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

2025-06-28

AI Summary

This work addresses the denoising problem for two emerging structured data types: multicolor barcodes (multivalued binary data) and Stiefel manifold-valued images. We propose a convexification framework that embeds these non-Euclidean, discrete-structured signals into Euclidean space and relaxes the rank constraint. Innovatively extending relaxed regularization to such data for the first time, we formulate tractable convex optimization models—combining total variation (TV) and Tikhonov regularization—that admit efficient numerical solution. Optimization is stabilized via positive semidefinite fixed-rank matrix encoding and convex analysis algorithms. Extensive experiments on synthetic and proof-of-concept data demonstrate substantial improvements in denoising accuracy and numerical stability: our method successfully recovers multicolor barcodes and Stiefel-valued images even under high noise levels. This work establishes a novel paradigm for solving inverse problems involving non-Euclidean and discrete-structured data.

Technology Category

• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Data transparency and provenance

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.

Problem

• Denoising multi-color QR codes using relaxed convexification
• Processing Stiefel-valued data for image and video recognition
• Extending TV and Tikhonov models for manifold-valued data

Innovation

• Relaxed rank constraints for convexification
• TV- and Tikhonov-based denoising models
• Euclidean embedding of manifold-valued data