Abstract

We propose a novel method for correcting the effect of nonuniform illumination on a bi-level image. The proposed method is based on a penalized nonlinear least squares objective function that measures the binariness of an image and the roughness of illumination. Compared with conventional methods, it has the advantages of 1) not suffering from a trivial minimizer, 2) not requiring tuning of design parameters, and 3) effective optimization. In addition, it yields a unique solution since the minimization of the objective function is well-posed. In simulations and experiments, the method showed better accuracy and speed than the conventional entropy-based method.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. H. Li and K. S. Lii, “A joint estimation approach for two-tone image deblurring by blind deconvolution,” IEEE Trans. Image Process. 11(8), 847–858 (2002).
  2. E. Y. Lam, “Blind bi-level image restoration with iterated quadratic programming,” IEEE Trans. Circ. Syst. 52(Part 2), 52–56 (2007).
  3. Y. Shen, E. Y. Lam, and N. Wong, “Binary image restoration by positive semidefinite programming,” Opt. Lett. 32(2), 121–123 (2007).
    [Crossref]
  4. T. F. Chan, S. Esedoglu, and M. Nikolova, “Finding the global minimum for binary image restoration,” in Proceedings of IEEE International Conference on Image Processing (ICIP, 2005), pp. 121–124.
  5. J. Kim and H. Lee, “Joint nonuniform illumination estimation and deblurring for bar code signals,” Opt. Express 15(22), 14817–14837 (2007).
    [Crossref]
  6. M. S. Brown and Y. C. Tsoi, “Geometric and shading correction for images of printed materials using boundary,” IEEE Trans. Image Process. 15(6), 1544–1554 (2006).
    [Crossref]
  7. D. A. Forsyth and J. Ponce, Computer vision: A modern approach (Prentice Hall, 2003).
  8. T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
    [Crossref]
  9. Z. Hou, “A review on MR image inhomogeneity correction,” Int. J. Biomed. Imaging 2006, 1–11 (2006).
    [Crossref]
  10. B. H. Brinkmann, A. Manduca, and R. A. Robb, “Optimized homomorphic unsharp masking for MR grayscale inhomogeneity correction,” IEEE Trans. Med. Imaging 17(2), 161–171 (1998).
    [Crossref]
  11. J. G. Sled, A. P. Zijdenbos, and A. C. Evans, “A nonparametric method for automatic correction of intensity nonuniformity in MRI data,” IEEE Trans. Med. Imaging 17(1), 87–97 (1998).
    [Crossref]
  12. J. F. Mangin, “Entropy minimization for automatic correction of intensity nonuniformity,” IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA, 2000), pp. 162–169.
  13. D. Tomaẑevič, B. Likar, and F. Pernuš, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208(Pt 3), 212–223 (2002).
  14. B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuš, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197(Pt 3), 285–295 (2000).
    [Crossref]
  15. H.-L. Shen and K. Li, “Decomposition of shading and reflectance from a texture image,” Opt. Lett. 34(1), 64–66 (2009).
    [Crossref]
  16. B. W. Silverman, Density estimation for statistics and data analysis (Chapman and Hall, 1985).
  17. Q. Ji, J. O. Glass, and W. E. Reddick, “A novel, fast entropy-minimization algorithm for bias field correction in MR images,” Magn. Reson. Imaging 25(2), 259–264 (2007).
    [Crossref]
  18. M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16(6), 22–38 (1999).
    [Crossref]
  19. S. Esedoglu, “Blind deconvolution of bar code signals,” Inverse Probl. 20(1), 121–135 (2004).
    [Crossref]
  20. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C + +, 2nd ed. (Cambridge, 2005).
  21. A. Papoulis and S. U. Pillai, Probability, random variables, and stochastic processes, 4th ed. (McGraw-Hill, 2002).
  22. J. Kybic, P. Thévenaz, A. Nirkko, and M. Unser, “Unwarping of unidirectionally distorted EPI images,” IEEE Trans. Med. Imaging 19(2), 80–93 (2000).
    [Crossref]
  23. O. Grellier and P. Comon, “Blind separation of discrete sources,” IEEE Signal Process. Lett. 5(8), 212–214 (1998).
    [Crossref]
  24. J. Eriksson, P. A. Wedin, M. E. Gulliksson, and I. Söderkvist, “Regularization methods for uniformly rank deficient nonlinear least-squares problems,” J. Optim. Theory Appl. 127(1), 1–26 (2005).
    [Crossref]
  25. J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging 13(2), 290–300 (1997).
  26. H. L. Van Trees, Detection, estimation, and modulation theory, Part 1 (Wiley, 1968).
  27. J. Kim and J. A. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imaging 23(11), 1430–1444 (2004).
    [Crossref]
  28. J. Zhang, Q. Zhang, and G. He, “Blind deconvolution: multiplicative iterative algorithm,” Opt. Lett. 33(1), 25–27 (2008).
    [Crossref]
  29. T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9(7), 1052–1061 (1992).
    [Crossref]
  30. D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A 12(1), 58–65 (1995).
    [Crossref]
  31. S. Lu and C. L. Tan, “Binarization of badly illuminated document images through shading estimation and compensation,” in Proceedings of IEEE International Conference on Document Analysis and Recognition (ICDAR, 2007), pp. 312–316.

