Abstract

Fourier-based deconvolution (FoD) techniques, such as modulation transfer function compensation, are commonly employed in remote sensing. However, the noise is strongly amplified by FoD and is colored, thus producing poor visual quality. We propose an adaptive wavelet-based deconvolution algorithm for remote sensing called wavelet denoise after Laplacian-regularized deconvolution (WDALRD) to overcome the colored noise and to preserve the textures of the restored image. This algorithm adaptively denoises the FoD result on a wavelet basis. The term “adaptive” means that the wavelet-based denoising procedure requires no parameter to be estimated or empirically set, and thus the inhomogeneous Laplacian prior and the Jeffreys hyperprior are proposed. Maximum a posteriori estimation based on such a prior and hyperprior leads us to an adaptive and efficient nonlinear thresholding estimator, and therefore WDALRD is computationally inexpensive and fast. Experimentally, textures and edges of the restored image are well preserved and sharp, while the homogeneous regions remain noise free, so WDALRD gives satisfactory visual quality.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. K. Patra, N. Mishra, R. Chandrakanth, and R. Ramachandran, “Image quality improvement through MTF compensation: a treatment to high resolution data,” Indian Cartogr. 22, 86-93 (2002).
  2. T. Choi, “IKONOS satellite on orbit modulation transfer function (MTF) measurement using edge and pulse method,” Ph.D. dissertation (South Dakota State University, 2002), pp. 41-89.
  3. M. Y. Zhou, Deconvolution and Signal Recovery (National Defence Industry Press, 2004).
  4. R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139-194 (2005).
    [CrossRef]
  5. I. M. Johnstone and B. W. Silverman, “Wavelet threshold estimators for data with correlated noise,” J. R. Statist. Soc. Ser. B 59, 319-351 (1997).
  6. J. Kalifa and S. Mallat, “Thresholding estimators for linear inverse problems and deconvolutions,” Ann. Statist. 31, 58-109 (2003).
  7. S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).
  8. D. L. Donoho, “Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition,” Appl. Comput. Harmon. Anal. 2, 101-126 (1995).
    [CrossRef]
  9. J. Kalifa, S. Mallat, and B. Rouge, “Image deconvolution in mirror wavelet bases,” in Proceedings of the 1998 International Conference on Image Processing (IEEE, 1998), pp. 565-569.
  10. J. Kalifa, S. Mallat, and B. Rouge, “Deconvolution by thresholding in mirror wavelet bases,” IEEE Trans. Image Process. 12, 446-457 (2003).
    [CrossRef]
  11. J. Kalifa, S. Mallat, and B. Rouge, “Minimax solution of inverse problems and deconvolution by mirror wavelet thresholding,” Proc. SPIE 3813, 42-57 (1999).
    [CrossRef]
  12. A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Satellite image deconvolution using complex wavelet packets,” INRIA Research Report 3955 (Institut National de Recherche en Informatique et en Automatique, 2000), pp. 7-73.
  13. R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418-433 (2004).
    [CrossRef]
  14. S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
    [CrossRef]
  15. J. M. Bioucas-Dias, M. A. Figueiredo, and J. P. Oliveira, “Total variation-based image deconvolution: a majorization-minimization approach,” in 2006 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 861-864.
  16. J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992-3004 (2007).
    [CrossRef]
  17. M. A. Figueiredo, J. M. Bioucas-Dias, and R. D. Nowak, “Majorization-minimization algorithms for wavelet-based image restoration,” IEEE Trans. Image Process. 16, 2980-2991 (2007).
    [CrossRef]
  18. T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” in Mathematical Models of Computer Vision (Springer-Verlag, 2005).
  19. D. Krishnan, P. Lin, and A. M. Yip, “A primal-dual active-set method for nonnegativity constrained total variation deblurring problems,” IEEE Trans. Image Process. 16, 2766-2777(2007).
    [CrossRef]
  20. M. A. Figueiredo and R. D. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906-916 (2003).
    [CrossRef]
  21. J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417-1429 (1995).
    [CrossRef]
  22. H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).
  23. M. A. Figueiredo and R. D. Nowak, “Wavelet-based image estimation: an empirical Bayes approach using Jeffreys' noninformative prior,” IEEE Trans. Image Process. 10, 1322-1331 (2001).
    [CrossRef]
  24. P. Moulin and J. Liu, “Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors,” IEEE Trans. Inf. Theory 45, 909-919 (1999).
    [CrossRef]
  25. D. K. Hammond and E. P. Simoncelli, “Image modeling and denoising with orientation-adapted Gaussian scale mixtures,” IEEE Trans. Image Process. 17, 2089-2101 (2008).
    [CrossRef]
  26. J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. 12, 1338-1351(2003).
    [CrossRef]
  27. S. G. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9, 1532-1546 (2000).
    [CrossRef]
  28. T. Leonard and J. S. J. Hsu, Bayesian Methods: an Analysis for Statisticians and Interdisciplinary Researchers (China Machine Press, 2006).
  29. J. E. Fowler, “The redundant discrete wavelet transform and additive noise,” IEEE Signal Process. Lett. 12, 629-632 (2005).
    [CrossRef]
  30. G. P. Nason and B. W. Silverman, “The stationary wavelet transform and some statistical applications,” Ph.D. dissertation (University of Bristol, 1995), pp. 1-19.
  31. M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10-12 (1996).
    [CrossRef]
  32. J. M. Bioucas-Dias, “Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors,” IEEE Trans. Image Process. 15, 937-951 (2006).
    [CrossRef]
  33. A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613-621 (2004).
    [CrossRef]
  34. A. D. Hillery and R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892-1899(1991).
    [CrossRef]
  35. F. Aghdasi and R. K. Ward, “Reduction of boundary artifacts in image restoration,” IEEE Trans. Image Process. 5, 611-618(1996).
    [CrossRef]
  36. R. Liu and J. Jia, “Reducing boundary artifacts in image deconvolution,” in Proceedings of 15th IEEE International Conference on Image Processing (IEEE, 2008), pp. 505-508.

