Denoising of Sphere- and SO(3)-Valued Data by Relaxed Tikhonov Regularization

TL;DR

This paper introduces a simplified convex relaxation approach for denoising manifold-valued data, including sphere- and SO(3)-valued data, using Tikhonov regularization and convex analysis techniques.

Contribution

The authors extend a convex relaxation method for circle-valued data to higher-dimensional spheres and SO(3), simplifying the model while maintaining solution accuracy.

Findings

Convex relaxation effectively denoises manifold-valued data.

The simplified model generalizes to higher dimensions and SO(3).

Numerical experiments show convergence to underlying manifold values.

Abstract

Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, a convex relaxation of the otherwise nonconvex Tikhonov-regularization for denoising circle-valued data has been proposed by Condat (2022). The circle constraints are here encoded in a series of low-dimensional, positive semi-definite matrices. Using Schur complement arguments, we show that the resulting variational model can be simplified while leading to the same solution. The simplified model can be generalized to higher dimensional spheres and to SO(3)-valued data, where we rely on the quaternion representation of the latter. Standard algorithms from convex analysis can be applied to solve the resulting convex minimization problem. As proof-of-the-concept, we use the alternating direction method of multipliers to demonstrate the denoising behavior of the proposed method. In a series of experiments, we demonstrate the numerical convergence of the signal- or image values to the underlying manifold.