# American Institute of Mathematical Sciences

August  2011, 5(3): 591-617. doi: 10.3934/ipi.2011.5.591

## 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, Canada 2 Department of Mathematics, University of California Los Angeles, Math Sciences Building 6363, 520 Portola Plaza, Los Angeles, California, 90095, United States 3 Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, 8888 University Drive, V5A 1S6, Canada

Received  July 2010 Revised  March 2011 Published  August 2011

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.
Citation: Rustum Choksi, Yves van Gennip, Adam Oberman. Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes. Inverse Problems & Imaging, 2011, 5 (3) : 591-617. doi: 10.3934/ipi.2011.5.591
##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar [3] B. Berkels, M. Burger, M. Droske, O. Nemitz and M. Rumpf, Cartoon extraction based on anisotropic image classification, in "Vision, Modeling, and Visualization," proceedings, November 22-24, 2006, Akademische Verlagsgesellschaft Aka GmbH, Berlin, (2006), 293-300. Google Scholar [4] E. Casas, K. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Appl. Math. Optim., 40 (1999), 229-257. doi: 10.1007/s002459900124.  Google Scholar [5] T. F. Chan and S. Esedoḡlu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837 (electronic). doi: 10.1137/040604297.  Google Scholar [6] T. F. Chan, S. Esedoḡlu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., 66 (2006), 1632-1648 (electronic). doi: 10.1137/040615286.  Google Scholar [7] T. F. Chan and J. Shen, "Image Processing and Analysis," Variational, PDE, Wavelet, and Stochastic Methods, SIAM (Society for Industrial and Applied Mathematics), Philadelphia, PA, 2005.  Google Scholar [8] R. Choksi and Y. van Gennip, Deblurring of one dimensional bar codes via total variation energy minimisation, SIAM J. Imaging Sci., 3 (2010), 735-764. doi: 10.1137/090773829.  Google Scholar [9] C.-H. Chu, D.-N. Yang and M.-S. Chen, Image stablization for 2d barcode in handheld devices, in "Proceedings of the 15th International Conference on Multimedia" (eds. R. Lienhart, A. R. Prasad, A. Hanjalic, S. Choi, B. P. Bailey and N. Sebe), Augsburg, Germany, September 24-29, 2007, ACM, (2007), 697-706. Google Scholar [10] I. Ekeland and R. Temam, "Convex Analysis and Variational Problems," Translated from the French, Studies in Mathematics and its Applications, 1, North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1976.  Google Scholar [11] S. Esedoḡlu, Blind deconvolution of bar code signals, Inverse Problems, 20 (2004), 121-135. doi: 10.1088/0266-5611/20/1/007.  Google Scholar [12] S. Esedoḡlu and S. J. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Comm. Pure Appl. Math., 57 (2004), 1609-1626. doi: 10.1002/cpa.20045.  Google Scholar [13] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar [14] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630. doi: 10.1137/060669358.  Google Scholar [15] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.  Google Scholar [16] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics, 80, Birkhäuser Verlag, Basel, 1984.  Google Scholar [17] M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs, in "Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences" (eds. V. Blondel, S. Boyd and H. Kimura), 371, Springer, London, (2008), 95-110. Available from: http://stanford.edu/ boyd/graph_dcp.html.  Google Scholar [18] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 1.21, May 2010., Available from: , ().   Google Scholar [19] Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, in "The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures," University Lecture Series, 22, American Mathematical Society, Providence, RI, 2001.  Google Scholar [20] J. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218. doi: 10.1007/s00208-004-0624-0.  Google Scholar [21] A. Mosek, Mosek: A Full Featured Software Package Intended for Solution of Large Scale Optimization Problems, 2008., Available from: , ().   Google Scholar [22] R. Palmer, "The Bar Code Book: A Comprehensive Guide to Reading, Printing, Specifying, Evaluating, and Using Bar Code and Other Machine-Readable Symbols," fifth edition, Trafford Publishing, 2007. Google Scholar [23] W. Ring, Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal., 34 (2000), 799-810. doi: 10.1051/m2an:2000104.  Google Scholar [24] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [25] J. E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588. doi: 10.1090/S0002-9904-1978-14499-1.  Google Scholar [26] W. Xu and S. McCloskey, 2D barcode localization and motion deblurring using a flutter shutter camera, in "2011 IEEE Workshop on Applications of Computer Vision (WACV)," (2011), 159-165. doi: 10.1109/WACV.2011.5711498.  Google Scholar

