Abstract

A novel scheme for blind deconvolution is presented. Adapted from our previous multiplicative iterative algorithm, a more general scheme—the asymmetric multiplicative iterative algorithm—is derived. Additionally, in order to obtain a meaningful estimate, a penalized asymmetric multiplicative iterative algorithm is used to further improve the performance of this scheme. Results of a numerical experiment with the asymmetric multiplicative iterative algorithm are presented.

© 2008 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 U. Press, 1923).
  2. M. Cannon, “Blind deconvolution of spatially invariant image blur with phase,” IEEE Trans. Acoust., Speech, Signal Process. 24, 58-63 (1976).
    [CrossRef]
  3. R. G. Lane and R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180-188 (1987).
    [CrossRef]
  4. D. C. Ghiglia, L. A. Romero, and G. A. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024-1036 (1993).
    [CrossRef]
  5. P. J. Bones, C. R. Parker, B. L. Satherly, and R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842-1857 (1995).
    [CrossRef]
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    [CrossRef]
  8. R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508-1514 (1992).
    [CrossRef]
  9. 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, 58-65 (1995).
    [CrossRef]
  10. H. Lantéri, C. Aime, H. Beaumont, and P. Gauchere, “Blind deconvolution using the Richardson-Lucy algorithm,” Proc. SPIE 2312, 182-192 (1994).
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  11. F. Tsumuraya, N. Miura, and N. Baba, “Iterative blind deconvolution method using Lucy's algorithm,” Astron. Astrophys. 282, 699-708 (1994).
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    [CrossRef] [PubMed]
  13. E. Thiébaut and J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485-492 (1995).
    [CrossRef]
  14. T. S. Zacchéo and R. A. Gonsalves, “Iterative maximum-likelihood estimators for positively constrained objects,” J. Opt. Soc. Am. A 13, 236-242 (1996).
    [CrossRef]
  15. R. G. Lane, “Methods for maximum-likelihood deconvolution,” J. Opt. Soc. Am. A 13, 1992-1998 (1996).
    [CrossRef]
  16. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064-1073 (1993).
    [CrossRef]
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  21. D. L. Snyder, T. J. Schulz, and J. A. O'Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143-1150 (1992).
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    [CrossRef]
  24. H. Lantéri, R. Soummer, and C. Aime, “Comparison between ISRA and RLA algorithms. Use of a Wiener filter based stopping criterion,” Astron. Astrophys. Suppl. Ser. 140, 235-246 (1999).
    [CrossRef]
  25. C. Xu, I. Assaoui, S. Jacquery, “Algebraic analysis of the Van Cittert iterative method of deconvolution with a general relaxation factor,” J. Opt. Soc. Am. A 11, 2804-2808 (1994).
    [CrossRef]
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    [CrossRef]
  27. J. A. Conchello, “Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images,” J. Opt. Soc. Am. A 15, 2609-2619 (1998).
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  28. http://www.cmnh.org/site/ClassesandProgramsLecturesFrontiersAstronomy.aspx.
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2008 (1)

2006 (1)

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Communic. Inf. Theory 2, 155-239 (2006).
[CrossRef]

2005 (1)

R. Vio, J. Bardsley, and W. Wamsteker, “Least-squares methods with Poissonian noise: an analysis and a comparison with the Richardson-Lucy algorithm,” Astron. Astrophys. 436, 741-755 (2005).
[CrossRef]

2000 (1)

1999 (1)

H. Lantéri, R. Soummer, and C. Aime, “Comparison between ISRA and RLA algorithms. Use of a Wiener filter based stopping criterion,” Astron. Astrophys. Suppl. Ser. 140, 235-246 (1999).
[CrossRef]

1998 (2)

1997 (1)

1996 (3)

1995 (3)

1994 (3)

H. Lantéri, C. Aime, H. Beaumont, and P. Gauchere, “Blind deconvolution using the Richardson-Lucy algorithm,” Proc. SPIE 2312, 182-192 (1994).
[CrossRef]

F. Tsumuraya, N. Miura, and N. Baba, “Iterative blind deconvolution method using Lucy's algorithm,” Astron. Astrophys. 282, 699-708 (1994).

