## Retrospective correction of nonuniform illumination on bi-level images

Optics Express, Vol. 17, Issue 26, pp. 23880-23893 (2009)

http://dx.doi.org/10.1364/OE.17.023880

Acrobat PDF (740 KB)

### 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

## 1. Introduction

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

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]

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]

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]

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

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]

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

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

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]

*a priori*information about the intrinsic image and/or the nonuniform illumination. The most frequently used

*a priori*information is the assumption that illumination is slowly changing in the spatial domain and that the number of different intensity values of an intrinsic image is small. The latter information is useful not only for brain MR images and some microscopic images but also for bi-level images, thereby making it possible to apply existing methods to bi-level images.

**15(22)**, 14817–14837 (2007). [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]

**15(22)**, 14817–14837 (2007). [CrossRef]

**2006**, 1–11 (2006). [CrossRef]

*a priori*facts noted above have shown improved performance compared with the homomorphic filtering method [13]. While dividing an acquired image using a smooth parameterized illumination model, methods that seek parameters that minimize the entropy of the corrected image [14

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]

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]

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]

## 2. Theory

### 2.1 Problem formulation

*M*×

*N*bi-level image acquired under nonuniform illumination as follows:

*x*,

_{i}*y*) represents a 2D discrete pixel grid on a image along the

_{j}*x*and

*y*axis,

*g*(

*x*,

_{i}*y*) denotes the acquired 2D image,

_{j}*s*(

*x*,

_{i}*y*) is unknown illumination,

_{j}*f*(

*x*,

_{i}*y*)∊{

_{j}*α*

_{1},

*α*

_{2}} is

*true*bi-level image, respectively. Note that the true image

*f*(

*x*,

_{i}*y*) has only two unknown intensity values

_{j}*α*

_{1}and

*α*

_{2}. The noise present in the acquired image follows Poisson distribution. However, it has been well known that the Poisson noise can be approximated by Gaussian noise if the arrival rate of photon is sufficiently large [21].

*s*=[

*s*(

*x*,

_{i}*y*),⋯,

_{j}*s*(

*x*,

_{M}*y*)] and

_{N}**f**=[

*f*(

*x*,

_{i}*y*),⋯,

_{j}*f*(

*x*,

_{M}*y*)] from g=[

_{N}*g*(

*x*,

_{i}*y*),⋯,

_{j}*g*(

*x*,

_{M}*y*)] [5

_{N}**15(22)**, 14817–14837 (2007). [CrossRef]

**s**and

**f**that minimize the square errors between the model and the acquired image. However, the least squares problem is ill-posed in the sense that the solution is not unique since the effect of

**s**and

**f**is interchangeable on

**g**[6

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]

*a priori*information about the true image. For example, a method that minimizes estimated entropy from a corrected image can be applied for bi-level images [12]. This method parameterizes the inverse of illumination

*h*(

*x*,

_{i}*y*) using a parameter vector

_{j}**b**, and searches the optimal parameter that minimizes the estimated entropy as follows [14

**197(Pt 3)**, 285–295 (2000). [CrossRef]

*p*(

_{g}*z*;

_{k}**b**) is the estimated PMF from

*g*(

*x*,

_{i}*y*)

_{j}*h*(

*x*,

_{i}*y*;

_{j}**b**),

*i*=1,⋯

*M*,

*j*=1,⋯,

*N*,

*R*(

**b**) is a roughness penalty function for acquiring smooth illumination field,

*λ*is a regularization parameter, and n

_{r}_{b}is the number of nonzero entries of the estimated PMF. One may apply the histogram based methods [14

**197(Pt 3)**, 285–295 (2000). [CrossRef]

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]

*a priori*information that the estimated entropy from a true bi-level image is small since the intensities can have only one of the two values. Based on the similar information, a method that maximizes the frequency content of the estimated PMF has also been investigated [11

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]

*z*(

_{k}*k*=1,⋯,

*n*),

_{b}*n*and kernel function. In addition, the methods suffer from the trivial minimizer of

_{b}*h*(

*x*,

_{i}*y*;