show all references

##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar [3] B. Berkels, M. Burger, M. Droske, O. Nemitz and M. Rumpf, Cartoon extraction based on anisotropic image classification, in "Vision, Modeling, and Visualization," proceedings, November 22-24, 2006, Akademische Verlagsgesellschaft Aka GmbH, Berlin, (2006), 293-300. Google Scholar [4] E. Casas, K. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Appl. Math. Optim., 40 (1999), 229-257. doi: 10.1007/s002459900124.  Google Scholar [5] T. F. Chan and S. Esedoḡlu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837 (electronic). doi: 10.1137/040604297.  Google Scholar [6] T. F. Chan, S. Esedoḡlu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., 66 (2006), 1632-1648 (electronic). doi: 10.1137/040615286.  Google Scholar [7] T. F. Chan and J. Shen, "Image Processing and Analysis," Variational, PDE, Wavelet, and Stochastic Methods, SIAM (Society for Industrial and Applied Mathematics), Philadelphia, PA, 2005.  Google Scholar [8] R. Choksi and Y. van Gennip, Deblurring of one dimensional bar codes via total variation energy minimisation, SIAM J. Imaging Sci., 3 (2010), 735-764. doi: 10.1137/090773829.  Google Scholar [9] C.-H. Chu, D.-N. Yang and M.-S. Chen, Image stablization for 2d barcode in handheld devices, in "Proceedings of the 15th International Conference on Multimedia" (eds. R. Lienhart, A. R. Prasad, A. Hanjalic, S. Choi, B. P. Bailey and N. Sebe), Augsburg, Germany, September 24-29, 2007, ACM, (2007), 697-706. Google Scholar [10] I. Ekeland and R. Temam, "Convex Analysis and Variational Problems," Translated from the French, Studies in Mathematics and its Applications, 1, North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1976.  Google Scholar [11] S. Esedoḡlu, Blind deconvolution of bar code signals, Inverse Problems, 20 (2004), 121-135. doi: 10.1088/0266-5611/20/1/007.  Google Scholar [12] S. Esedoḡlu and S. J. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Comm. Pure Appl. Math., 57 (2004), 1609-1626. doi: 10.1002/cpa.20045.  Google Scholar [13] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar [14] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630. doi: 10.1137/060669358.  Google Scholar [15] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.  Google Scholar [16] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics, 80, Birkhäuser Verlag, Basel, 1984.  Google Scholar [17] M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs, in "Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences" (eds. V. Blondel, S. Boyd and H. Kimura), 371, Springer, London, (2008), 95-110. Available from: http://stanford.edu/ boyd/graph_dcp.html.  Google Scholar [18] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 1.21, May 2010., Available from: , ().   Google Scholar [19] Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, in "The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures," University Lecture Series, 22, American Mathematical Society, Providence, RI, 2001.  Google Scholar [20] J. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218. doi: 10.1007/s00208-004-0624-0.  Google Scholar [21] A. Mosek, Mosek: A Full Featured Software Package Intended for Solution of Large Scale Optimization Problems, 2008., Available from: , ().   Google Scholar [22] R. Palmer, "The Bar Code Book: A Comprehensive Guide to Reading, Printing, Specifying, Evaluating, and Using Bar Code and Other Machine-Readable Symbols," fifth edition, Trafford Publishing, 2007. Google Scholar [23] W. Ring, Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal., 34 (2000), 799-810. doi: 10.1051/m2an:2000104.  Google Scholar [24] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [25] J. E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588. doi: 10.1090/S0002-9904-1978-14499-1.  Google Scholar [26] W. Xu and S. McCloskey, 2D barcode localization and motion deblurring using a flutter shutter camera, in "2011 IEEE Workshop on Applications of Computer Vision (WACV)," (2011), 159-165. doi: 10.1109/WACV.2011.5711498.  Google Scholar
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