2008 (1)

D. K. Hammond and E. P. Simoncelli, “Image modeling and denoising with orientation-adapted Gaussian scale mixtures,” IEEE Trans. Image Process. 17, 2089-2101 (2008).
[CrossRef]

2007 (3)

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

M. A. Figueiredo, J. M. Bioucas-Dias, and R. D. Nowak, “Majorization-minimization algorithms for wavelet-based image restoration,” IEEE Trans. Image Process. 16, 2980-2991 (2007).
[CrossRef]

D. Krishnan, P. Lin, and A. M. Yip, “A primal-dual active-set method for nonnegativity constrained total variation deblurring problems,” IEEE Trans. Image Process. 16, 2766-2777(2007).
[CrossRef]

2006 (1)

J. M. Bioucas-Dias, “Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors,” IEEE Trans. Image Process. 15, 937-951 (2006).
[CrossRef]

2005 (2)

J. E. Fowler, “The redundant discrete wavelet transform and additive noise,” IEEE Signal Process. Lett. 12, 629-632 (2005).
[CrossRef]

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139-194 (2005).
[CrossRef]

2004 (2)

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

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613-621 (2004).
[CrossRef]

2003 (4)

J. Kalifa, S. Mallat, and B. Rouge, “Deconvolution by thresholding in mirror wavelet bases,” IEEE Trans. Image Process. 12, 446-457 (2003).
[CrossRef]

J. Kalifa and S. Mallat, “Thresholding estimators for linear inverse problems and deconvolutions,” Ann. Statist. 31, 58-109 (2003).

M. A. Figueiredo and R. D. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906-916 (2003).
[CrossRef]

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. 12, 1338-1351(2003).
[CrossRef]

2002 (1)

S. K. Patra, N. Mishra, R. Chandrakanth, and R. Ramachandran, “Image quality improvement through MTF compensation: a treatment to high resolution data,” Indian Cartogr. 22, 86-93 (2002).