_{j}**b**)=0. One may try to avoid the trivial minimizer by including an additional penalty function. For example, an objective function that penalizes the difference between average pixel value of the observed image and the corrected image has been investigated [12]:

*λ*and

_{m}*λ*are regularization parameters. However, the method requires the tuning of two regularization parameters

_{r}*λ*and

_{m}*λ*as well as design parameters in estimating the PMF. In addition, an effective optimization of the entropy is challenging [17

_{r}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]

## 2.2 Proposed method

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

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]

*β*

^{3}(·) is a third order B-spline function centered at zero,

**b**=[

*b*

_{1,1},

*b*

_{1,2}.,

*b*

_{P,Q}] denotes the coefficients for the tensor product of the 1D B-spline function,

*P*and

*Q*are the numbers of the 1D B-spline functions along the

*x*and

*y*axis,

*h*and

_{x}*h*are knot spacing along the

_{y}*x*and

*y*axis, and

*l*,

_{k}*l*,

_{l}*k*=1,⋯,

*P*,

*l*=1,⋯,

*Q*denote uniformly distributed center locations of the 1D B-spline functions along the

*x*and

*y*axis, respectively. Note that

*h*and

_{x}*h*controls the spacing between B-spline functions [22

_{y}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]

**15(22)**, 14817–14837 (2007). [CrossRef]

*α*

_{1},

*α*

_{2}are unknown. Even if

*α*

_{1},

*α*

_{2}are known, the solution requires a binary optimization since the value of

*f*(

*x*,

_{i}*y*) should be determined by either

_{j}*α*

_{1}or

*α*

_{2}on every (

*x*,

_{i}*y*),

_{j}*i*=1,…,

*N*,

*j*=1,…,,

*M*. Note that the binary optimization is challenging since the constraint is not a convex set [2]. In addition, the problem is ill-posed in the sense that the minimizer is not unique. If

*f*(

*x*,

_{i}*y*) consisted of two intensity values

_{j}*α*

_{1}and

*α*

_{2}and

*h*(

*x*,

_{i}*y*;

_{j}**b**) are a minimizer of the problem defined in Eq. (5), then, for any arbitrary constant

*c*>0, two intensity values

*cα*

_{1}and

*cα*

_{2}and

*h*(

*x*,

_{i}*y*;

_{j}*c*

**b**) would be also a minimizer. To overcome this difficulty, we solve a minimization problem that includes only one intensity value as follows:

**b**and

*α*based on the minimization problem defined in Eq. (6) is equivalent to the following:

*a*,

*b*} denotes the minimum of a and b. We approximate the argument minimization problem defined in Eq. (7) by the argument minimization problem defined in Eq. (8) under low now noise assumption. We argue the validity of this approximation under low noise assumption using the similar argument developed previously [23

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

*h*(

*x*,

_{i}*y*;

_{j}**b**̂) to

*g*(

*x*,

_{i}*y*),

_{j}*i.e.*, multiplying the estimated inverse of illumination to the acquired image. We further incorporate the important

*a priori*information that the illumination is slowly varying in the spatial domain by a roughness penalty function. The proposed method with the roughness penalty function is defined as an estimation of (

*PQ*+1)×1 parameter vector

*θ*=[

**b**,

*α*]

*by nonlinear least squares as follows:*

^{T}*R*(

*θ*) is a roughness penalty function that measures the roughness of the illumination and

*λ*>0 is a regularization parameter. One possible choice of the penalty function can be defined as follows:

_{r}**R**denotes the matrix representation of the roughness penalty function.

*h*(

*x*,

_{i}*y*;

_{j}**b**)2 term in the objective function prevents the illumination from being zero. In addition, it does not require the tuning of design parameters except for one regularization parameter, which is also required in the existing methods.