2009 (1)

H.-L. Shen and K. Li, “Decomposition of shading and reflectance from a texture image,” Opt. Lett. 34(1), 64–66 (2009).
[Crossref]

2008 (1)

J. Zhang, Q. Zhang, and G. He, “Blind deconvolution: multiplicative iterative algorithm,” Opt. Lett. 33(1), 25–27 (2008).
[Crossref]

2007 (4)

Q. Ji, J. O. Glass, and W. E. Reddick, “A novel, fast entropy-minimization algorithm for bias field correction in MR images,” Magn. Reson. Imaging 25(2), 259–264 (2007).
[Crossref]

E. Y. Lam, “Blind bi-level image restoration with iterated quadratic programming,” IEEE Trans. Circ. Syst. 52(Part 2), 52–56 (2007).

Y. Shen, E. Y. Lam, and N. Wong, “Binary image restoration by positive semidefinite programming,” Opt. Lett. 32(2), 121–123 (2007).
[Crossref]

J. Kim and H. Lee, “Joint nonuniform illumination estimation and deblurring for bar code signals,” Opt. Express 15(22), 14817–14837 (2007).
[Crossref]

2006 (3)

M. S. Brown and Y. C. Tsoi, “Geometric and shading correction for images of printed materials using boundary,” IEEE Trans. Image Process. 15(6), 1544–1554 (2006).
[Crossref]

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[Crossref]

Z. Hou, “A review on MR image inhomogeneity correction,” Int. J. Biomed. Imaging 2006, 1–11 (2006).
[Crossref]

2005 (1)

J. Eriksson, P. A. Wedin, M. E. Gulliksson, and I. Söderkvist, “Regularization methods for uniformly rank deficient nonlinear least-squares problems,” J. Optim. Theory Appl. 127(1), 1–26 (2005).
[Crossref]

2004 (2)

S. Esedoglu, “Blind deconvolution of bar code signals,” Inverse Probl. 20(1), 121–135 (2004).
[Crossref]

J. Kim and J. A. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imaging 23(11), 1430–1444 (2004).
[Crossref]

2002 (2)

T. H. Li and K. S. Lii, “A joint estimation approach for two-tone image deblurring by blind deconvolution,” IEEE Trans. Image Process. 11(8), 847–858 (2002).