2001 (1)

M. A. Figueiredo and R. D. Nowak, “Wavelet-based image estimation: an empirical Bayes approach using Jeffreys' noninformative prior,” IEEE Trans. Image Process. 10, 1322-1331 (2001).
[CrossRef]

2000 (1)

S. G. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9, 1532-1546 (2000).
[CrossRef]

1999 (2)

P. Moulin and J. Liu, “Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors,” IEEE Trans. Inf. Theory 45, 909-919 (1999).
[CrossRef]

J. Kalifa, S. Mallat, and B. Rouge, “Minimax solution of inverse problems and deconvolution by mirror wavelet thresholding,” Proc. SPIE 3813, 42-57 (1999).
[CrossRef]

1997 (2)

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

I. M. Johnstone and B. W. Silverman, “Wavelet threshold estimators for data with correlated noise,” J. R. Statist. Soc. Ser. B 59, 319-351 (1997).

1996 (2)

F. Aghdasi and R. K. Ward, “Reduction of boundary artifacts in image restoration,” IEEE Trans. Image Process. 5, 611-618(1996).
[CrossRef]

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10-12 (1996).
[CrossRef]

1995 (2)

D. L. Donoho, “Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition,” Appl. Comput. Harmon. Anal. 2, 101-126 (1995).
[CrossRef]

J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417-1429 (1995).
[CrossRef]

1991 (1)

A. D. Hillery and R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892-1899(1991).
[CrossRef]

Zhou, M. Y.

M. Y. Zhou, Deconvolution and Signal Recovery (National Defence Industry Press, 2004).

Aghdasi, F.

F. Aghdasi and R. K. Ward, “Reduction of boundary artifacts in image restoration,” IEEE Trans. Image Process. 5, 611-618(1996).
[CrossRef]

Andrews, H.

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Baraniuk, R.

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

Baraniuk, R. G.

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

Bioucas-Dias, J. M.

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

M. A. Figueiredo, J. M. Bioucas-Dias, and R. D. Nowak, “Majorization-minimization algorithms for wavelet-based image restoration,” IEEE Trans. Image Process. 16, 2980-2991 (2007).
[CrossRef]

J. M. Bioucas-Dias, “Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors,” IEEE Trans. Image Process. 15, 937-951 (2006).
[CrossRef]

J. M. Bioucas-Dias, M. A. Figueiredo, and J. P. Oliveira, “Total variation-based image deconvolution: a majorization-minimization approach,” in 2006 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 861-864.

Blanc-Féraud, L.

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613-621 (2004).
[CrossRef]

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Satellite image deconvolution using complex wavelet packets,” INRIA Research Report 3955 (Institut National de Recherche en Informatique et en Automatique, 2000), pp. 7-73.

Burrus, C. S.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10-12 (1996).
[CrossRef]

Chan, T.

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

Chandrakanth, R.

S. K. Patra, N. Mishra, R. Chandrakanth, and R. Ramachandran, “Image quality improvement through MTF compensation: a treatment to high resolution data,” Indian Cartogr. 22, 86-93 (2002).

Chang, S. G.

S. G. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9, 1532-1546 (2000).
[CrossRef]

Chin, R. T.

A. D. Hillery and R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892-1899(1991).
[CrossRef]

Choi, H.

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

Choi, T.

T. Choi, “IKONOS satellite on orbit modulation transfer function (MTF) measurement using edge and pulse method,” Ph.D. dissertation (South Dakota State University, 2002), pp. 41-89.

Donoho, D. L.

D. L. Donoho, “Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition,” Appl. Comput. Harmon. Anal. 2, 101-126 (1995).
[CrossRef]

Esedoglu, S.

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

Fessler, J.

J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417-1429 (1995).
[CrossRef]

Figueiredo, M. A.

