function. It yields a natural generalization of the total variation to the roto-translation space. As our main result, we show that the proposed convex representation is tight for characteristic functions of smooth shapes. We also discuss cases where this representation fails. For numerical solution, we propose a staggered grid discretization based on an averaged Raviart-Thomas finite elements approximation. This discretization is consistent, up to minor details, with the underlying continuous model. The resulting non-smooth convex optimization problem is solved using a first-order primal-dual algorithm. We illustrate the results of our numerical algorithm on various problems from shape-and image processing.