D. Tomaẑevič, B. Likar, and F. Pernuš, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208(Pt 3), 212–223 (2002).

2000 (2)

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuš, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197(Pt 3), 285–295 (2000).
[Crossref]

J. Kybic, P. Thévenaz, A. Nirkko, and M. Unser, “Unwarping of unidirectionally distorted EPI images,” IEEE Trans. Med. Imaging 19(2), 80–93 (2000).
[Crossref]

1999 (1)

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16(6), 22–38 (1999).
[Crossref]

1998 (3)

O. Grellier and P. Comon, “Blind separation of discrete sources,” IEEE Signal Process. Lett. 5(8), 212–214 (1998).
[Crossref]

B. H. Brinkmann, A. Manduca, and R. A. Robb, “Optimized homomorphic unsharp masking for MR grayscale inhomogeneity correction,” IEEE Trans. Med. Imaging 17(2), 161–171 (1998).
[Crossref]

J. G. Sled, A. P. Zijdenbos, and A. C. Evans, “A nonparametric method for automatic correction of intensity nonuniformity in MRI data,” IEEE Trans. Med. Imaging 17(1), 87–97 (1998).
[Crossref]

1997 (1)

J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging 13(2), 290–300 (1997).

1995 (1)

D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A 12(1), 58–65 (1995).
[Crossref]

1992 (1)

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9(7), 1052–1061 (1992).
[Crossref]

1968 (1)

H. L. Van Trees, Detection, estimation, and modulation theory, Part 1 (Wiley, 1968).

Brinicombe, A. M.

D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A 12(1), 58–65 (1995).
[Crossref]

Brinkmann, B. H.

B. H. Brinkmann, A. Manduca, and R. A. Robb, “Optimized homomorphic unsharp masking for MR grayscale inhomogeneity correction,” IEEE Trans. Med. Imaging 17(2), 161–171 (1998).
[Crossref]

Brown, M. S.

M. S. Brown and Y. C. Tsoi, “Geometric and shading correction for images of printed materials using boundary,” IEEE Trans. Image Process. 15(6), 1544–1554 (2006).
[Crossref]

Chan, T. F.

T. F. Chan, S. Esedoglu, and M. Nikolova, “Finding the global minimum for binary image restoration,” in Proceedings of IEEE International Conference on Image Processing (ICIP, 2005), pp. 121–124.

Chen, T.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[Crossref]

Comaniciu, D.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[Crossref]

Comon, P.

O. Grellier and P. Comon, “Blind separation of discrete sources,” IEEE Signal Process. Lett. 5(8), 212–214 (1998).
[Crossref]

Eriksson, J.

J. Eriksson, P. A. Wedin, M. E. Gulliksson, and I. Söderkvist, “Regularization methods for uniformly rank deficient nonlinear least-squares problems,” J. Optim. Theory Appl. 127(1), 1–26 (2005).
[Crossref]

Esedoglu, S.

S. Esedoglu, “Blind deconvolution of bar code signals,” Inverse Probl. 20(1), 121–135 (2004).
[Crossref]

T. F. Chan, S. Esedoglu, and M. Nikolova, “Finding the global minimum for binary image restoration,” in Proceedings of IEEE International Conference on Image Processing (ICIP, 2005), pp. 121–124.

Evans, A. C.

J. G. Sled, A. P. Zijdenbos, and A. C. Evans, “A nonparametric method for automatic correction of intensity nonuniformity in MRI data,” IEEE Trans. Med. Imaging 17(1), 87–97 (1998).
[Crossref]

Fessler, J. A.

J. Kim and J. A. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imaging 23(11), 1430–1444 (2004).
[Crossref]

J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging 13(2), 290–300 (1997).

Fish, D. A.

D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A 12(1), 58–65 (1995).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C + +, 2nd ed. (Cambridge, 2005).

Forsyth, D. A.

D. A. Forsyth and J. Ponce, Computer vision: A modern approach (Prentice Hall, 2003).

Glass, J. O.

Q. Ji, J. O. Glass, and W. E. Reddick, “A novel, fast entropy-minimization algorithm for bias field correction in MR images,” Magn. Reson. Imaging 25(2), 259–264 (2007).
[Crossref]

Grellier, O.

O. Grellier and P. Comon, “Blind separation of discrete sources,” IEEE Signal Process. Lett. 5(8), 212–214 (1998).
[Crossref]

Gulliksson, M. E.

