Abstract

In this paper we propose an approach for handling noise in deconvolution algorithm based on multidirectional filters. Most image deconvolution techniques are sensitive to the noise. Even a small amount of noise will degrade the quality of image estimation dramatically. We found that by applying a directional low-pass filter to the blurred image, we can reduce the noise level while preserving the blur information in the orthogonal direction to the filter. So we apply a series of directional filters at different orientations to the blurred image, and a guided filter based edge-preserving image deconvolution is used to estimate an accurate Radon transform of the clear image from each filtered image. Finally, we reconstruct the original image using the inverse Radon transform. We compare our deconvolution algorithm with many competitive deconvolution techniques in terms of the improvement in signal-to-noise ratio and visual quality.

© 2013 Optical Society of America

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  1. A. K. Jain, Fundamental of Digital Image Processing (Prentice-Hall, 1989), pp. 267–341.
  2. A. K. Katsaggelos, ed., Digital Image Restoration (Springer, 1991), pp. 24–41.
  3. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
  4. R. Neelamani, H. Choi, and R. G. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418–433 (2004).
    [CrossRef]
  5. J. Starck, M. K. Nguyen, and F. Murtagh, “Wavelets and curvelets for image deconvolution: a combined approach,” Signal Process. 83, 2279–2283 (2003).
    [CrossRef]
  6. V. M. Patel, G. R. Easley, and D. M. Healy, “Shearlet-based deconvolution,” IEEE Trans. Image Process. 18, 2673–2685 (2009).
    [CrossRef]
  7. H. Yang and Z. Zhang, “Fusion of wave atom-based wiener shrinkage filter and joint non-local means filter for texture-preserving image deconvolution” Opt. Eng. 51, 067009 (2012).
    [CrossRef]
  8. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
    [CrossRef]
  9. T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Handbook of Mathematical Models in Computer Vision (Springer, 2005).
  10. Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imag. Sci. 1, 248–272 (2008).
    [CrossRef]
  11. J. Oliveira, J. M. Bioucas-Dias, and M. A. Figueiredo, “Adaptive total variation image deblurring: a majorization-minimization approach,” Signal Process. 89, 1683–1693 (2009).
    [CrossRef]
  12. O. V. Michailovich, “An iterative shrinkage approach to total-variation image restoration,” IEEE Trans. Image Process. 20, 1281–1299 (2011).
    [CrossRef]
  13. I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457 (2004).
    [CrossRef]
  14. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imag. Sci. 2, 183–202 (2009).
    [CrossRef]
  15. J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
    [CrossRef]
  16. J. A. Guerrero-Colon, L. Mancera, and J. Portilla, “Image restoration using space-variant Gaussian scale mixtures in overcomplete pyramids,” IEEE Trans. Image Process. 17, 27–41 (2008).
    [CrossRef]
  17. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
    [CrossRef]
  18. K. Dabove, A. Foi, V. Katkovnik, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).
    [CrossRef]
  19. J. Mairal, M. Elad, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
    [CrossRef]
  20. R. Rubinstein, A. M. Bruckstein, and M. Elad, “Dictionaries for sparse representation modeling,” Proc. IEEE 98, 1045–1057 (2010).
    [CrossRef]
  21. L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Progressive inter-scale and intra-scale non-blind image deconvolution,” ACM Trans. Graph. 27, 74 (2008).
    [CrossRef]
  22. J. Portilla, “Image restoration through l0 analysis-based sparse optimization in tight frames,” in Proceedings of the International Conference on Image Processing (IEEE, 2009), pp. 3909–3912.
  23. Z. Ming, Z. Wei, and W. g. Zhile, “Satellite image deconvolution based on nonlocal means,” Appl. Opt. 49, 6286–6294 (2010).
    [CrossRef]
  24. J. Ni, P. Turaga, V. M. Patel, and R. Chellappa, “Example-driven manifold priors for image deconvolution,” IEEE Trans. Image Process. 20, 3086–3096 (2011).
    [CrossRef]
  25. F. Xue, F. Luisier, and T. Blu, “Multi-Wiener SURE-LET deconvolution,” IEEE Trans. Image Process. 22, 1954–1968 (2013).
    [CrossRef]
  26. H. Yang, M. Zhu, Z. Zhang, and H. Huang, “Guided filter based edge-preserving image non-blind deconvolution,” in Proceedings of 20th IEEE International Conference on Image Processing (IEEE, 2013).
  27. P. Toft, “The Radon transform—theory and implementation,” Ph.D. thesis (Technical University of Denmark, 1996).
  28. K. He, J. Sun, and X. Tang, “Guided image filtering,” in Proceedings of the 11th European Conference on Computer Vision: Part I, Heidelberg (2010), pp. 1–14.

