Priors with Coupled First and Second Order Differences for Manifold-Valued Image Processing

TL;DR

This paper extends variational image restoration models involving first and second order derivatives to manifold-valued images, proposing extrinsic and intrinsic methods, and demonstrating their effectiveness on various manifolds.

Contribution

It introduces the first generalization of coupled first and second order difference priors to manifold-valued images, including new intrinsic models and algorithms.

Findings

Extrinsic models formulated with Euclidean embedding and ADMM optimization.

Intrinsic models based on Lie group structures and Riemannian differences.

Numerical examples show effective restoration on multiple manifolds.

Abstract

Recently variational models with priors involving first and second order derivatives resp. differences were successfully applied for image restoration. There are several ways to incorporate the derivatives of first and second order into the prior, for example additive coupling or using infimal convolution (IC), as well as the more general model of total generalized variation (TGV). The later two methods give also decompositions of the restored images into image components with distinct "smoothness" properties which are useful in applications. This paper is the first attempt to generalize these models to manifold-valued images. We propose both extrinsic and intrinsic approaches. The extrinsic approach is based on embedding the manifold into an Euclidean space of higher dimension. Models following this approach can be formulated within the Euclidean space with a constraint restricting them to the manifold. Then alternating direction methods of multipliers can be employed for finding minima. However, the components within the infimal convolution or total generalized variation decomposition live in the embedding space rather than on the manifold which makes their interpretation difficult. Therefore we also investigate two intrinsic approaches. For manifolds which are Lie groups we propose three priors which exploit the group operation, an additive one, another with IC coupling and a third TGV like one. For computing the minimizers of the intrinsic models we apply gradient descent algorithms. For general Riemannian manifolds we further define a model for infimal convolution based on the recently developed second order differences. We demonstrate by numerical examples that our approaches works well for the circle, the 2-sphere, the rotation group, and the manifold of positive definite matrices with the affine invariant metric.