Abstract

We develop a unified algorithm for performing blind deconvolution of a noisy degraded image. By incorporating a low-pass filter into the asymmetric multiplicative iterative algorithm and extending it to multiframe blind deconvolution, this algorithm accomplishes the blind deconvolution and noise removal concurrently. We report numerical experiments of applying the algorithm to the restoration of short-exposure atmosphere turbulence degraded images. These experiments evidently demonstrate that the unified algorithm has both good blind deconvolution performance and high-resolution image restoration.

© 2009 Optical Society of America

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References

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  1. J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale Univ. Press, 1923).
  2. R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139-194 (2005).
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  6. L. Bar, N. Kiryati, and N. Sochen, “Image deblurring in the presence of impulse noise,” Int. J. Comput. Vis. 70, 279-298(2006).
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    [CrossRef]
  8. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).
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    [CrossRef] [PubMed]
  10. S. S. Young, R. G. Driggers, B. P. Teaney, and E. L. Jacobs, “Adaptive deblurring of noisy images,” Appl. Opt. 46, 744-752(2007).
    [CrossRef] [PubMed]
  11. R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002).
  12. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  13. S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862-874 (1993).
    [CrossRef]
  14. J. W. Goodman, Statistical Optics (Wiley, 2000).
  15. D. G. Sheppard, B. R. Hunt, and M. W. Marcellin, “Iterative multiframe superresolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A 15, 978-992(1998).
    [CrossRef]
  16. M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).
  17. J. Canny, “A computational approach for edge detection,” IEEE Trans. Pattern Anal. Machine Intell. pami-8, 679-698 (1986).
    [CrossRef]

2008 (1)

2007 (1)

2006 (1)

L. Bar, N. Kiryati, and N. Sochen, “Image deblurring in the presence of impulse noise,” Int. J. Comput. Vis. 70, 279-298(2006).
[CrossRef]

2005 (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]

2002 (1)

1998 (2)

1996 (1)

A. van der Schaaf and J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

1993 (1)

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862-874 (1993).
[CrossRef]

1986 (1)

J. Canny, “A computational approach for edge detection,” IEEE Trans. Pattern Anal. Machine Intell. pami-8, 679-698 (1986).
[CrossRef]

Bar, L.

L. Bar, N. Kiryati, and N. Sochen, “Image deblurring in the presence of impulse noise,” Int. J. Comput. Vis. 70, 279-298(2006).
[CrossRef]

Campisi, P.

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

Canny, J.

J. Canny, “A computational approach for edge detection,” IEEE Trans. Pattern Anal. Machine Intell. pami-8, 679-698 (1986).
[CrossRef]

Christou, J. C.

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862-874 (1993).
[CrossRef]

Conhello, J.-A.

Driggers, R. G.

Egiazarian, K.

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

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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]

Hadamard, J.

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale Univ. Press, 1923).

He, G.

Hunt, B. R.

Jacobs, E. L.

Jefferies, S. M.

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862-874 (1993).
[CrossRef]

Kiryati, N.

L. Bar, N. Kiryati, and N. Sochen, “Image deblurring in the presence of impulse noise,” Int. J. Comput. Vis. 70, 279-298(2006).
[CrossRef]

Marcellin, M. W.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Prasad, S.

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]

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).

Sheppard, D. G.

Sochen, N.

L. Bar, N. Kiryati, and N. Sochen, “Image deblurring in the presence of impulse noise,” Int. J. Comput. Vis. 70, 279-298(2006).
[CrossRef]

Teaney, B. P.

van der Schaaf, A.

A. van der Schaaf and J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

van Hateren, J. H.

A. van der Schaaf and J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002).

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]

Young, S. S.

Zhang, J.

Zhang, Q.

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. Opt. (1)

Astrophys. J. (1)

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862-874 (1993).
[CrossRef]

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

J. Canny, “A computational approach for edge detection,” IEEE Trans. Pattern Anal. Machine Intell. pami-8, 679-698 (1986).
[CrossRef]

Int. J. Comput. Vis. (1)

L. Bar, N. Kiryati, and N. Sochen, “Image deblurring in the presence of impulse noise,” Int. J. Comput. Vis. 70, 279-298(2006).
[CrossRef]

J. Opt. Soc. Am. A (4)

Vision Res. (1)

A. van der Schaaf and J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

Other (7)

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).

J. W. Goodman, Statistical Optics (Wiley, 2000).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale Univ. Press, 1923).

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Figures (7)

Fig. 1
Fig. 1

Restoration of Saturn image: (a) original image, (b) one of the five noisy degraded images with noise being 0.1 intensity of image variance, (c) image restored by the AMIA extended for multiframe blind deconvolution, and (d) image restored by the UMBD-LPF.