2013 (1)

F. Xue, F. Luisier, and T. Blu, “Multi-Wiener SURE-LET deconvolution,” IEEE Trans. Image Process. 22, 1954–1968 (2013).
[CrossRef]

2012 (1)

H. Yang and Z. Zhang, “Fusion of wave atom-based wiener shrinkage filter and joint non-local means filter for texture-preserving image deconvolution” Opt. Eng. 51, 067009 (2012).
[CrossRef]

2011 (2)

O. V. Michailovich, “An iterative shrinkage approach to total-variation image restoration,” IEEE Trans. Image Process. 20, 1281–1299 (2011).
[CrossRef]

J. Ni, P. Turaga, V. M. Patel, and R. Chellappa, “Example-driven manifold priors for image deconvolution,” IEEE Trans. Image Process. 20, 3086–3096 (2011).
[CrossRef]

2010 (2)

Z. Ming, Z. Wei, and W. g. Zhile, “Satellite image deconvolution based on nonlocal means,” Appl. Opt. 49, 6286–6294 (2010).
[CrossRef]

R. Rubinstein, A. M. Bruckstein, and M. Elad, “Dictionaries for sparse representation modeling,” Proc. IEEE 98, 1045–1057 (2010).
[CrossRef]

2009 (3)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imag. Sci. 2, 183–202 (2009).
[CrossRef]

V. M. Patel, G. R. Easley, and D. M. Healy, “Shearlet-based deconvolution,” IEEE Trans. Image Process. 18, 2673–2685 (2009).
[CrossRef]

J. Oliveira, J. M. Bioucas-Dias, and M. A. Figueiredo, “Adaptive total variation image deblurring: a majorization-minimization approach,” Signal Process. 89, 1683–1693 (2009).
[CrossRef]

2008 (5)

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imag. Sci. 1, 248–272 (2008).
[CrossRef]

J. A. Guerrero-Colon, L. Mancera, and J. Portilla, “Image restoration using space-variant Gaussian scale mixtures in overcomplete pyramids,” IEEE Trans. Image Process. 17, 27–41 (2008).
[CrossRef]

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Progressive inter-scale and intra-scale non-blind image deconvolution,” ACM Trans. Graph. 27, 74 (2008).
[CrossRef]

K. Dabove, A. Foi, V. Katkovnik, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).
[CrossRef]

J. Mairal, M. Elad, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

2007 (2)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

2004 (2)

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457 (2004).
[CrossRef]

R. Neelamani, H. Choi, and R. G. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418–433 (2004).
[CrossRef]

2003 (1)

J. Starck, M. K. Nguyen, and F. Murtagh, “Wavelets and curvelets for image deconvolution: a combined approach,” Signal Process. 83, 2279–2283 (2003).
[CrossRef]

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Baraniuk, R. G.

R. Neelamani, H. Choi, and R. G. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418–433 (2004).
[CrossRef]

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imag. Sci. 2, 183–202 (2009).
[CrossRef]

Bioucas-Dias, J.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

Bioucas-Dias, J. M.

J. Oliveira, J. M. Bioucas-Dias, and M. A. Figueiredo, “Adaptive total variation image deblurring: a majorization-minimization approach,” Signal Process. 89, 1683–1693 (2009).
[CrossRef]

Blu, T.

F. Xue, F. Luisier, and T. Blu, “Multi-Wiener SURE-LET deconvolution,” IEEE Trans. Image Process. 22, 1954–1968 (2013).
[CrossRef]

Bruckstein, A. M.