C. Xu, I. Assaoui, S. Jacquery, “Algebraic analysis of the Van Cittert iterative method of deconvolution with a general relaxation factor,” J. Opt. Soc. Am. A 11, 2804-2808 (1994).
[CrossRef]

1993 (3)

1992 (3)

1989 (1)

B. L. K. Davey, R. G. Lane, and R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353-356 (1989).
[CrossRef]

1988 (1)

1987 (1)

1976 (1)

M. Cannon, “Blind deconvolution of spatially invariant image blur with phase,” IEEE Trans. Acoust., Speech, Signal Process. 24, 58-63 (1976).
[CrossRef]

Astron. Astrophys. (2)

F. Tsumuraya, N. Miura, and N. Baba, “Iterative blind deconvolution method using Lucy's algorithm,” Astron. Astrophys. 282, 699-708 (1994).

R. Vio, J. Bardsley, and W. Wamsteker, “Least-squares methods with Poissonian noise: an analysis and a comparison with the Richardson-Lucy algorithm,” Astron. Astrophys. 436, 741-755 (2005).
[CrossRef]

Astron. Astrophys. Suppl. Ser. (1)

H. Lantéri, R. Soummer, and C. Aime, “Comparison between ISRA and RLA algorithms. Use of a Wiener filter based stopping criterion,” Astron. Astrophys. Suppl. Ser. 140, 235-246 (1999).
[CrossRef]

Found. Trends Communic. Inf. Theory (1)

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Communic. Inf. Theory 2, 155-239 (2006).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (1)

M. Cannon, “Blind deconvolution of spatially invariant image blur with phase,” IEEE Trans. Acoust., Speech, Signal Process. 24, 58-63 (1976).
[CrossRef]

IEEE Trans. Signal Process. (1)

D. L. Snyder, T. J. Schulz, and J. A. O'Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143-1150 (1992).
[CrossRef]

J. Opt. Soc. Am. A (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]

E. Y. Lam and J. W. Goodman, “Iterative statistical approach to blind image deconvolution,” J. Opt. Soc. Am. A 17, 1177-1184 (2000).
[CrossRef]

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052-1061 (1992).
[CrossRef] [PubMed]

E. Thiébaut and J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485-492 (1995).
[CrossRef]

T. S. Zacchéo and R. A. Gonsalves, “Iterative maximum-likelihood estimators for positively constrained objects,” J. Opt. Soc. Am. A 13, 236-242 (1996).
[CrossRef]

R. G. Lane, “Methods for maximum-likelihood deconvolution,” J. Opt. Soc. Am. A 13, 1992-1998 (1996).
[CrossRef]

T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064-1073 (1993).
[CrossRef]

R. G. Lane and R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180-188 (1987).
[CrossRef]

D. C. Ghiglia, L. A. Romero, and G. A. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024-1036 (1993).
[CrossRef]

P. J. Bones, C. R. Parker, B. L. Satherly, and R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842-1857 (1995).
[CrossRef]

R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508-1514 (1992).
[CrossRef]

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, 58-65 (1995).
[CrossRef]

C. Xu, I. Assaoui, S. Jacquery, “Algebraic analysis of the Van Cittert iterative method of deconvolution with a general relaxation factor,” J. Opt. Soc. Am. A 11, 2804-2808 (1994).
[CrossRef]

P. J. Sementilli, B. R. Hunt, and M. S. Nadar, “Analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265-2276 (1993).
[CrossRef]

J. A. Conchello, “Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images,” J. Opt. Soc. Am. A 15, 2609-2619 (1998).
[CrossRef]

Opt. Commun. (2)

B. L. K. Davey, R. G. Lane, and R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353-356 (1989).
[CrossRef]

N. F. Law and R. G. Lane, “Blind deconvolution using least squares minimisation,” Opt. Commun. 128, 341-352 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (1)

H. Lantéri, C. Aime, H. Beaumont, and P. Gauchere, “Blind deconvolution using the Richardson-Lucy algorithm,” Proc. SPIE 2312, 182-192 (1994).
[CrossRef]

Other (4)

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

http://www.cmnh.org/site/ClassesandProgramsLecturesFrontiersAstronomy.aspx.

