Abstract

This paper proposes a new approach for blindly deconvolving images that are contaminated by Poisson noise. The proposed approach incorporates a new prior, that is the L0 sparse analysis prior, together with the total variation constraint into the maximum a posteriori (MAP) framework for deconvolution. A greedy analysis pursuit numerical scheme is exploited to solve the L0 regularized MAP problem. Experimental results show that our approach not only produces smooth results substantially suppressing artifacts and noise, but also preserves intensity changes sharply. Both quantitative and qualitative comparisons to the specialized state-of-the-art algorithms demonstrate its superiority.

© 2014 Optical Society of America

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  1. D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Proc. Mag. 11, 61–63 (1996).
    [CrossRef]
  2. P. Campisi, K. Egiazarian, Blind Image Deconvolution: Theory and Applications (CRC Press, 2007).
    [CrossRef]
  3. M. Demenikov, A. R. Harvey, “Parametric blind-deconvolution algorithm to remove image artifacts in hybrid imaging systems,” Opt. Express 18(17), 18035–18040 (2010).
    [CrossRef] [PubMed]
  4. S. V. Vorontsov, V. N. Strakhov, S. M. Jefferies, K. J. Borelli, “Deconvolution of astronomical images using SOR with adaptive relaxation,” Opt. Express 19(14), 13509–13524 (2011)
    [CrossRef] [PubMed]
  5. F. Dupe, J. M. Fadili, J. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18(2), 310–321 (2009).
    [CrossRef] [PubMed]
  6. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  7. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  8. D. A. Fish, A. M. Brinicombe, E. R. Pike, J. G. Walker, “Blind deconvolution by means of the richardson-lucy algorithm,” J. Opt. Soc. Am. 12, 58–65 (1995).
    [CrossRef]
  9. Z. Xu, E. Lam, “Maximum a posteriori blind image deconvolution with huber-Markov random-field regularization,” Opt. Lett. 34, 1453–1455 (2009).
    [CrossRef] [PubMed]
  10. L. Yan, H. Fang, S. Zhong, “Blind image deconvolution with spatially adaptive total variation regularization,” Opt. Lett. 37, 2778–2780 (2012).
    [CrossRef] [PubMed]
  11. S. Nam, M. E. Davies, M. Elad, R. Gribonval, “The cosparse analysis model and algorithms,” Appl. Comput. Harmon. A. 34, 30–56 (2013).
    [CrossRef]
  12. J.-L. Starck, E. J. Candes, D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
    [CrossRef]
  13. J. Cai, H. Ji, C. Liu, Z. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process. 21, 562–572 (2012).
    [CrossRef]
  14. L. Xu, S. Zheng, J. Jia, “Unnatural l0 sparse representation for natural image deblurring,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, 1107–1114 (2013).
  15. http://www.imageprocessingplace.com/root_files_V3/image_databases.htm
  16. H. Fang, L. Yan, H. Liu, Y. Chang, “Blind poissonian images deconvolution with framelet regularization,” Opt. Lett. 38, 389–391 (2013).
    [CrossRef] [PubMed]
  17. T. Chan, C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370–375 (1998).
    [CrossRef]

2013 (2)

S. Nam, M. E. Davies, M. Elad, R. Gribonval, “The cosparse analysis model and algorithms,” Appl. Comput. Harmon. A. 34, 30–56 (2013).
[CrossRef]

H. Fang, L. Yan, H. Liu, Y. Chang, “Blind poissonian images deconvolution with framelet regularization,” Opt. Lett. 38, 389–391 (2013).
[CrossRef] [PubMed]

2012 (2)

J. Cai, H. Ji, C. Liu, Z. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process. 21, 562–572 (2012).
[CrossRef]

L. Yan, H. Fang, S. Zhong, “Blind image deconvolution with spatially adaptive total variation regularization,” Opt. Lett. 37, 2778–2780 (2012).
[CrossRef] [PubMed]

2011 (1)

2010 (1)

2009 (2)

F. Dupe, J. M. Fadili, J. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18(2), 310–321 (2009).
[CrossRef] [PubMed]

Z. Xu, E. Lam, “Maximum a posteriori blind image deconvolution with huber-Markov random-field regularization,” Opt. Lett. 34, 1453–1455 (2009).
[CrossRef] [PubMed]

2002 (1)

J.-L. Starck, E. J. Candes, D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

1998 (1)

T. Chan, C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370–375 (1998).
[CrossRef]

1996 (1)

D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Proc. Mag. 11, 61–63 (1996).
[CrossRef]

1995 (1)

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

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1972 (1)

Borelli, K. J.

Brinicombe, A. M.

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

Cai, J.

J. Cai, H. Ji, C. Liu, Z. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process. 21, 562–572 (2012).
[CrossRef]

Campisi, P.

P. Campisi, K. Egiazarian, Blind Image Deconvolution: Theory and Applications (CRC Press, 2007).
[CrossRef]

Candes, E. J.

J.-L. Starck, E. J. Candes, D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

Chan, T.

T. Chan, C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370–375 (1998).
[CrossRef]

Chang, Y.

Davies, M. E.

S. Nam, M. E. Davies, M. Elad, R. Gribonval, “The cosparse analysis model and algorithms,” Appl. Comput. Harmon. A. 34, 30–56 (2013).
[CrossRef]

Demenikov, M.

Donoho, D. L.

J.-L. Starck, E. J. Candes, D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

Dupe, F.

F. Dupe, J. M. Fadili, J. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18(2), 310–321 (2009).
[CrossRef] [PubMed]

Egiazarian, K.

P. Campisi, K. Egiazarian, Blind Image Deconvolution: Theory and Applications (CRC Press, 2007).
[CrossRef]

Elad, M.