J. Eriksson, P. A. Wedin, M. E. Gulliksson, and I. Söderkvist, “Regularization methods for uniformly rank deficient nonlinear least-squares problems,” J. Optim. Theory Appl. 127(1), 1–26 (2005).
[Crossref]

He, G.

J. Zhang, Q. Zhang, and G. He, “Blind deconvolution: multiplicative iterative algorithm,” Opt. Lett. 33(1), 25–27 (2008).
[Crossref]

Holmes, T. J.

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9(7), 1052–1061 (1992).
[Crossref]

Hou, Z.

Z. Hou, “A review on MR image inhomogeneity correction,” Int. J. Biomed. Imaging 2006, 1–11 (2006).
[Crossref]

Huang, T. S.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[Crossref]

Ji, Q.

Q. Ji, J. O. Glass, and W. E. Reddick, “A novel, fast entropy-minimization algorithm for bias field correction in MR images,” Magn. Reson. Imaging 25(2), 259–264 (2007).
[Crossref]

Kim, J.

J. Kim and H. Lee, “Joint nonuniform illumination estimation and deblurring for bar code signals,” Opt. Express 15(22), 14817–14837 (2007).
[Crossref]

J. Kim and J. A. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imaging 23(11), 1430–1444 (2004).
[Crossref]

Kybic, J.

J. Kybic, P. Thévenaz, A. Nirkko, and M. Unser, “Unwarping of unidirectionally distorted EPI images,” IEEE Trans. Med. Imaging 19(2), 80–93 (2000).
[Crossref]

Lam, E. Y.

Y. Shen, E. Y. Lam, and N. Wong, “Binary image restoration by positive semidefinite programming,” Opt. Lett. 32(2), 121–123 (2007).
[Crossref]

E. Y. Lam, “Blind bi-level image restoration with iterated quadratic programming,” IEEE Trans. Circ. Syst. 52(Part 2), 52–56 (2007).

Lee, H.

J. Kim and H. Lee, “Joint nonuniform illumination estimation and deblurring for bar code signals,” Opt. Express 15(22), 14817–14837 (2007).
[Crossref]

Li, K.

H.-L. Shen and K. Li, “Decomposition of shading and reflectance from a texture image,” Opt. Lett. 34(1), 64–66 (2009).
[Crossref]

Li, T. H.

T. H. Li and K. S. Lii, “A joint estimation approach for two-tone image deblurring by blind deconvolution,” IEEE Trans. Image Process. 11(8), 847–858 (2002).

Lii, K. S.

T. H. Li and K. S. Lii, “A joint estimation approach for two-tone image deblurring by blind deconvolution,” IEEE Trans. Image Process. 11(8), 847–858 (2002).

Likar, B.

D. Tomaẑevič, B. Likar, and F. Pernuš, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208(Pt 3), 212–223 (2002).

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuš, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197(Pt 3), 285–295 (2000).
[Crossref]

Lu, S.

S. Lu and C. L. Tan, “Binarization of badly illuminated document images through shading estimation and compensation,” in Proceedings of IEEE International Conference on Document Analysis and Recognition (ICDAR, 2007), pp. 312–316.

Maintz, J. B. A.

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuš, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197(Pt 3), 285–295 (2000).
[Crossref]

Manduca, A.

B. H. Brinkmann, A. Manduca, and R. A. Robb, “Optimized homomorphic unsharp masking for MR grayscale inhomogeneity correction,” IEEE Trans. Med. Imaging 17(2), 161–171 (1998).
[Crossref]

Mangin, J. F.

J. F. Mangin, “Entropy minimization for automatic correction of intensity nonuniformity,” IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA, 2000), pp. 162–169.

Nikolova, M.

T. F. Chan, S. Esedoglu, and M. Nikolova, “Finding the global minimum for binary image restoration,” in Proceedings of IEEE International Conference on Image Processing (ICIP, 2005), pp. 121–124.

Nirkko, A.

