Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes

Rustum Choksi; Yves van Gennip; Adam Oberman

doi:10.3934/ipi.2011.5.591

Article Contents

2011, Volume 5, Issue 3: 591-617. Doi: 10.3934/ipi.2011.5.591

This issue Previous Article Alpha divergences based mass transport models for image matching problems Next Article Errors of regularisation under range inclusions using variable Hilbert scales

Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes

1.
Department of Mathematics and Statistics, McGill University, Burnside Hall, Room 1005, 805 Sherbrooke Street West, Montreal, Quebec, H3A 2K6
2.
Department of Mathematics, University of California Los Angeles, Math Sciences Building 6363, 520 Portola Plaza, Los Angeles, California, 90095
3.
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, 8888 University Drive, V5A 1S6

Received: June 30, 2010

Revised: February 28, 2011

Published: July 2011

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Abstract

We consider variations of the Rudin-Osher-Fatemi functional which are particularly well-suited to denoising and deblurring of 2D bar codes. These functionals consist of an anisotropic total variation favoring rectangles and a fidelity term which measure the $L^1$ distance to the signal, both with and without the presence of a deconvolution operator. Based upon the existence of a certain associated vector field, we find necessary and sufficient conditions for a function to be a minimizer. We apply these results to 2D bar codes to find explicit regimes -- in terms of the fidelity parameter and smallest length scale of the bar codes -- for which the perfect bar code is attained via minimization of the functionals. Via a discretization reformulated as a linear program, we perform numerical experiments for all functionals demonstrating their denoising and deblurring capabilities.

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Mathematics Subject Classification: 49N45, 94A08.

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