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

M. A. Figueiredo, J. M. Bioucas-Dias, and R. D. Nowak, “Majorization-minimization algorithms for wavelet-based image restoration,” IEEE Trans. Image Process. 16, 2980-2991 (2007).
[CrossRef]

M. A. Figueiredo and R. D. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906-916 (2003).
[CrossRef]

M. A. Figueiredo and R. D. Nowak, “Wavelet-based image estimation: an empirical Bayes approach using Jeffreys' noninformative prior,” IEEE Trans. Image Process. 10, 1322-1331 (2001).
[CrossRef]

J. M. Bioucas-Dias, M. A. Figueiredo, and J. P. Oliveira, “Total variation-based image deconvolution: a majorization-minimization approach,” in 2006 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 861-864.

Fowler, J. E.

J. E. Fowler, “The redundant discrete wavelet transform and additive noise,” IEEE Signal Process. Lett. 12, 629-632 (2005).
[CrossRef]

Ghael, S. P.

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

Gosnell, T. R.

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139-194 (2005).
[CrossRef]

Guo, H.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10-12 (1996).
[CrossRef]

Hammond, D. K.

D. K. Hammond and E. P. Simoncelli, “Image modeling and denoising with orientation-adapted Gaussian scale mixtures,” IEEE Trans. Image Process. 17, 2089-2101 (2008).
[CrossRef]

Hero, A.

J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417-1429 (1995).
[CrossRef]

Hillery, A. D.

A. D. Hillery and R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892-1899(1991).
[CrossRef]

Hsu, J. S. J.

T. Leonard and J. S. J. Hsu, Bayesian Methods: an Analysis for Statisticians and Interdisciplinary Researchers (China Machine Press, 2006).

Hunt, B.

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Jalobeanu, A.

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613-621 (2004).
[CrossRef]

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Satellite image deconvolution using complex wavelet packets,” INRIA Research Report 3955 (Institut National de Recherche en Informatique et en Automatique, 2000), pp. 7-73.

Jia, J.

R. Liu and J. Jia, “Reducing boundary artifacts in image deconvolution,” in Proceedings of 15th IEEE International Conference on Image Processing (IEEE, 2008), pp. 505-508.

Johnstone, I. M.

I. M. Johnstone and B. W. Silverman, “Wavelet threshold estimators for data with correlated noise,” J. R. Statist. Soc. Ser. B 59, 319-351 (1997).

Kalifa, J.

J. Kalifa, S. Mallat, and B. Rouge, “Deconvolution by thresholding in mirror wavelet bases,” IEEE Trans. Image Process. 12, 446-457 (2003).
[CrossRef]

J. Kalifa and S. Mallat, “Thresholding estimators for linear inverse problems and deconvolutions,” Ann. Statist. 31, 58-109 (2003).

J. Kalifa, S. Mallat, and B. Rouge, “Minimax solution of inverse problems and deconvolution by mirror wavelet thresholding,” Proc. SPIE 3813, 42-57 (1999).
[CrossRef]

J. Kalifa, S. Mallat, and B. Rouge, “Image deconvolution in mirror wavelet bases,” in Proceedings of the 1998 International Conference on Image Processing (IEEE, 1998), pp. 565-569.

Krishnan, D.

D. Krishnan, P. Lin, and A. M. Yip, “A primal-dual active-set method for nonnegativity constrained total variation deblurring problems,” IEEE Trans. Image Process. 16, 2766-2777(2007).
[CrossRef]

Lang, M.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10-12 (1996).
[CrossRef]

Leonard, T.

T. Leonard and J. S. J. Hsu, Bayesian Methods: an Analysis for Statisticians and Interdisciplinary Researchers (China Machine Press, 2006).

Lin, P.

D. Krishnan, P. Lin, and A. M. Yip, “A primal-dual active-set method for nonnegativity constrained total variation deblurring problems,” IEEE Trans. Image Process. 16, 2766-2777(2007).
[CrossRef]

Liu, J.

P. Moulin and J. Liu, “Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors,” IEEE Trans. Inf. Theory 45, 909-919 (1999).
[CrossRef]

Liu, R.

