Denoising Multi-Color QR Codes and Stiefel-Valued Data by Relaxed Regularizations

Abstract

The handling of manifold-valued data, for instance, plays a central role in color restoration tasks relying on circle- or sphere-valued color models, in the study of rotational or directional information related to the special orthogonal group, and in Gaussian image processing, where the pixel statistics are interpreted as values on the hyperbolic sheet. Especially, to denoise these kind of data, there have been proposed several generalizations of total variation (TV) and Tikhonov-type denoising models incorporating the underlying manifolds. Recently, a novel, numerically efficient denoising approach has been introduced, where the data are embedded in an Euclidean ambient space, the non-convex manifolds are encoded by a series of positive semi-definite, fixed-rank matrices, and the rank constraint is relaxed to obtain a convexification that can be solved using standard algorithms from convex analysis. The aim of the present paper is to extent this approach to new kinds of data like multi-binary and Stiefel-valued data. Multi-binary data can, for instance, be used to model multi-color QR codes whereas Stiefel-valued data occur in image and video-based recognition. For both new data types, we propose TV- and Tikhonov-based denoising modelstogether with easy-to-solve convexification. All derived methods are evaluated on proof-of-concept, synthetic experiments.