## 2.3 Well-posedness of the proposed method

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]

*J*(

*θ*) is

*MN*×(

*PQ*+1) Jacobian matrix of 1D column-stacked version of

*f*(

*x*,

_{i}*y*;

_{j}*θ*). Each element of

*J*(

*θ*) is defined as follows:

*k*=(

*j*-1)

*M*+

*i*. From Eq. (12), unless all the elements of

**b**have the same value, the Hessian is positive definite since

*θ*

^{T}**R**

*θ*>0. Therefore, it is sufficient to show that the Hessian is still positive definite even for the case of

**b**=[

_{u}*b*,.,

_{u}*b*] such that

_{u}*h*(

*x*,

_{i}*y*;

_{j}**b**)=

_{u}*c*,

*i.e.*, uniform illumination in the spatial domain. In this case, the kth element of 1×

*MN*vector

*θ*(

^{T}J*θ*)

*is defined as follows:*

^{T}*k*=(

*j*-1)

*M*+

*i*. Note that unless

*θ*(

^{T}J*θ*)

*=0 for ∀(*

^{T}*x*,

_{i}*y*), the Hessian is positive definite. It is not difficult to show that the condition θ

_{j}^{T}

*J*(

*θ*)

*=0 for ∀(*

^{T}*x*,

_{i}*y*) can be rewritten as follows:

_{j}*g*(

*x*,

_{i}*y*) should be the solution of the quadratic equation. Therefore, the nonlinear least squares problem is well-posed for this case. On the other hand, even if the observation has only two different intensity values, Eq. (15) is satisfied only for the following case:

_{j}*g*(

*x*,

_{i}*y*) is a constant,

_{j}*i.e.*, a uniform image, the nonlinear least squares problem is well-posed and its solution is unique. Note that if the acquired image is a uniform image, separating illumination and the true image from the observed image is, by nature, not possible.

## 3. Results

### 3.1 Simulations

*h*=16 and

_{x}*h*=16) for Figs. 1(a) and 1(b), respectively. We empirically determined the number of B-spline functions (equivalently, the spacing between B-spline functions) considering the speed of optimization and accuracy. If the number of B-spline functions is big, then optimization becomes more complicated [2]. On the contrary, if the number of B-spline functions is too small, the B-spline based model may not be able to estimate the nonuniform illumination correctly. Figures 2(a) and 2(b) show the corrected images of Figs. 1(a) and 1(b), respectively, and Figs. 2(e) and 2(f) show the estimated inverse of the illumination for the two images. One can see that the estimated inverses of the illuminations have similar shapes to the inverses of the true illuminations shown in Figs. 1(e) and 1(f). Figures 2(c) and 2(d) show the results of binarization using the corrected images shown in Figs. 2(a) and 2(b). It is clear that the results are greatly improved compared with the results shown in Figs. 1(c) and 1(d).

_{y}*K*(·) is a Gaussian kernel density function with standard deviation

_{σ}*σ*. Note that there exist several design parameters including

*σ*, the minimum and the maximum of the range of the PMF, and the number of nonzero PMF entries

*n*, etc.

_{b}*σ*=0.2,

*λ*=1

_{m}*e*

^{-5},

*λ*=1

_{r}*e*

^{-3}). Due to ineffectively chosen design parameters, the corrected image shown in Fig. 3(a), its binarization shown in Fig. 3(c), and the estimated illumination field shown in Fig. 3(e) are significantly different from the true images and illumination. We observe a similar phenomenon for the image shown in Fig. 1(b) (

*σ*=0.01,

*λ*=1

_{m}*e*

^{-5},

*λ*=1

_{r}*e*

^{-3}). Neither the corrected image shown in Fig. 3(b) nor its binarization shown in Fig. 3(d) is close to the true image since the estimated inverse of the illumination shown in Fig. 3(f) is significantly different from the true field.

*z*(

_{k}*k*=1,⋯,

*n*),

_{b}*n*and

_{b}*λ*were determined by trial and error in a way such that the performance becomes the best. Note that it is challenging to find optimal design parameters in practice since the ground truth image is not known.

_{m}*σ*=0.1). For such a case, although the proposed method yielded more similar image to the true image, the BER of the proposed method was slightly higher than that of the entropy based method. We suspect that this is due to the nonlinear nature of binary thresholding.

### 3.2 Experiments

## 4. Discussion

**15(22)**, 14817–14837 (2007). [CrossRef]

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

## 5. Conclusion

## Appendix A

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

*ε*=

*h*(

*x*

_{i},y_{j};**b**)

*g*(

*x*)-

_{i},y_{j}*α*. Then the following quantity defined in Eq. (A.2) can be rewritten by Eq. (A.3):

*ε*≈0, since Eq. (A.3) can be approximated by Eq. (A.1), we have the following approximation:

*i.e*., low noise case.