R. Liu and J. Jia, “Reducing boundary artifacts in image deconvolution,” in Proceedings of 15th IEEE International Conference on Image Processing (IEEE, 2008), pp. 505-508.

Mallat, S.

J. Kalifa and S. Mallat, “Thresholding estimators for linear inverse problems and deconvolutions,” Ann. Statist. 31, 58-109 (2003).

J. Kalifa, S. Mallat, and B. Rouge, “Deconvolution by thresholding in mirror wavelet bases,” IEEE Trans. Image Process. 12, 446-457 (2003).
[CrossRef]

J. Kalifa, S. Mallat, and B. Rouge, “Minimax solution of inverse problems and deconvolution by mirror wavelet thresholding,” Proc. SPIE 3813, 42-57 (1999).
[CrossRef]

J. Kalifa, S. Mallat, and B. Rouge, “Image deconvolution in mirror wavelet bases,” in Proceedings of the 1998 International Conference on Image Processing (IEEE, 1998), pp. 565-569.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

Mishra, N.

S. K. Patra, N. Mishra, R. Chandrakanth, and R. Ramachandran, “Image quality improvement through MTF compensation: a treatment to high resolution data,” Indian Cartogr. 22, 86-93 (2002).

Moulin, P.

P. Moulin and J. Liu, “Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors,” IEEE Trans. Inf. Theory 45, 909-919 (1999).
[CrossRef]

Nason, G. P.

G. P. Nason and B. W. Silverman, “The stationary wavelet transform and some statistical applications,” Ph.D. dissertation (University of Bristol, 1995), pp. 1-19.

Neelamani, R.

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

Nowak, R. D.

M. A. Figueiredo, J. M. Bioucas-Dias, and R. D. Nowak, “Majorization-minimization algorithms for wavelet-based image restoration,” IEEE Trans. Image Process. 16, 2980-2991 (2007).
[CrossRef]

M. A. Figueiredo and R. D. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906-916 (2003).
[CrossRef]

M. A. Figueiredo and R. D. Nowak, “Wavelet-based image estimation: an empirical Bayes approach using Jeffreys' noninformative prior,” IEEE Trans. Image Process. 10, 1322-1331 (2001).
[CrossRef]

Odegard, J. E.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10-12 (1996).
[CrossRef]

Oliveira, J. P.

J. M. Bioucas-Dias, M. A. Figueiredo, and J. P. Oliveira, “Total variation-based image deconvolution: a majorization-minimization approach,” in 2006 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 861-864.

Park, F.

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

Patra, S. K.

S. K. Patra, N. Mishra, R. Chandrakanth, and R. Ramachandran, “Image quality improvement through MTF compensation: a treatment to high resolution data,” Indian Cartogr. 22, 86-93 (2002).

Portilla, J.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. 12, 1338-1351(2003).
[CrossRef]

Puetter, R. C.

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139-194 (2005).
[CrossRef]

Ramachandran, R.

S. K. Patra, N. Mishra, R. Chandrakanth, and R. Ramachandran, “Image quality improvement through MTF compensation: a treatment to high resolution data,” Indian Cartogr. 22, 86-93 (2002).

Rouge, B.

J. Kalifa, S. Mallat, and B. Rouge, “Deconvolution by thresholding in mirror wavelet bases,” IEEE Trans. Image Process. 12, 446-457 (2003).
[CrossRef]

J. Kalifa, S. Mallat, and B. Rouge, “Minimax solution of inverse problems and deconvolution by mirror wavelet thresholding,” Proc. SPIE 3813, 42-57 (1999).
[CrossRef]

J. Kalifa, S. Mallat, and B. Rouge, “Image deconvolution in mirror wavelet bases,” in Proceedings of the 1998 International Conference on Image Processing (IEEE, 1998), pp. 565-569.

Sayeed, A. M.

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

Silverman, B. W.

I. M. Johnstone and B. W. Silverman, “Wavelet threshold estimators for data with correlated noise,” J. R. Statist. Soc. Ser. B 59, 319-351 (1997).