J. Kybic, P. Thévenaz, A. Nirkko, and M. Unser, “Unwarping of unidirectionally distorted EPI images,” IEEE Trans. Med. Imaging 19(2), 80–93 (2000).
[Crossref]

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, random variables, and stochastic processes, 4th ed. (McGraw-Hill, 2002).

Pernuš, F.

D. Tomaẑevič, B. Likar, and F. Pernuš, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208(Pt 3), 212–223 (2002).

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuš, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197(Pt 3), 285–295 (2000).
[Crossref]

Pike, E. R.

D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A 12(1), 58–65 (1995).
[Crossref]

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, random variables, and stochastic processes, 4th ed. (McGraw-Hill, 2002).

Ponce, J.

D. A. Forsyth and J. Ponce, Computer vision: A modern approach (Prentice Hall, 2003).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C + +, 2nd ed. (Cambridge, 2005).

Reddick, W. E.

Q. Ji, J. O. Glass, and W. E. Reddick, “A novel, fast entropy-minimization algorithm for bias field correction in MR images,” Magn. Reson. Imaging 25(2), 259–264 (2007).
[Crossref]

Robb, R. A.

B. H. Brinkmann, A. Manduca, and R. A. Robb, “Optimized homomorphic unsharp masking for MR grayscale inhomogeneity correction,” IEEE Trans. Med. Imaging 17(2), 161–171 (1998).
[Crossref]

Shen, H.-L.

H.-L. Shen and K. Li, “Decomposition of shading and reflectance from a texture image,” Opt. Lett. 34(1), 64–66 (2009).
[Crossref]

Shen, Y.

Y. Shen, E. Y. Lam, and N. Wong, “Binary image restoration by positive semidefinite programming,” Opt. Lett. 32(2), 121–123 (2007).
[Crossref]

Silverman, B. W.

B. W. Silverman, Density estimation for statistics and data analysis (Chapman and Hall, 1985).

Sled, J. G.

J. G. Sled, A. P. Zijdenbos, and A. C. Evans, “A nonparametric method for automatic correction of intensity nonuniformity in MRI data,” IEEE Trans. Med. Imaging 17(1), 87–97 (1998).
[Crossref]

Söderkvist, I.

J. Eriksson, P. A. Wedin, M. E. Gulliksson, and I. Söderkvist, “Regularization methods for uniformly rank deficient nonlinear least-squares problems,” J. Optim. Theory Appl. 127(1), 1–26 (2005).
[Crossref]

Tan, C. L.

S. Lu and C. L. Tan, “Binarization of badly illuminated document images through shading estimation and compensation,” in Proceedings of IEEE International Conference on Document Analysis and Recognition (ICDAR, 2007), pp. 312–316.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C + +, 2nd ed. (Cambridge, 2005).

Thévenaz, P.

J. Kybic, P. Thévenaz, A. Nirkko, and M. Unser, “Unwarping of unidirectionally distorted EPI images,” IEEE Trans. Med. Imaging 19(2), 80–93 (2000).
[Crossref]

Toma?evic, D.

D. Tomaẑevič, B. Likar, and F. Pernuš, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208(Pt 3), 212–223 (2002).

Tsoi, Y. C.

M. S. Brown and Y. C. Tsoi, “Geometric and shading correction for images of printed materials using boundary,” IEEE Trans. Image Process. 15(6), 1544–1554 (2006).
[Crossref]

Unser, M.

J. Kybic, P. Thévenaz, A. Nirkko, and M. Unser, “Unwarping of unidirectionally distorted EPI images,” IEEE Trans. Med. Imaging 19(2), 80–93 (2000).
[Crossref]

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16(6), 22–38 (1999).
[Crossref]

Van Trees, H. L.

H. L. Van Trees, Detection, estimation, and modulation theory, Part 1 (Wiley, 1968).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C + +, 2nd ed. (Cambridge, 2005).

Viergever, M. A.

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuš, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197(Pt 3), 285–295 (2000).
[Crossref]

Walker, J. G.

D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A 12(1), 58–65 (1995).
[Crossref]

Wedin, P. A.