R. Rubinstein, A. M. Bruckstein, and M. Elad, “Dictionaries for sparse representation modeling,” Proc. IEEE 98, 1045–1057 (2010).
[CrossRef]

Chan, T.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Handbook of Mathematical Models in Computer Vision (Springer, 2005).

Chellappa, R.

J. Ni, P. Turaga, V. M. Patel, and R. Chellappa, “Example-driven manifold priors for image deconvolution,” IEEE Trans. Image Process. 20, 3086–3096 (2011).
[CrossRef]

Choi, H.

R. Neelamani, H. Choi, and R. G. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418–433 (2004).
[CrossRef]

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

Dabove, K.

K. Dabove, A. Foi, V. Katkovnik, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).
[CrossRef]

Daubechies, I.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457 (2004).
[CrossRef]

De Mol, C.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457 (2004).
[CrossRef]

Defrise, M.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457 (2004).
[CrossRef]

Easley, G. R.

V. M. Patel, G. R. Easley, and D. M. Healy, “Shearlet-based deconvolution,” IEEE Trans. Image Process. 18, 2673–2685 (2009).
[CrossRef]

Egiazarian, K.

K. Dabove, A. Foi, V. Katkovnik, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).
[CrossRef]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

Elad, M.

R. Rubinstein, A. M. Bruckstein, and M. Elad, “Dictionaries for sparse representation modeling,” Proc. IEEE 98, 1045–1057 (2010).
[CrossRef]

J. Mairal, M. Elad, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

Esedoglu, S.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Handbook of Mathematical Models in Computer Vision (Springer, 2005).

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Figueiredo, M.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

Figueiredo, M. A.

J. Oliveira, J. M. Bioucas-Dias, and M. A. Figueiredo, “Adaptive total variation image deblurring: a majorization-minimization approach,” Signal Process. 89, 1683–1693 (2009).
[CrossRef]

Foi, A.

K. Dabove, A. Foi, V. Katkovnik, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).
[CrossRef]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

Guerrero-Colon, J. A.

J. A. Guerrero-Colon, L. Mancera, and J. Portilla, “Image restoration using space-variant Gaussian scale mixtures in overcomplete pyramids,” IEEE Trans. Image Process. 17, 27–41 (2008).
[CrossRef]

Hansen, P. C.

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).

He, K.

K. He, J. Sun, and X. Tang, “Guided image filtering,” in Proceedings of the 11th European Conference on Computer Vision: Part I, Heidelberg (2010), pp. 1–14.

Healy, D. M.

V. M. Patel, G. R. Easley, and D. M. Healy, “Shearlet-based deconvolution,” IEEE Trans. Image Process. 18, 2673–2685 (2009).
[CrossRef]

Huang, H.

H. Yang, M. Zhu, Z. Zhang, and H. Huang, “Guided filter based edge-preserving image non-blind deconvolution,” in Proceedings of 20th IEEE International Conference on Image Processing (IEEE, 2013).

Jain, A. K.

A. K. Jain, Fundamental of Digital Image Processing (Prentice-Hall, 1989), pp. 267–341.

Katkovnik, V.

K. Dabove, A. Foi, V. Katkovnik, and K. Egiazarian, “Image restoration by sparse 3D transform-domain collaborative filtering,” Proc. SPIE 6812, 681207 (2008).
[CrossRef]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

Luisier, F.

F. Xue, F. Luisier, and T. Blu, “Multi-Wiener SURE-LET deconvolution,” IEEE Trans. Image Process. 22, 1954–1968 (2013).
[CrossRef]

Mairal, J.

J. Mairal, M. Elad, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

Mancera, L.

J. A. Guerrero-Colon, L. Mancera, and J. Portilla, “Image restoration using space-variant Gaussian scale mixtures in overcomplete pyramids,” IEEE Trans. Image Process. 17, 27–41 (2008).
[CrossRef]

Michailovich, O. V.

O. V. Michailovich, “An iterative shrinkage approach to total-variation image restoration,” IEEE Trans. Image Process. 20, 1281–1299 (2011).
[CrossRef]

Ming, Z.