G. P. Nason and B. W. Silverman, “The stationary wavelet transform and some statistical applications,” Ph.D. dissertation (University of Bristol, 1995), pp. 1-19.

Simoncelli, E. P.

D. K. Hammond and E. P. Simoncelli, “Image modeling and denoising with orientation-adapted Gaussian scale mixtures,” IEEE Trans. Image Process. 17, 2089-2101 (2008).
[CrossRef]

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. 12, 1338-1351(2003).
[CrossRef]

Strela, V.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. 12, 1338-1351(2003).
[CrossRef]

Vetterli, M.

S. G. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9, 1532-1546 (2000).
[CrossRef]

Wainwright, M. J.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. 12, 1338-1351(2003).
[CrossRef]

Ward, R. K.

F. Aghdasi and R. K. Ward, “Reduction of boundary artifacts in image restoration,” IEEE Trans. Image Process. 5, 611-618(1996).
[CrossRef]

Wells, R. O.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10-12 (1996).
[CrossRef]

Yahil, A.

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139-194 (2005).
[CrossRef]

Yip, A.

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

Yip, A. M.

D. Krishnan, P. Lin, and A. M. Yip, “A primal-dual active-set method for nonnegativity constrained total variation deblurring problems,” IEEE Trans. Image Process. 16, 2766-2777(2007).
[CrossRef]

Yu, B.

S. G. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9, 1532-1546 (2000).
[CrossRef]

Zerubia, J.

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613-621 (2004).
[CrossRef]

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Satellite image deconvolution using complex wavelet packets,” INRIA Research Report 3955 (Institut National de Recherche en Informatique et en Automatique, 2000), pp. 7-73.

Ann. Statist. (1)

J. Kalifa and S. Mallat, “Thresholding estimators for linear inverse problems and deconvolutions,” Ann. Statist. 31, 58-109 (2003).

Annu. Rev. Astron. Astrophys. (1)

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139-194 (2005).
[CrossRef]

Appl. Comput. Harmon. Anal. (1)

D. L. Donoho, “Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition,” Appl. Comput. Harmon. Anal. 2, 101-126 (1995).
[CrossRef]

IEEE Signal Process. Lett. (2)

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10-12 (1996).
[CrossRef]

J. E. Fowler, “The redundant discrete wavelet transform and additive noise,” IEEE Signal Process. Lett. 12, 629-632 (2005).
[CrossRef]

IEEE Trans. Image Process. (13)

F. Aghdasi and R. K. Ward, “Reduction of boundary artifacts in image restoration,” IEEE Trans. Image Process. 5, 611-618(1996).
[CrossRef]

J. M. Bioucas-Dias, “Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors,” IEEE Trans. Image Process. 15, 937-951 (2006).
[CrossRef]

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613-621 (2004).
[CrossRef]

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

M. A. Figueiredo, J. M. Bioucas-Dias, and R. D. Nowak, “Majorization-minimization algorithms for wavelet-based image restoration,” IEEE Trans. Image Process. 16, 2980-2991 (2007).
[CrossRef]

M. A. Figueiredo and R. D. Nowak, “Wavelet-based image estimation: an empirical Bayes approach using Jeffreys' noninformative prior,” IEEE Trans. Image Process. 10, 1322-1331 (2001).
[CrossRef]

D. K. Hammond and E. P. Simoncelli, “Image modeling and denoising with orientation-adapted Gaussian scale mixtures,” IEEE Trans. Image Process. 17, 2089-2101 (2008).
[CrossRef]

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. 12, 1338-1351(2003).
[CrossRef]

S. G. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9, 1532-1546 (2000).
[CrossRef]

J. Kalifa, S. Mallat, and B. Rouge, “Deconvolution by thresholding in mirror wavelet bases,” IEEE Trans. Image Process. 12, 446-457 (2003).
[CrossRef]

D. Krishnan, P. Lin, and A. M. Yip, “A primal-dual active-set method for nonnegativity constrained total variation deblurring problems,” IEEE Trans. Image Process. 16, 2766-2777(2007).
[CrossRef]