## Acknowledgments

## References and links

1. | T. H. Li and K. S. Lii, “A joint estimation approach for two-tone image deblurring by blind deconvolution,” IEEE Trans. Image Process. |

2. | E. Y. Lam, “Blind bi-level image restoration with iterated quadratic programming,” IEEE Trans. Circ. Syst. |

3. | Y. Shen, E. Y. Lam, and N. Wong, “Binary image restoration by positive semidefinite programming,” Opt. Lett. |

4. | T. F. Chan, S. Esedoglu, and M. Nikolova, “Finding the global minimum for binary image restoration,” in |

5. | J. Kim and H. Lee, “Joint nonuniform illumination estimation and deblurring for bar code signals,” Opt. Express |

6. | M. S. Brown and Y. C. Tsoi, “Geometric and shading correction for images of printed materials using boundary,” IEEE Trans. Image Process. |

7. | D. A. Forsyth and J. Ponce, |

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. |

9. | Z. Hou, “A review on MR image inhomogeneity correction,” Int. J. Biomed. Imaging |

10. | B. H. Brinkmann, A. Manduca, and R. A. Robb, “Optimized homomorphic unsharp masking for MR grayscale inhomogeneity correction,” IEEE Trans. Med. Imaging |

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 |

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. |

14. | B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuš, “Retrospective shading correction based on entropy minimization,” J. Microsc. |

15. | H.-L. Shen and K. Li, “Decomposition of shading and reflectance from a texture image,” Opt. Lett. |

16. | B. W. Silverman, |

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 |

18. | M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. |

19. | S. Esedoglu, “Blind deconvolution of bar code signals,” Inverse Probl. |

20. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

21. | A. Papoulis and S. U. Pillai, |

22. | J. Kybic, P. Thévenaz, A. Nirkko, and M. Unser, “Unwarping of unidirectionally distorted EPI images,” IEEE Trans. Med. Imaging |

23. | O. Grellier and P. Comon, “Blind separation of discrete sources,” IEEE Signal Process. Lett. |

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. |

25. | J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging |

26. | H. L. Van Trees, |

27. | J. Kim and J. A. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imaging |

28. | J. Zhang, Q. Zhang, and G. He, “Blind deconvolution: multiplicative iterative algorithm,” Opt. Lett. |

29. | T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A |

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 |

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. |

**OCIS Codes**

(100.3020) Image processing : Image reconstruction-restoration

(100.3190) Image processing : Inverse problems

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 30, 2009

Revised Manuscript: October 30, 2009

Manuscript Accepted: November 16, 2009

Published: December 15, 2009

**Citation**

Hana Lee and Jeongtae Kim, "Retrospective correction of nonuniform
illumination on bi-level images," Opt. Express **17**, 23880-23893 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23880

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### References

- 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).
- 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]
- 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.
- J. Kim and H. Lee, "Joint nonuniform illumination estimation and deblurring for bar code signals," Opt. Express 15(22), 14817-14837 (2007). [CrossRef]
- 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]
- D. A. Forsyth, and J. Ponce, Computer vision: A modern approach (Prentice Hall, 2003).
- 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]
- 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. F. Mangin, "Entropy minimization for automatic correction of intensity nonuniformity," IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA, 2000), pp. 162-169.
- D. Tomaževič, 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]
- H.-L. Shen, and K. Li, "Decomposition of shading and reflectance from a texture image," Opt. Lett. 34(1), 64-66 (2009). [CrossRef]
- B. W. Silverman, Density estimation for statistics and data analysis (Chapman and Hall, 1985).
- 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]
- M. Unser, "Splines: A perfect fit for signal and image processing," IEEE Signal Process. Mag. 16(6), 22-38 (1999). [CrossRef]
- S. Esedoglu, "Blind deconvolution of bar code signals," Inverse Probl. 20(1), 121-135 (2004). [CrossRef]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C + +, 2nd ed. (Cambridge, 2005).
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