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.

[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).

[3]

G. Dal Maso, "An Introduction to Gamma Convergence,'', Progress in Nonlinear Differential Equations and their Applications, 8 (1993).

[4]

S. Esedoglu, Blind deconvolution of bar code signals,, Inverse Problems, 20 (2004), 121. doi: 10.1088/0266-5611/20/1/007.

[5]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'', Studies in Advanced Mathematics, (1992).

[6]

E. Isaacson and H. B. Keller, "Analysis of Numerical Methods,'', Corrected reprint of the 1966 original [Wiley, (1966).

[7]

L. Modica and S. Mortola, Un esempio di gamma-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285.

[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.

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.

[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).

[3]

G. Dal Maso, "An Introduction to Gamma Convergence,'', Progress in Nonlinear Differential Equations and their Applications, 8 (1993).

[4]

S. Esedoglu, Blind deconvolution of bar code signals,, Inverse Problems, 20 (2004), 121. doi: 10.1088/0266-5611/20/1/007.

[5]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'', Studies in Advanced Mathematics, (1992).

[6]

E. Isaacson and H. B. Keller, "Analysis of Numerical Methods,'', Corrected reprint of the 1966 original [Wiley, (1966).

[7]

L. Modica and S. Mortola, Un esempio di gamma-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285.

[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.

[1]

Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159

[2]

Konstantinos Chrysafinos. Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077

[3]

Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99

[4]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961

[5]

Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473

[6]

Gisella Croce, Nikos Katzourakis, Giovanni Pisante. $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266

[7]

Ivar Ekeland. From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1101-1119. doi: 10.3934/dcds.2010.28.1101

[8]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

[9]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481

[10]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[11]

Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709

[12]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551

[13]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010

[14]

Marcone C. Pereira, Ricardo P. Silva. Error estimates for a Neumann problem in highly oscillating thin domains. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 803-817. doi: 10.3934/dcds.2013.33.803

[15]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577

[16]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

[17]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[18]

Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673

[19]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541

[20]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]