Murtagh, F.

J. Starck, M. K. Nguyen, and F. Murtagh, “Wavelets and curvelets for image deconvolution: a combined approach,” Signal Process. 83, 2279–2283 (2003).
[CrossRef]

Neelamani, R.

R. Neelamani, H. Choi, and R. G. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418–433 (2004).
[CrossRef]

Nguyen, M. K.

J. Starck, M. K. Nguyen, and F. Murtagh, “Wavelets and curvelets for image deconvolution: a combined approach,” Signal Process. 83, 2279–2283 (2003).
[CrossRef]

Ni, J.

J. Ni, P. Turaga, V. M. Patel, and R. Chellappa, “Example-driven manifold priors for image deconvolution,” IEEE Trans. Image Process. 20, 3086–3096 (2011).
[CrossRef]

Oliveira, J.

J. Oliveira, J. M. Bioucas-Dias, and M. A. Figueiredo, “Adaptive total variation image deblurring: a majorization-minimization approach,” Signal Process. 89, 1683–1693 (2009).
[CrossRef]

Osher, S.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Park, F.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Handbook of Mathematical Models in Computer Vision (Springer, 2005).

Patel, V. M.

J. Ni, P. Turaga, V. M. Patel, and R. Chellappa, “Example-driven manifold priors for image deconvolution,” IEEE Trans. Image Process. 20, 3086–3096 (2011).
[CrossRef]

V. M. Patel, G. R. Easley, and D. M. Healy, “Shearlet-based deconvolution,” IEEE Trans. Image Process. 18, 2673–2685 (2009).
[CrossRef]

Portilla, J.

J. A. Guerrero-Colon, L. Mancera, and J. Portilla, “Image restoration using space-variant Gaussian scale mixtures in overcomplete pyramids,” IEEE Trans. Image Process. 17, 27–41 (2008).
[CrossRef]

J. Portilla, “Image restoration through l0 analysis-based sparse optimization in tight frames,” in Proceedings of the International Conference on Image Processing (IEEE, 2009), pp. 3909–3912.

Quan, L.

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Progressive inter-scale and intra-scale non-blind image deconvolution,” ACM Trans. Graph. 27, 74 (2008).
[CrossRef]

Rubinstein, R.

R. Rubinstein, A. M. Bruckstein, and M. Elad, “Dictionaries for sparse representation modeling,” Proc. IEEE 98, 1045–1057 (2010).
[CrossRef]

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Sapiro, G.

J. Mairal, M. Elad, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

Shum, H.-Y.

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Progressive inter-scale and intra-scale non-blind image deconvolution,” ACM Trans. Graph. 27, 74 (2008).
[CrossRef]

Starck, J.

J. Starck, M. K. Nguyen, and F. Murtagh, “Wavelets and curvelets for image deconvolution: a combined approach,” Signal Process. 83, 2279–2283 (2003).
[CrossRef]

Sun, J.

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Progressive inter-scale and intra-scale non-blind image deconvolution,” ACM Trans. Graph. 27, 74 (2008).
[CrossRef]

K. He, J. Sun, and X. Tang, “Guided image filtering,” in Proceedings of the 11th European Conference on Computer Vision: Part I, Heidelberg (2010), pp. 1–14.

Tang, X.

K. He, J. Sun, and X. Tang, “Guided image filtering,” in Proceedings of the 11th European Conference on Computer Vision: Part I, Heidelberg (2010), pp. 1–14.

Teboulle, M.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imag. Sci. 2, 183–202 (2009).
[CrossRef]

Toft, P.

P. Toft, “The Radon transform—theory and implementation,” Ph.D. thesis (Technical University of Denmark, 1996).

Turaga, P.

J. Ni, P. Turaga, V. M. Patel, and R. Chellappa, “Example-driven manifold priors for image deconvolution,” IEEE Trans. Image Process. 20, 3086–3096 (2011).
[CrossRef]

Wang, Y.

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imag. Sci. 1, 248–272 (2008).
[CrossRef]

Wei, Z.

Xue, F.

