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

arXiv: https://arxiv.org/abs/2307.10980 · HTML: https://mlsanity.com/paper/2307.10980/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2023.

## Built on by (most cited first, 1 of 1)

- [Denoising Multi-Color QR Codes and Stiefel-Valued Data by Relaxed Regularizations](https://mlsanity.com/paper/2506.22826/) (2025)

## Abstract (arXiv)

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.