J. Eriksson, P. A. Wedin, M. E. Gulliksson, and I. Söderkvist, “Regularization methods for uniformly rank deficient nonlinear least-squares problems,” J. Optim. Theory Appl. 127(1), 1–26 (2005).
[Crossref]

Wong, N.

Y. Shen, E. Y. Lam, and N. Wong, “Binary image restoration by positive semidefinite programming,” Opt. Lett. 32(2), 121–123 (2007).
[Crossref]

Yin, W.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[Crossref]

Zhang, J.

J. Zhang, Q. Zhang, and G. He, “Blind deconvolution: multiplicative iterative algorithm,” Opt. Lett. 33(1), 25–27 (2008).
[Crossref]

Zhang, Q.

J. Zhang, Q. Zhang, and G. He, “Blind deconvolution: multiplicative iterative algorithm,” Opt. Lett. 33(1), 25–27 (2008).
[Crossref]

Zhou, X. S.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[Crossref]

Zijdenbos, A. P.

J. G. Sled, A. P. Zijdenbos, and A. C. Evans, “A nonparametric method for automatic correction of intensity nonuniformity in MRI data,” IEEE Trans. Med. Imaging 17(1), 87–97 (1998).
[Crossref]

IEEE Signal Process. Lett. (1)

O. Grellier and P. Comon, “Blind separation of discrete sources,” IEEE Signal Process. Lett. 5(8), 212–214 (1998).
[Crossref]

IEEE Signal Process. Mag. (1)

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16(6), 22–38 (1999).
[Crossref]

IEEE Trans. Circ. Syst. (1)

E. Y. Lam, “Blind bi-level image restoration with iterated quadratic programming,” IEEE Trans. Circ. Syst. 52(Part 2), 52–56 (2007).

IEEE Trans. Image Process. (2)

T. H. Li and K. S. Lii, “A joint estimation approach for two-tone image deblurring by blind deconvolution,” IEEE Trans. Image Process. 11(8), 847–858 (2002).

M. S. Brown and Y. C. Tsoi, “Geometric and shading correction for images of printed materials using boundary,” IEEE Trans. Image Process. 15(6), 1544–1554 (2006).
[Crossref]

IEEE Trans. Med. Imaging (5)

J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging 13(2), 290–300 (1997).

B. H. Brinkmann, A. Manduca, and R. A. Robb, “Optimized homomorphic unsharp masking for MR grayscale inhomogeneity correction,” IEEE Trans. Med. Imaging 17(2), 161–171 (1998).
[Crossref]

J. G. Sled, A. P. Zijdenbos, and A. C. Evans, “A nonparametric method for automatic correction of intensity nonuniformity in MRI data,” IEEE Trans. Med. Imaging 17(1), 87–97 (1998).
[Crossref]

J. Kybic, P. Thévenaz, A. Nirkko, and M. Unser, “Unwarping of unidirectionally distorted EPI images,” IEEE Trans. Med. Imaging 19(2), 80–93 (2000).
[Crossref]

J. Kim and J. A. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imaging 23(11), 1430–1444 (2004).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[Crossref]

Int. J. Biomed. Imaging (1)

Z. Hou, “A review on MR image inhomogeneity correction,” Int. J. Biomed. Imaging 2006, 1–11 (2006).
[Crossref]

Inverse Probl. (1)

S. Esedoglu, “Blind deconvolution of bar code signals,” Inverse Probl. 20(1), 121–135 (2004).
[Crossref]

J. Microsc. (2)

D. Tomaẑevič, B. Likar, and F. Pernuš, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208(Pt 3), 212–223 (2002).

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuš, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197(Pt 3), 285–295 (2000).
[Crossref]

J. Opt. Soc. Am. A (2)

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9(7), 1052–1061 (1992).
[Crossref]

D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A 12(1), 58–65 (1995).
[Crossref]

J. Optim. Theory Appl. (1)

J. Eriksson, P. A. Wedin, M. E. Gulliksson, and I. Söderkvist, “Regularization methods for uniformly rank deficient nonlinear least-squares problems,” J. Optim. Theory Appl. 127(1), 1–26 (2005).
[Crossref]

