Abstract

An alternative approach to the maximum-likelihood solution of deconvolution problems is presented. The resulting algorithms are faster converging than the conventional Richardson–Lucy and clean algorithms, as well as being more flexible when one is dealing with different types of noise. The performance of the algorithms on both Poisson and independent sensor noise is quantified.

© 1996 Optical Society of America

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References

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  1. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
  2. B. Lucy, “An iterative method for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
  3. J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 415–426 (1974).
  4. K. A. Marsh, J. M. Richardson, “The objective function implicit in the clean algorithm,” Astron. Astrophys. 182, 174–178 (1987).
  5. R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
  6. E. Thiébaut, J.-M. Conan, “Strict a prioriconstraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
  7. T. J. Holmes, “Blind deconvolution of quantum-limited imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
  8. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
  9. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).
  10. U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (method clean),” Astron. Astrophys. 65, 345–356 (1978).
  11. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
  12. D. S. C. Biggs, M. Andrews, “Conjugate gradient acceleration of maximum likelihood image reconstruction,” Electron. Lett. 31, 1985–1986 (1995).

1995 (2)

E. Thiébaut, J.-M. Conan, “Strict a prioriconstraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).

D. S. C. Biggs, M. Andrews, “Conjugate gradient acceleration of maximum likelihood image reconstruction,” Electron. Lett. 31, 1985–1986 (1995).

1993 (1)

1992 (3)

1987 (1)

K. A. Marsh, J. M. Richardson, “The objective function implicit in the clean algorithm,” Astron. Astrophys. 182, 174–178 (1987).

1978 (1)

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (method clean),” Astron. Astrophys. 65, 345–356 (1978).

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

1974 (2)

B. Lucy, “An iterative method for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 415–426 (1974).

1972 (1)

Andrews, M.

D. S. C. Biggs, M. Andrews, “Conjugate gradient acceleration of maximum likelihood image reconstruction,” Electron. Lett. 31, 1985–1986 (1995).

Biggs, D. S. C.

D. S. C. Biggs, M. Andrews, “Conjugate gradient acceleration of maximum likelihood image reconstruction,” Electron. Lett. 31, 1985–1986 (1995).

Conan, J.-M.

Dainty, J. C.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

Glindemann, A.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).

Högbom, J. A.

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 415–426 (1974).

Holmes, T. J.

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

Lane, R. G.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).

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

Lucy, B.

B. Lucy, “An iterative method for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).

Marsh, K. A.

K. A. Marsh, J. M. Richardson, “The objective function implicit in the clean algorithm,” Astron. Astrophys. 182, 174–178 (1987).

Richardson, J. M.

K. A. Marsh, J. M. Richardson, “The objective function implicit in the clean algorithm,” Astron. Astrophys. 182, 174–178 (1987).

Richardson, W. H.

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

Schulz, T. J.

Schwarz, U. J.

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (method clean),” Astron. Astrophys. 65, 345–356 (1978).

Thiébaut, E.

Astron. Astrophys. (3)

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 415–426 (1974).

K. A. Marsh, J. M. Richardson, “The objective function implicit in the clean algorithm,” Astron. Astrophys. 182, 174–178 (1987).

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (method clean),” Astron. Astrophys. 65, 345–356 (1978).

Astron. J. (1)

B. Lucy, “An iterative method for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).

Electron. Lett. (1)

D. S. C. Biggs, M. Andrews, “Conjugate gradient acceleration of maximum likelihood image reconstruction,” Electron. Lett. 31, 1985–1986 (1995).

J. Opt. Soc. Am. (1)

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

J. R. Stat. Soc. (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. 39, 1–38 (1977).

Waves Random Media (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).

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Figures (8)

Fig. 1
Fig. 1

(a) Test image used for comparison of the proposed algorithms. Total photon count was assumed to equal 10,000. (b) Distorted speckle image formed by convolving Fig. 1(a) with a point-spread function resulting from atmospheric turbulence with D/r0 = 10.

Fig. 2
Fig. 2

Noise-corrupted images of Fig. 1(b). (a) Image-dependent Poisson noise only, (b) independent Poisson noise modeled as a Poisson process with a mean of ten events per pixel, (c) image-dependent Poisson noise and independent Poisson noise with a mean of 10 events per pixel.

