# American Institute of Mathematical Sciences

September  2012, 17(6): 1889-1902. doi: 10.3934/dcdsb.2012.17.1889

## Error estimates for a bar code reconstruction method

 1 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, United States 2 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received  August 2011 Revised  February 2012 Published  May 2012

We analyze a variational method for reconstructing a bar code signal from a blurry and noisy measurement. The bar code is modeled as a binary function with a finite number of transitions and a parameter controlling minimal feature size. The measured signal is the convolution of this binary function with a Gaussian kernel. In this work, we assume that the blur kernel is known and establish conditions (involving noise level and variance of the convolution kernel) under which the variational method considered recovers essentially the correct bar code.
Citation: Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
##### References:
 [1] R. Choksi and Y. van Gennip, Deblurring of one dimensional bar codes via total variation energy minimization,, SIAM J. on Imaging Sciences, 3 (2010), 735. Google Scholar [2] R. Choksi, Y. van Gennip and A. Oberman, Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes,, Technical report, (2010). Google Scholar [3] G. Dal Maso, "An Introduction to Gamma Convergence,'', Progress in Nonlinear Differential Equations and their Applications, 8 (1993). Google Scholar [4] S. Esedoglu, Blind deconvolution of bar code signals,, Inverse Problems, 20 (2004), 121. doi: 10.1088/0266-5611/20/1/007. Google Scholar [5] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'', Studies in Advanced Mathematics, (1992). Google Scholar [6] E. Isaacson and H. B. Keller, "Analysis of Numerical Methods,'', Corrected reprint of the 1966 original [Wiley, (1966). Google Scholar [7] L. Modica and S. Mortola, Un esempio di gamma-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285. Google Scholar [8] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar

show all references

##### References:
 [1] R. Choksi and Y. van Gennip, Deblurring of one dimensional bar codes via total variation energy minimization,, SIAM J. on Imaging Sciences, 3 (2010), 735. Google Scholar [2] R. Choksi, Y. van Gennip and A. Oberman, Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes,, Technical report, (2010). Google Scholar [3] G. Dal Maso, "An Introduction to Gamma Convergence,'', Progress in Nonlinear Differential Equations and their Applications, 8 (1993). Google Scholar [4] S. Esedoglu, Blind deconvolution of bar code signals,, Inverse Problems, 20 (2004), 121. doi: 10.1088/0266-5611/20/1/007. Google Scholar [5] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'', Studies in Advanced Mathematics, (1992). Google Scholar [6] E. Isaacson and H. B. Keller, "Analysis of Numerical Methods,'', Corrected reprint of the 1966 original [Wiley, (1966). Google Scholar [7] L. Modica and S. Mortola, Un esempio di gamma-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285. Google Scholar [8] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar
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