F. Xue, F. Luisier, and T. Blu, “Multi-Wiener SURE-LET deconvolution,” IEEE Trans. Image Process. 22, 1954–1968 (2013).
[CrossRef]

Yang, H.

H. Yang and Z. Zhang, “Fusion of wave atom-based wiener shrinkage filter and joint non-local means filter for texture-preserving image deconvolution” Opt. Eng. 51, 067009 (2012).
[CrossRef]

H. Yang, M. Zhu, Z. Zhang, and H. Huang, “Guided filter based edge-preserving image non-blind deconvolution,” in Proceedings of 20th IEEE International Conference on Image Processing (IEEE, 2013).

Yang, J.

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imag. Sci. 1, 248–272 (2008).
[CrossRef]

Yin, W.

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imag. Sci. 1, 248–272 (2008).
[CrossRef]

Yip, A.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Handbook of Mathematical Models in Computer Vision (Springer, 2005).

Yuan, L.

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Progressive inter-scale and intra-scale non-blind image deconvolution,” ACM Trans. Graph. 27, 74 (2008).
[CrossRef]

Zhang, Y.

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imag. Sci. 1, 248–272 (2008).
[CrossRef]

Zhang, Z.

H. Yang and Z. Zhang, “Fusion of wave atom-based wiener shrinkage filter and joint non-local means filter for texture-preserving image deconvolution” Opt. Eng. 51, 067009 (2012).
[CrossRef]

H. Yang, M. Zhu, Z. Zhang, and H. Huang, “Guided filter based edge-preserving image non-blind deconvolution,” in Proceedings of 20th IEEE International Conference on Image Processing (IEEE, 2013).

Zhile, W. g.

Zhu, M.

H. Yang, M. Zhu, Z. Zhang, and H. Huang, “Guided filter based edge-preserving image non-blind deconvolution,” in Proceedings of 20th IEEE International Conference on Image Processing (IEEE, 2013).

ACM Trans. Graph. (1)

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Progressive inter-scale and intra-scale non-blind image deconvolution,” ACM Trans. Graph. 27, 74 (2008).
[CrossRef]

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457 (2004).
[CrossRef]

IEEE Trans. Image Process. (8)

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

J. A. Guerrero-Colon, L. Mancera, and J. Portilla, “Image restoration using space-variant Gaussian scale mixtures in overcomplete pyramids,” IEEE Trans. Image Process. 17, 27–41 (2008).
[CrossRef]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

V. M. Patel, G. R. Easley, and D. M. Healy, “Shearlet-based deconvolution,” IEEE Trans. Image Process. 18, 2673–2685 (2009).
[CrossRef]

J. Ni, P. Turaga, V. M. Patel, and R. Chellappa, “Example-driven manifold priors for image deconvolution,” IEEE Trans. Image Process. 20, 3086–3096 (2011).
[CrossRef]

F. Xue, F. Luisier, and T. Blu, “Multi-Wiener SURE-LET deconvolution,” IEEE Trans. Image Process. 22, 1954–1968 (2013).
[CrossRef]

J. Mairal, M. Elad, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

O. V. Michailovich, “An iterative shrinkage approach to total-variation image restoration,” IEEE Trans. Image Process. 20, 1281–1299 (2011).
[CrossRef]

IEEE Trans. Signal Process. (1)

R. Neelamani, H. Choi, and R. G. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418–433 (2004).
[CrossRef]

Opt. Eng. (1)

H. Yang and Z. Zhang, “Fusion of wave atom-based wiener shrinkage filter and joint non-local means filter for texture-preserving image deconvolution” Opt. Eng. 51, 067009 (2012).
[CrossRef]

Physica D (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Proc. IEEE (1)

R. Rubinstein, A. M. Bruckstein, and M. Elad, “Dictionaries for sparse representation modeling,” Proc. IEEE 98, 1045–1057 (2010).
[CrossRef]

Proc. SPIE (1)

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Figures (7)

Fig. 1.
Fig. 1.

Directional filter and the Radon transform. (a) Original image, (b) Radon transform of original image along the θ=3π/4, (c) filtered image by apply the directional filter (θ=π/4), and (d) Radon transform of filtered image along the θ=3π/4. Note that the Radon transform of the filtered image is the same as the original unfiltered image.