Fig. 2
Fig. 2

Corresponding edges of images in Fig. 1: (a) edges of the original image, (b) edges of the noisy degraded image, (c) edges of the image restored by the AMIA, and (d) edges of the image restored by the UMBD-LPF.

Fig. 3
Fig. 3

Corresponding spectra of images in Fig. 1: (a) spectra of the original image, (b) spectra of the noisy degraded image, (c) spectra of the image restored by the AMIA, and (d) spectra of the image restored by the UMBD-LPF.

Fig. 4
Fig. 4

Restoration of a general image: (a) original image, (b) one of the five noisy degraded images with noise being 0.15 intensity of image variance, and (c) image restored by the UMBD-LPF.

Fig. 5
Fig. 5

Corresponding edges of images in Fig. 4: (a) edges of the original image, (b) edges of the noisy degraded image, and (c) edges of the image restored by the UMBD-LPF.

Fig. 6
Fig. 6

Corresponding spectra of images in Fig. 4: (a) spectra of the original image, (b) spectra of the noisy degraded image, and (c) spectra of the image restored by the UMBD-LPF.

Fig. 7
Fig. 7

Natural logarithms of normalized Fourier-transform magnitudes along the horizontal axis of the corresponding original image, restored image, and blurred image in Fig. 4.

Tables (1)

Tables Icon

Table 1 Normalized-Mean-Square Errors of the Blurred Images and the Restored Images in Figs. 1, 4

Equations (21)

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g ( x ) = h ( x ) * o ( x ) + n ( x ) ,
G ( u ) = H ( u ) O ( u ) + N ( u ) ,
J ( o , h ) = x [ g ( x ) h ( x ) * o ( x ) ] 2 2 .
J ( O , H ) = 1 2 G ( u ) H ( u ) O ( u ) 2 ,
J ( O , H ) = 1 2 W ( u ) [ G ( u ) H ( u ) O ( u ) ] 2 .
J ( o , h ) = x 1 2 { w ( x ) * [ g ( x ) h ( x ) * o ( x ) ] } 2 .
J ( o , h ) o = w ( x ) * h c ( x ) * [ g ( x ) h ( x ) * o ( x ) ] ,
J ( o , h ) h = w ( x ) * o c ( x ) * [ g ( x ) h ( x ) * o ( x ) ] ,
w ( x ) * h c ( x ) * [ g ( x ) h ( x ) * o ( x ) ] = 0 ,
w ( x ) * o c ( x ) * [ g ( x ) h ( x ) * o ( x ) ] = 0.
exp { λ 1 w ( x ) * h c ( x ) * [ g ( x ) h ( x ) * o ( x ) ] } = 1 , λ 1 > 0 ,
exp { λ 2 w ( x ) * o c ( x ) * [ g ( x ) h ( x ) * o ( x ) ] } = 1 , λ 2 > 0.
o k + 1 ( x ) = o k ( x ) exp { λ 1 w ( x ) * h k c ( x ) ) * [ g ( x ) h k ( x ) * o k ( x ) ] } o k + 1 ( x ) = o k + 1 ( x ) / [ x o k + 1 ( x ) ] , λ 1 > 0 ,
h k + 1 ( x ) = h k ( x ) exp { λ 2 w ( x ) * o k + 1 c ( x ) * [ g ( x ) - h k ( x ) * o k + 1 ( x ) ] } h k + 1 ( x ) = h k + 1 ( x ) / [ x h k + 1 ( x ) ] , λ 2 > 0.
W ( u ) = exp [ u 2 / ( 2 D 0 2 ) ] ,
N A = ( 1.22 λ l D ) 1 N μ ,
N c = N N A = ( D 1.22 λ l ) N μ N .
J ( o , h | 1 L ) = i = 1 L x 1 2 { w ( x ) * [ g i ( x ) h i ( x ) * o ( x ) ] } 2 .
o k + 1 ( x ) = o k ( x ) exp { λ 1 i = 1 L w ( x ) * h i , k c ( x ) * [ g i ( x ) h i , k ( x ) * o k ( x ) ] } o k + 1 ( x ) = o k + 1 ( x ) / [ x o k + 1 ( x ) ] , λ 1 > 0 ,
h i , k + 1 ( x ) = h i , k ( x ) exp { λ 2 w ( x ) * o k + 1 c ( x ) * [ g i ( x ) h i , k ( x ) * o k + 1 ( x ) ] } , h i , k + 1 ( x ) = h i , k + 1 ( x ) / [ x h i , k + 1 ( x ) ] , λ 2 > 0 for     i = 1 , 2 , , L .
NMSE = x [ o ( x ) g ( x ) ] 2 x o ( x ) 2 .

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