S. Nam, M. E. Davies, M. Elad, R. Gribonval, “The cosparse analysis model and algorithms,” Appl. Comput. Harmon. A. 34, 30–56 (2013).
[CrossRef]

Fadili, J. M.

F. Dupe, J. M. Fadili, J. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18(2), 310–321 (2009).
[CrossRef] [PubMed]

Fang, H.

Fish, D. A.

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

Gribonval, R.

S. Nam, M. E. Davies, M. Elad, R. Gribonval, “The cosparse analysis model and algorithms,” Appl. Comput. Harmon. A. 34, 30–56 (2013).
[CrossRef]

Harvey, A. R.

Hatzinakos, D.

D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Proc. Mag. 11, 61–63 (1996).
[CrossRef]

Jefferies, S. M.

Ji, H.

J. Cai, H. Ji, C. Liu, Z. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process. 21, 562–572 (2012).
[CrossRef]

Jia, J.

L. Xu, S. Zheng, J. Jia, “Unnatural l0 sparse representation for natural image deblurring,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, 1107–1114 (2013).

Kundur, D.

D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Proc. Mag. 11, 61–63 (1996).
[CrossRef]

Lam, E.

Liu, C.

J. Cai, H. Ji, C. Liu, Z. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process. 21, 562–572 (2012).
[CrossRef]

Liu, H.

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Nam, S.

S. Nam, M. E. Davies, M. Elad, R. Gribonval, “The cosparse analysis model and algorithms,” Appl. Comput. Harmon. A. 34, 30–56 (2013).
[CrossRef]

Pike, E. R.

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

Richardson, W. H.

Shen, Z.

J. Cai, H. Ji, C. Liu, Z. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process. 21, 562–572 (2012).
[CrossRef]

Starck, J.

F. Dupe, J. M. Fadili, J. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18(2), 310–321 (2009).
[CrossRef] [PubMed]

Starck, J.-L.

J.-L. Starck, E. J. Candes, D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

Strakhov, V. N.

Vorontsov, S. V.

Walker, J. G.

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

Wong, C.

T. Chan, C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370–375 (1998).
[CrossRef]

Xu, L.

L. Xu, S. Zheng, J. Jia, “Unnatural l0 sparse representation for natural image deblurring,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, 1107–1114 (2013).

Xu, Z.

Yan, L.

Zheng, S.

L. Xu, S. Zheng, J. Jia, “Unnatural l0 sparse representation for natural image deblurring,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, 1107–1114 (2013).

Zhong, S.

Appl. Comput. Harmon. A. (1)

S. Nam, M. E. Davies, M. Elad, R. Gribonval, “The cosparse analysis model and algorithms,” Appl. Comput. Harmon. A. 34, 30–56 (2013).
[CrossRef]

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

IEEE Signal Proc. Mag. (1)

D. Kundur, D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Proc. Mag. 11, 61–63 (1996).
[CrossRef]

IEEE Trans. Image Process. (4)

J.-L. Starck, E. J. Candes, D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

J. Cai, H. Ji, C. Liu, Z. Shen, “Framelet-based blind motion deblurring from a single image,” IEEE Trans. Image Process. 21, 562–572 (2012).
[CrossRef]

F. Dupe, J. M. Fadili, J. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18(2), 310–321 (2009).
[CrossRef] [PubMed]

T. Chan, C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370–375 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (3)

Other (3)

L. Xu, S. Zheng, J. Jia, “Unnatural l0 sparse representation for natural image deblurring,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, 1107–1114 (2013).

http://www.imageprocessingplace.com/root_files_V3/image_databases.htm

P. Campisi, K. Egiazarian, Blind Image Deconvolution: Theory and Applications (CRC Press, 2007).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Restoration of the degraded images. (a) is the degraded image. (b), (c), (d) are the results restored by the RLTV, BPIDFR, and TVL0 approaches respectively. From top to bottom, the scenarios are Cameraman, satellite, brain, and nebula.

Fig. 2
Fig. 2

Restoration of the degraded images. (a) is the degraded image. (b), (c), (d) are the images restored by the RLTV, BPIDFR, and TVL0 approaches respectively. The top is the Saturn and the middle is neuron. The bottom shows a zoomed part of the neuron image.

Tables (2)

Tables Icon

Algorithm 1: GAP for Blind Image Deconvolution

Tables Icon

Table 1 NMSE of the degraded images and the restored images obtained by different algorithms. Here, TVL0 refers to our proposed approach.

Equations (10)

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

y = 𝒫 ( Hx ) = 𝒫 ( Xh ) ,
p ( y | x , h ) = i = 1 n ( Hx ) i y i exp ( Hx ) i y i ! .
p ( x ) exp ( α x T V ) exp ( β Ω x 0 ) ,
arg min { x , h } J ( x , H , y ) + α x T V + β Ω x 0 , subject to x 0
{ x ^ 0 , h ^ 0 } = arg min { x , h } J ( x , H , y ) + α x T V + β Ω Λ ^ 0 x 2 subject to x 0 .
arg min { x , h } J ( x , H , y ) + α x T V + β Ω Λ ^ 0 x 2 subject to x 0 .
arg min { x , h } J ( x , H , y ) + α x T V + β Ω Λ ^ 0 x 2 subject to x 0 .
h k + 1 = h k { X * y X h k } h k + 1 = h k + 1 h k + 1 2 ,
x k + 1 = x k 1 α div ( x x 2 ) + β Ω Λ T Ω Λ x { H * y H x k } , x k + 1 = max { x k + 1 , 0 }
NMSE = x ^ x 2 2 x 2 2 ,

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