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

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

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Figures (5)

Fig. 1
Fig. 1

(a) Astronomical image (based on [28]) and (b) its Fourier-transform magnitude.

Fig. 2
Fig. 2

Image restoration by the AMIA. (a) image of Fig. 1a blurred by computer-simulated atmosphere turbulence; (b) image restored from (a) after 300 iterations by the AMIA with λ 1 = 2.0 × 10 4 , λ 2 = 0.8 × 10 2 , μ = 0.1 ; (c), (d) Fourier-transform magnitudes of (a) and (b), respectively.

Fig. 3
Fig. 3

Natural logarithms of normalized Fourier-transform magnitudes along the horizontal axis of the original image, restored image after 100 iterations, and blurred image.

Fig. 4
Fig. 4

Natural logarithms of normalized Fourier-transform magnitudes along the horizontal axis of the original image, restored image after 200 iterations, and blurred image.

Fig. 5
Fig. 5

Natural logarithms of normalized Fourier-transform magnitudes along the horizontal axis of the original image, restored image after 300 iteration, and blurred image.

Equations (59)

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g ( x ) = h ( x ) * o ( x ) + n ( x ) ,
L ( o , h ) = x [ g ( x ) h ( x ) * o ( x ) ] 2 2 + C ,
L ( o , h ) = x [ g ( x ) h ( x ) * o ( x ) ] 2 2 = 1 2 g ( x ) h ( x ) * o ( x ) 2 ,
o k + 1 ( x ) = o k ( x ) exp { λ 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 > 0
h k + 1 ( x ) = h k ( x ) exp { λ 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 > 0
o k + 1 ( x ) = o k ( x ) exp { λ 1 h k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] } , λ 1 > 0 ,
h k + 1 ( x ) = h k ( x ) exp { λ 2 o k + 1 c ( x ) * [ g ( x ) h k ( x ) * o k + 1 ( x ) ] } , λ 2 > 0 .
o ( x ) = o k + 1 ( x ) [ x o k + 1 ( x ) ] ,
h ( x ) = h k + 1 ( x ) [ x h k + 1 ( x ) ] .
L ( o k + 1 , h k + 1 ) L ( o k , h k ) > 0 .
0 < λ 1 < min x { min { 2 ( e 1 ) o k ( x ) [ x h k ( x ) ] 2 , 1 h k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] } } ,
0 < λ 2 < min x { min { 2 ( e 1 ) h k ( x ) [ x o k ( x ) ] 2 , 1 o k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] } } .
L ( o k + 1 , h k + 1 ) L ( o k , h k ) > 0 , when L ( o k , h k ) 0 . ( 12 )
L ( o k + 1 , h k ) L ( o k , h k ) > 0 ,
L ( o k + 1 , h k + 1 ) L ( o k + 1 , h k ) > 0 .
L ( o k + 1 , h k + 1 ) L ( o k , h k ) > 0 .
r ( x ) = h k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] ,
s ( x ) = exp [ λ 1 r ( x ) ] .
o k + 1 ( x ) = o k ( x ) s ( x ) .
o ̂ k + 1 ( u ) = o ̂ k ( u ) * s ̂ ( u ) ,
r ( x ) = h k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] ,
r ( x ) 0 ,
s ( x ) = exp [ λ 1 r ( x ) ] ,
s ( x ) 1 ,
L R ( o , h ) = L ( o , h ) μ φ ( h ) , μ 0 ,
φ ( h ) = 1 2 ( x h 2 ( x ) 1 ) .