M. A. Figueiredo and R. D. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906-916 (2003).
[CrossRef]

J. Fessler and A. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417-1429 (1995).
[CrossRef]

IEEE Trans. Inf. Theory (1)

P. Moulin and J. Liu, “Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors,” IEEE Trans. Inf. Theory 45, 909-919 (1999).
[CrossRef]

IEEE Trans. Signal Process. (2)

A. D. Hillery and R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 39, 1892-1899(1991).
[CrossRef]

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

Indian Cartogr. (1)

S. K. Patra, N. Mishra, R. Chandrakanth, and R. Ramachandran, “Image quality improvement through MTF compensation: a treatment to high resolution data,” Indian Cartogr. 22, 86-93 (2002).

J. R. Statist. Soc. Ser. B (1)

I. M. Johnstone and B. W. Silverman, “Wavelet threshold estimators for data with correlated noise,” J. R. Statist. Soc. Ser. B 59, 319-351 (1997).

Proc. SPIE (2)

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

J. Kalifa, S. Mallat, and B. Rouge, “Minimax solution of inverse problems and deconvolution by mirror wavelet thresholding,” Proc. SPIE 3813, 42-57 (1999).
[CrossRef]

Other (11)

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Satellite image deconvolution using complex wavelet packets,” INRIA Research Report 3955 (Institut National de Recherche en Informatique et en Automatique, 2000), pp. 7-73.

J. M. Bioucas-Dias, M. A. Figueiredo, and J. P. Oliveira, “Total variation-based image deconvolution: a majorization-minimization approach,” in 2006 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 861-864.

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

T. Choi, “IKONOS satellite on orbit modulation transfer function (MTF) measurement using edge and pulse method,” Ph.D. dissertation (South Dakota State University, 2002), pp. 41-89.

M. Y. Zhou, Deconvolution and Signal Recovery (National Defence Industry Press, 2004).

J. Kalifa, S. Mallat, and B. Rouge, “Image deconvolution in mirror wavelet bases,” in Proceedings of the 1998 International Conference on Image Processing (IEEE, 1998), pp. 565-569.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

R. Liu and J. Jia, “Reducing boundary artifacts in image deconvolution,” in Proceedings of 15th IEEE International Conference on Image Processing (IEEE, 2008), pp. 505-508.

G. P. Nason and B. W. Silverman, “The stationary wavelet transform and some statistical applications,” Ph.D. dissertation (University of Bristol, 1995), pp. 1-19.

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

T. Leonard and J. S. J. Hsu, Bayesian Methods: an Analysis for Statisticians and Interdisciplinary Researchers (China Machine Press, 2006).

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

Proposed deconvolution algorithm (WDALRD).

Fig. 2
Fig. 2

SNR performances of Wiener (solid curve), ForWaRD (dotted curve) and WDALRD (dash–dot curve) as a function of regularization parameter τ obtained from (a) the first experiment and (b) the second experiment.

Fig. 3
Fig. 3

Deconvolution experiment on the remote sensing image: (a) original remote sensing image, (b) degraded image ( BSNR = 30 dB ), (c) Wiener filter estimate ( SNR = 22.75 dB ), (d) ForWaRD estimate ( SNR = 23.24 dB ), and (e) WDALRD estimate ( SNR = 23.53 dB ).

Fig. 4
Fig. 4

Zoom on an 84 × 128 area extracted from the image of Fig. 3: (a) original, (b) Wiener, (c) ForWaRD, and (d) WDALRD.

Fig. 5
Fig. 5

Deconvolution experiment on the cameraman image: (a) original cameraman image, (b) degraded image ( BSNR = 40 dB ), (c) Wiener filter estimate ( SNR = 20.98 dB ), (d) ForWaRD estimate ( SNR = 22.53 dB ), and (e) WDALRD estimate ( SNR = 22.77 dB ).