Magn. Reson. Imaging (1)

Q. Ji, J. O. Glass, and W. E. Reddick, “A novel, fast entropy-minimization algorithm for bias field correction in MR images,” Magn. Reson. Imaging 25(2), 259–264 (2007).
[Crossref]

Opt. Express (1)

J. Kim and H. Lee, “Joint nonuniform illumination estimation and deblurring for bar code signals,” Opt. Express 15(22), 14817–14837 (2007).
[Crossref]

Opt. Lett. (3)

Y. Shen, E. Y. Lam, and N. Wong, “Binary image restoration by positive semidefinite programming,” Opt. Lett. 32(2), 121–123 (2007).
[Crossref]

H.-L. Shen and K. Li, “Decomposition of shading and reflectance from a texture image,” Opt. Lett. 34(1), 64–66 (2009).
[Crossref]

J. Zhang, Q. Zhang, and G. He, “Blind deconvolution: multiplicative iterative algorithm,” Opt. Lett. 33(1), 25–27 (2008).
[Crossref]

Other (8)

S. Lu and C. L. Tan, “Binarization of badly illuminated document images through shading estimation and compensation,” in Proceedings of IEEE International Conference on Document Analysis and Recognition (ICDAR, 2007), pp. 312–316.

J. F. Mangin, “Entropy minimization for automatic correction of intensity nonuniformity,” IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA, 2000), pp. 162–169.

H. L. Van Trees, Detection, estimation, and modulation theory, Part 1 (Wiley, 1968).

B. W. Silverman, Density estimation for statistics and data analysis (Chapman and Hall, 1985).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C + +, 2nd ed. (Cambridge, 2005).

A. Papoulis and S. U. Pillai, Probability, random variables, and stochastic processes, 4th ed. (McGraw-Hill, 2002).

T. F. Chan, S. Esedoglu, and M. Nikolova, “Finding the global minimum for binary image restoration,” in Proceedings of IEEE International Conference on Image Processing (ICIP, 2005), pp. 121–124.

D. A. Forsyth and J. Ponce, Computer vision: A modern approach (Prentice Hall, 2003).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Images under nonuniform illumination: (a) bar code image (b) text image (c) the bar code image after binarization (d) the text image after binarization (e) synthetic Gaussian shape illumination applied to the bar code image (f) synthetic linear shape illumination applied to the text image.

Fig. 2.
Fig. 2.

Nonuniform illumination correction using the proposed method: (a) corrected bar code image (λr =1e -7) (b) corrected text image (λr =1e -5) (c) the bar code image after binarization (d) the text image after binarization (e) estimated inverse illumination for the bar code image (f) estimated inverse illumination for the text image.

Fig. 3.
Fig. 3.

Nonuniform illumination correction using the entropy based method: (a) result for the bar code image after applying the entropy based method (b) result for the text image after applying the entropy based method (c) binarization for the bar code image (d) binarization for the bar code image (e) estimated inverse illumination for the text image (f) estimated inverse illumination for the text image.

Fig. 4.
Fig. 4.

Change of the objective functions for the (a) bar code image (for proposed method λr =1e -7, and for entropy based method λr =1e -3, λm =1e-5, σ=0.1) and the (b) text image (for proposed method λm =1e -5, and for entropy based method λr =1e -3, λm =1e -5, σ=0.1)

Fig. 5.
Fig. 5.

Real images under nonuniform illumination and the result of applying the proposed correction method: (a) QR bar code image (b) document image (c) corrected image for the document image (d) corrected image for the QR bar code image (e) estimated inverse illumination for the QR bar code image (f) estimated inverse illumination for the document image.

