  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.