Fig. 2.
Fig. 2.

Illustration of applying directional filters for image deblurring. (a) Blurred noisy image. (b) We apply directional filters in different orientations to the blurred noisy image. (c) From each filtered image, a deblurred image is computed first, then (d) projected along the orthogonal direction to generate the correct Radon transform of the true image. (e) Final true image u is reconstructed using inverse Radon transform.

Fig. 3.
Fig. 3.

Images used in this paper for different experiments. (a) Cameraman image, (b) Lena image, (c) house image, and (d) boat image.

Fig. 4.
Fig. 4.

Details of the image deconvolution experiment with a Cameraman image. (a) Original image, (b) blurred image, (c) ForWaRD result, ISNR=4.88dB, (d) TVS result, ISNR=5.72dB, (e) L0-AbS result, ISNR=5.45dB, and (f) our result, ISNR=6.16dB.

Fig. 5.
Fig. 5.

Details of the image deconvolution experiment with a Lena image. (a) Original image, (b) blurred image, (c) TVS result, ISNR=4.12dB, (d) L0-AbS result, ISNR=4.42dB, (e) SURE-LET result, ISNR=4.80dB, and (f) our result, ISNR=5.01dB.

Fig. 6.
Fig. 6.

Details of the image deconvolution experiment with a House image. (a) Original image, (b) blurred image, (c) TVS result, ISNR=4.51dB, (d) L0-AbS result, ISNR=4.55dB, (e) SURE-LET result, ISNR=4.02dB, and (f) Our result, ISNR=4.99dB.

Fig. 7.
Fig. 7.

Visual comparison of Boat image in Exp. 6. (a) Crop from Boat image, (b) blurred image, (c) ForWaRD result, ISNR=4.75dB, (d) L0-AbS result, ISNR=5.25dB, (e) SURE-LET result, ISNR=5.41dB, and (f) our result, ISNR=5.61dB.

Tables (3)

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Algorithm 1 Noise-aware Image Deconvolution Algorithm

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Table 1. Description of the Observation Parameters for the Six Experiments

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Table 2. ISNR for Different Experiments

Equations (18)

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g(n1,n2)=Huorig(n1,n2)+γ(n1,n2)=(h*uorig)(n1,n2)+γ(n1,n2),
G(k1,k2)=H(k1,k2)·Uorig(k1,k2)+Γ(k1,k2),
(I*lθ)(n1,n2)=1W+w(t)I(n1+tcosθ,n2+tsinθ)dt,
gθ=g*lθ=h*u*lθ+γ*lθ,
Ru(θ,ρ)=u(x,y)δ(ρxcosθysinθ)dxdy,
Ruθ(θ+π2,ρ)=Ru*lθ(θ+π2,ρ).
Rf*g(θ,ρ)=(Rf(θ,·)*Rg(θ,·))(ρ),
Ruθ(θ+π2,ρ)=(Ru(θ+π2,·)*Rlθ(θ+π2,·))(ρ).
Rlθ(θ+π2,ρ)=lθ(x,y)δ(ρxcos(θ+π2)ysin(θ+π2))dxdy=1W+w(t)δ(ρ+tcosθsinθtsinθcosθ)dt=δ(ρ).
Ruθ(θ+π2,ρ)=Ru(θ+π2,ρ).
minuθgθh*uθ2+λuθGuid(pθ,uθ)2,
uθ=argminuθgθh*uθ2+λuθu¯θ2,
pθ=argminpθgθh*pθ2+λθpθθu¯θ2,
u¯θ=Guid(pθ,uθ).
F(uθ)=F(h)*·F(gθ)+λF(u¯θ)|F(h)|2+λ,
F(pθ)=F(h)*·F(gθ)+λ|F(θ)|2·F(u¯θ)|F(h)|2+λ|F(θ)|2,
|F(θ)|2=|(F(x),F(y))×Rθ|2=|F(x)cosθF(y)sinθ|2+|F(x)sinθ+F(y)cosθ|2,
ISNR=10log10(uorigg22uorigu^22),

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