Fig. 3
Fig. 3

Reconstructions by use of ML1 of (a) Fig. 2(a), (b) Fig. 2(b), and (c) Fig. 2(c).

Fig. 4
Fig. 4

Reconstructions by use of ML2 of (a) Fig. 2(a), (b) Fig. 2(b), and (c) Fig. 2(c).

Fig. 5
Fig. 5

Reconstructions by use of ML3 of (a) Fig. 2(a), (b) Fig. 2(b), and (c) Fig. 2(c).

Fig. 6
Fig. 6

Estimated convolution ĝ(x, y) corresponding to (a) Fig. 3(b), (b) Fig. 4(b), and (c) Fig. 4(c).

Fig. 7
Fig. 7

Comparison of the convergence of ML1 against the conventional Richardson–Lucy algorithm. Results averaged over 100 different random noise processes. (a) Convergence of the log likelihood. Upper trace, ML1; lower trace, conventional Richardson–Lucy algorithm. (b) Convergence of the true error E as a function of iteration. Error bars show ±3σ. Upper trace, conventional Richardson–Lucy algorithm; lower trace, ML1.

Fig. 8
Fig. 8

Reconstruction of Fig. 2(a) by use of the Richardson–Lucy algorithm.

Tables (1)

Tables Icon

Table 1 Performance of the Modified Richardson–Lucy Algorithm for Reconstructions of the Images Shown in Fig. 2a

Equations (18)

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f ( x , y ) F ( u , v ) = - + - + f ( x , y ) exp [ - i 2 π ( u v + v y ) ] d x d y ,
g ( x , y ) = f ( x , y ) h ( x , y ) F ( u , v ) H ( u , v ) ,
d ( x , y ) = g ( x , y ) + n ( x , y ) ,
f ^ max Pr [ d ( x , y ) f ^ ( x , y ) ] ,
Pr [ d ( x , y ) f ^ ( x , y ) ] = x , y g ^ ( x , y ) d ( x , y ) exp [ - g ^ ( x , y ) ] d ( x , y ) ! ,
x , y d ( x , y ) log [ g ^ ( x , y ) ] - g ^ ( x , y ) ,
f ^ k + 1 ( x , y ) = f ^ k ( x , y ) [ d ( x , y ) g ^ k ( x , y ) * h ( x , y ) ] ,
f 0 ( x , y ) = 0 f k ( x , y ) = 0 k .
f 0 ( x , y ) > 0 f k ( x , y ) > 0 k .
log Pr [ d ( x , y ) f ^ ( x , y ) ] f ^ ( x , y ) = n ^ ( x , y ) g ^ ( x , y ) * h ( x , y ) ,
f ^ ( x , y ) = ψ ^ 2 ( x , y ) .
log Pr [ d ( x , y ) ψ ^ ( x , y ) ] ψ ^ ( x , y ) = n ^ ( x , y ) g ^ ( x , y ) * h ( x , y ) 2 × ψ ^ ( x , y ) .
x , y d ( x , y ) log [ g ^ ( x , y ) + ζ ] - g ^ ( x , y ) - ζ ,
log Pr [ d ( x , y ) ψ ^ ( x , y ) ] ψ ^ ( x , y ) = [ d ( x , y ) - g ^ ( x , y ) - ζ g ^ ( x , y ) + ζ ] * h ( x , y ) 2 × ψ ^ ( x , y ) .
x , y - [ d ( x , y ) - g ^ ( x , y ) ] 2 2 σ 2 ,
log Pr [ d ( x , y ) / ψ ^ ( x , y ) ] + ψ ^ ( x , y ) = n ^ ( x , y ) σ 2 * h ( x , y ) 2 × ψ ^ ( x , y ) .
E = 1 100 n = 1 100 x , y [ f ^ ( x , y ) - f ( x , y ) ] 2 x , y f ( x , y ) 2
log P r [ d ( x , y ) / f ^ ( x , y ) ] f ^ ( x , y ) = n ^ ( x , y ) σ 2 * h ( x , y ) .

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