Denoising Sphere-Valued Data by Relaxed Total Variation Regularization

TL;DR

This paper introduces a novel variational approach for denoising sphere-valued data using relaxed total variation regularization, providing efficient algorithms and proving tightness in special cases, with applications in imaging and signal recovery.

Contribution

It proposes a new convex relaxation method for sphere-valued data denoising, utilizing inner product-based data fidelity and ADMM, with proven tightness for binary signals and demonstrated effectiveness in various applications.

Findings

The relaxation is provably tight for binary signals.

The method effectively denoises color and SO(3)-valued data.

Numerical experiments confirm tightness and efficiency.

Abstract

Circle- and sphere-valued data play a significant role in inverse problems like magnetic resonance phase imaging and radar interferometry, in the analysis of directional information, and in color restoration tasks. In this paper, we aim to restore $(d-1)$-sphere-valued signals exploiting the classical anisotropic total variation on the surrounding $d$-dimensional Euclidean space. For this, we propose a novel variational formulation, whose data fidelity is based on inner products instead of the usually employed squared norms. Convexifying the resulting non-convex problem and using ADMM, we derive an efficient and fast numerical denoiser. In the special case of binary (0-sphere-valued) signals, the relaxation is provable tight, i.e. the relaxed solution can be used to construct a solution of the original non-convex problem. Moreover, the tightness can be numerically observed for barcode and QR code denoising as well as in higher dimensional experiments like the color restoration using hue and chromaticity and the recovery of SO(3)-valued signals.