Anisotropic total variation regularized L¹ approximation and denoising/deblurring of 2D bar codes

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 L¹ 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.