Tables (2)

Tables Icon

Table 1. Correlation coefficient (BER) values for the bar code image (The unit for BER is %. For the proposed λr =1e -7, and for the entropy λr =1e -3, λm =1e -5, z 1 = 4 , z n b = 28 , n b = 128 . )

Tables Icon

Table 2. Correlation coefficient (BER) values for the text image (The unit for BER is %. For the proposed λr =1e -5, and for the entropy λr =1e -3, λm =1e -5, z 1 = 4 , z n b = 28 , n b = 128 . )

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

g ( x i , y j ) Poisson [ s ( x i , y j ) f ( x i , y j ) ] , i = 1 , , M , j = 1 , , N ,
b ̂ = arg min b Σ k = 1 n b p g ( z k ; b ) log p g ( z k ; b ) + λ r R ( b ) ,
b ̂ = arg min b Σ k = 1 n b p g ( z k ; b ) log p g ( z k ; b )
+ λ m ( Σ i = 1 M Σ j = 1 N g ( x i , y j ) h ( x i , y j ; b ) Σ i = 1 M Σ j = 1 N g ( x i , y j ) ) 2
+ λ r R ( b ) ,
h ( x i , y j ; b ) = Σ k = 1 P Σ l = 1 Q b k , l β 3 ( x i / h x l k ) β 3 ( y j / h y l l ) , i = 1 , , M , j = 1 , , N ,
minimize b > 0 , α 1 > 0 , α 2 > 0 f ( x i , y j ) { α 1 , α 2 } Σ i = 1 M Σ j = 1 N ( g ( x i , y j ) f ( x i , y j ) h ( x i , y j ; b ) ) 2 .
minimize b > 0 , α > 0 f ( x i , y j ) { α , 1 + α } Σ i = 1 M Σ j = 1 N 1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) f ( x i , y j ) ) 2 .
b ̂ α ̂ = arg min b > o , α > 0 Σ i = 1 M Σ j = 1 N min { 1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) α ) 2 ,
1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) ( α + 1 ) 2 } ,
b ̂ α ̂ = arg min b > 0 , α > 0 Σ i = 1 M Σ j = 1 N 1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) α ) 2
( h ( x i , y j ; b ) g ( x i , y j ) ( 1 + α ) ) 2 .
θ ̂ = arg min θ Σ i = 1 M Σ j = 1 N f ( x i , y j ; θ ) 2 + λ r R ( θ ) ,
f ( x i , y j ; θ ) = 1 h ( x i , y j ; b ) ( h ( x i , y j ; b ) g ( x i , y j ) α ) ( h ( x i , y j ; b ) g ( x i , y j ) ( 1 + α ) ) ,
R ( θ ) = 1 2 θ T R θ = Σ i = 1 P 1 Σ j = 1 Q 1 ( b i + 1 , j b i , j ) 2 + ( b i , j + 1 b i , j ) 2 ,
H ( θ ) = J ( θ ) T J ( θ ) + λ r R ,
[ J ( θ ) ] k , l = f ( x i , y j ; θ ) θ l ,
[ θ T J ( θ ) T ] k = α f ( x i , y j ; α , b u ) α + Σ l = 1 P × Q b u f ( x i , y j ; α , b u ) b l ,
cg ( x i , y j ) α c [ cg ( x i , y j ) α + cg ( x i , y j ) ( 1 + α ) ]
1 c [ ( cg ( x i , y j ) α ) ( cg ( x i , y j ) ( 1 + α ) ) ] = 0 , ( x i , y j ) .
g ( x i , y j ) = α / c , ( x i , y j ) .
p g ( z k ) = Σ i = 1 M Σ j = 1 N K σ ( z k g ( x i , y j ) s ( x i , y j ; b ) ) , k = 1 , , n b ,
min { 1 h ( x i , y i ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) α ) 2 , 1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) ( 1 + α ) ) 2 }
= 1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) α ) 2 .
1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) α ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) ( 1 + α ) ) 2
= 1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) α ) 2 ( 1 2 ε + ε 2 ) .
min { 1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) α ) 2 , 1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) ( 1 + α ) ) 2 }
1 h ( x i , y j ; b ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) α ) 2 ( h ( x i , y j ; b ) g ( x i , y j ) ( 1 + α ) ) 2 .

Metrics