o k + 1 ( x ) = o k ( x ) exp { λ 1 L R ( o , h ) o } , λ 1 > 0 ,
o k + 1 ( x ) = o k ( x ) exp { λ 1 h k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] } ,
h k + 1 ( x ) = h k ( x ) exp { λ 2 L R ( o , h ) h } , λ 2 > 0 ,
h k + 1 ( x ) = h k ( x ) exp { λ 2 o k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] λ 2 μ h ( x ) } .
0 < λ 1 < min x { min { 2 ( e 1 ) o k ( x ) [ x h k ( x ) ] 2 , 1 h k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] } } .
0 < λ 2 < min x { min { 2 ( e 1 ) h k ( x ) [ ( x o k ( x ) ) 2 + μ ] , 1 o k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] μ h ( x ) } } .
L R ( o k + 1 , h k + 1 ) L R ( o k , h k ) > 0 , when L R ( o k , h k ) 0 .
g = H o + n .
L ( o k + 1 , h k ) L ( o k , h k ) = 1 2 g ( x ) h k ( x ) * o k + 1 ( x ) 2 + 1 2 g ( x ) h k ( x ) * o k ( x ) 2 .
L ( o k + 1 , h k ) L ( o k , h k ) = 1 2 ( g H k o k + 1 ) T ( g H k o k + 1 ) + 1 2 ( g H k o k ) T ( g H k o k ) .
L ( o k + 1 , h k ) L ( o k , h k ) = 1 2 ( o k + 1 o k ) T H k T H k ( o k + 1 o k ) + ( g H k o k ) T H k ( o k + 1 o k ) = 1 2 ( o k + 1 o k ) T H k T H k ( o k + 1 o k ) + ( H k T g H k T H k o k ) T ( o k + 1 o k ) .
o k + 1 ( x ) = o k ( x ) exp [ λ 1 r ( x ) ] .
q ( x ) = { [ exp ( x ) 1 ] x x 0 1 x = 0 .
exp ( x ) = q ( x ) x + 1 .
o k + 1 ( x ) o k ( x ) = o k ( x ) { exp [ λ 1 r ( x ) ] 1 } = o k ( x ) q [ λ 1 r ( x ) ] λ 1 r ( x ) = λ 1 o k ( x ) q [ λ 1 r ( x ) ] { h k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] } .
o k + 1 o k = D r ,
D = diag { λ 1 o k ( x ) q [ λ 1 r ( x ) ] } ,
L ( o k + 1 , h k ) L ( o k , h k ) = 1 2 ( D r ) T H k T H k D r + r T D r = r T ( D 1 2 D T H k T H k D ) r .
q [ λ 1 r ( x ) ] e 1 , for any x ,
0 < λ 1 < min x { min { 2 ( e 1 ) η h o k ( x ) , 1 h k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] } }
L ( o k + 1 , h k ) L ( o k , h k ) > 0 , for any r 0 .
η h = [ x h k ( x ) ] 2 .
g = O h + n
L R ( o k , h k + 1 ) L R ( o k , h k ) = 1 2 ( g O k h k + 1 ) T ( g O k h k + 1 ) + 1 2 ( g O k h k ) T ( g O k h k ) 1 2 [ μ h k + 1 T h k + 1 μ h k T h k ] = 1 2 ( h k + 1 h k ) T [ O k T O k + μ ] ( h k + 1 h k ) + ( O k T g O k T O k h k μ h k ) T ( h k + 1 h k ) .
h k + 1 ( x ) = h k ( x ) exp [ λ 2 r h ( x ) ] = h k ( x ) { λ 2 q [ λ 2 r h ( x ) ] r h ( x ) + 1 } .
h k + 1 h k = D h r h ,
D h = diag { λ 2 h k ( x ) q [ λ 2 r h ( x ) ] } ,
L R ( o k + 1 , h k ) L R ( o k , h k ) = 1 2 ( D h r h ) T ( O k T O k + μ ) D h r h + r h T D h r h = r h T [ D h 1 2 D h T ( O k T O k + μ ) D h ] r h .
0 < λ 2 < min x { min { 2 ( e 1 ) ( η o + μ ) h k ( x ) , 1 o k c ( x ) * [ g ( x ) h k ( x ) * o k ( x ) ] μ h ( x ) } }
L R ( o k , h k + 1 ) L R ( o k , h k ) > 0 , for any r 0 .
η o = [ x o k ( x ) ] 2 .

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