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

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.

Coupling Image Restoration and Segmentation: A Generalized Linear Model/Bregman Perspective

“…5d). The second uses the same data-fitting energy, but an anisotropic TV regularizer 14 , as proposed by Choksi et al (2011) (Fig. 5e).…”

“…In Algorithm 1, we only need to modify the w 2 -subproblem: Instead of a coupled soft shrinkage applied to w 2 , two soft shrinkages are separately applied to each component 15 of w 2 . The third model follows closely Choksi et al (2011), but explicitly includes the inpainting problem: An L 1 data-fitting energy is used with w d encoding the inpainting combined with an anisotroptic TV regularizer (Fig. 5f).…”

“…One such application is barcode decoding. Choksi et al (2011) applied a variational image-restoration framework to denoise and deblur 2D barcodes. We show that this problem can be reformulated and solved as a joint task.…”

A few remarks on variational models for denoising

“…In particular, for L p fidelity terms it has been noted that a signal f is left invariant by energy minimization (for λ sufficiently large) if and only if there exists a suitably regular, TV-realizing vector field (see [27,16,30,7,18]). Let us first note here that the sufficiency of such a vector field immediately follows from our comparison estimate (4.1).…”

“…Since the pioneering work of Rudin, Osher and Fatemi [31], much attention has been devoted to T V -regularization in the context of image denoising. Analytical work has mainly targeted the study of fast and efficient numerical implementations and comparison with other models (see [11,15,17,24,25,28] and the references therein), the understanding of the structure of possible minimizers (see [1,2,3,4,6,7,8,13,14,16,18,19,22,27,30,34]), and the analysis of the effects of T V -regularization in terms of edge, block, texture, and scale preservation (see [20,25,27,32]). …”