Tables (1)

Tables Icon

Table 1 Computational Time in Different Image Sizes (in Seconds)

Equations (21)

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

y = H x + n = ( h x ) + n ,
Y ˜ ( ν ) = H ˜ ( ν ) X ˜ ( ν ) + N ˜ ( ν ) ,
X ^ ( ν ) = H ˜ * ( ν ) | H ˜ ( ν ) | 2 + τ | C ˜ ( ν ) | 2 Y ˜ ( ν ) = | H ˜ ( ν ) | 2 | H ˜ ( ν ) | 2 + τ | C ˜ ( ν ) | 2 X ˜ ( ν ) + H ˜ * ( ν ) | H ˜ ( ν ) | 2 + τ | C ˜ ( ν ) | 2 N ˜ ( ν ) = X ˜ λ ( ν ) + H ˜ * ( ν ) | H ˜ ( ν ) | 2 + τ | C ˜ ( ν ) | 2 N ˜ ( ν ) ,
x ^ = x λ + γ ,
P Γ ( ν ) = | H ˜ ( ν ) | 2 ( | H ˜ ( ν ) | 2 + τ | C ˜ ( ν ) | 2 ) 2 σ 2 .
ξ s , m = θ s , m + z s , m ,
σ s , m 2 = E { | z s , m | 2 } = 1 N ν P Γ ( ν ) | ψ ˜ s , m ( ν ) | 2 .
σ s , m 2 = 1 N ν P Γ ( ν ) | ψ ˜ s , 0 ( ν ) | 2 = σ s 2 .
π ( ξ s , m | θ s , m ) N ( 0 , σ s 2 ) exp ( ( ξ s , m θ s , m ) 2 2 σ s 2 ) .
π ( θ s , m ; σ θ ( s , m ) ) = 1 2 σ θ ( s , m ) exp ( ( | θ s , m | σ θ ( s , m ) ) ) ,
π ( σ θ ( s , m ) ) | F ( σ θ ( s , m ) ) | 1 / 2 ,
F ( σ θ ( s , m ) ) = E [ 2 log π ( θ s , m ; σ θ ( s , m ) ) 2 σ θ ( s , m ) ]
2 log π ( θ s , m ; σ θ ( s , m ) ) 2 σ θ ( s , m ) = 2 | θ s , m | ( σ θ ( s , m ) ) 3 1 ( σ θ ( s , m ) ) 2 .
E { | θ s , m | } = + | θ s , m | π ( θ s , m ; σ θ ( s , m ) ) d θ s , m = + | θ s , m | 1 2 σ θ ( s , m ) exp ( ( | θ s , m | σ θ ( s , m ) ) ) d θ s , m = σ θ ( s , m ) ,
π ( σ θ ( s , m ) ) 1 σ θ ( s , m ) .
π ( θ s , m , σ θ ( s , m ) ) = π ( θ s , m ; σ θ ( s , m ) ) π ( σ θ ( s , m ) ) 1 ( σ θ ( s , m ) ) 2 exp ( ( | θ s , m | σ θ ( s , m ) ) ) .
π ( θ s , m , σ θ ( s , m ) | ξ s , m ) π ( ξ s , m | θ s , m , σ θ ( s , m ) ) π ( θ s , m , σ θ ( s , m ) ) 1 ( σ θ ( s , m ) ) 2 exp ( ( ξ s , m θ s , m ) 2 2 σ s 2 ( | θ s , m | σ θ ( s , m ) ) ) .
π ( θ s , m , σ θ ( s , m ) | ξ s , m ) θ s , m = 0 θ s , m = sgn ( ξ s , m ) ( | ξ s , m | σ s 2 σ θ ( s , m ) ) + ,
π ( θ s , m , σ θ ( s , m ) | ξ s , m ) σ θ ( s , m ) = 0 σ θ ( s , m ) = | θ s , m | 2 ,
θ s , m = { sgn ( ξ s , m ) ( | ξ s , m | + ξ s , m 2 8 σ s 2 2 ) ξ s , m 2 2 σ s 0 ξ s , m < 2 2 σ s .
SNR = 10 log ( x 2 x λ x 2 ) .

Metrics