Abstract

Data acquired with a CCD camera are modeled as an additive Poisson–Gaussian mixture, with the Poisson component representing cumulative counts of object-dependent photoelectrons, object-independent photoelectrons, bias electrons, and thermoelectrons and the Gaussian component representing readout noise. Two methods are examined for compensating for readout noise. One method relies on approximating the Gaussian readout noise by a Poisson noise and then using a modified Richardson–Lucy algorithm to effect the compensation. This method has been used for restoring images acquired with CCD’s in the original Wide-Field/Planetary Camera aboard the Hubble Space Telescope. The second method directly uses the expectation-maximization algorithm derived for the Poisson–Gaussian mixture data. This requires the determination of the conditional-mean estimate of the Poisson component of the mixture, which is accomplished by the evaluation of a nonlinear function of the data. The second method requires more computation than the first but is more accurate mathematically and yields modest improvements in the quality of the restorations, particularly for fainter objects. As a specific example, we compare the two methods in restorations of images representative of those acquired with that camera; they contain excess blurring that is due to spherical aberration and a rms readout noise level of 13 electrons.

© 1995 Optical Society of America

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References

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  1. D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
    [CrossRef] [PubMed]
  2. J. Llacer, J. Nunez, “Iterative maximum-likelihood and Bayesian algorithms for image reconstruction in astronomy,” in Restoration of Hubble Space Telescope Images, R. L. White, R. J. Allen, eds. (Space Telescope Science Institute, Baltimore, Md., 1990), pp. 62–69.
  3. D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imaging 10, 82–89 (1991).
    [CrossRef] [PubMed]
  4. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. A 62, 55–59 (1972).
    [CrossRef]
  5. L. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  6. L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–121 (1982).
    [CrossRef]
  7. W. Feller, Introduction to Probability Theory and Its Applications (Wiley, New York, 1968), pp. 190, 245.
  8. C. Helstrom, “Approximate evaluation of detection probabilities in radar and optical communications,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 630–640 (1978).
    [CrossRef]
  9. G. C. Newton, L. A. Gould, J. F. Kaiser, Analytical Design of Linear Feedback Controls (Wiley, New York, 1961), p. 239.

1993 (1)

1991 (1)

D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imaging 10, 82–89 (1991).
[CrossRef] [PubMed]

1982 (1)

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–121 (1982).
[CrossRef]

1978 (1)

C. Helstrom, “Approximate evaluation of detection probabilities in radar and optical communications,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 630–640 (1978).
[CrossRef]

1974 (1)

L. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1972 (1)

W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. A 62, 55–59 (1972).
[CrossRef]

Feller, W.

W. Feller, Introduction to Probability Theory and Its Applications (Wiley, New York, 1968), pp. 190, 245.

Gould, L. A.

G. C. Newton, L. A. Gould, J. F. Kaiser, Analytical Design of Linear Feedback Controls (Wiley, New York, 1961), p. 239.

Hammoud, A. M.

Helstrom, C.

C. Helstrom, “Approximate evaluation of detection probabilities in radar and optical communications,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 630–640 (1978).
[CrossRef]

Kaiser, J. F.

G. C. Newton, L. A. Gould, J. F. Kaiser, Analytical Design of Linear Feedback Controls (Wiley, New York, 1961), p. 239.

Llacer, J.

J. Llacer, J. Nunez, “Iterative maximum-likelihood and Bayesian algorithms for image reconstruction in astronomy,” in Restoration of Hubble Space Telescope Images, R. L. White, R. J. Allen, eds. (Space Telescope Science Institute, Baltimore, Md., 1990), pp. 62–69.

Lucy, L.

L. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Newton, G. C.

G. C. Newton, L. A. Gould, J. F. Kaiser, Analytical Design of Linear Feedback Controls (Wiley, New York, 1961), p. 239.

Nunez, J.

J. Llacer, J. Nunez, “Iterative maximum-likelihood and Bayesian algorithms for image reconstruction in astronomy,” in Restoration of Hubble Space Telescope Images, R. L. White, R. J. Allen, eds. (Space Telescope Science Institute, Baltimore, Md., 1990), pp. 62–69.

Politte, D. G.

D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imaging 10, 82–89 (1991).
[CrossRef] [PubMed]

Richardson, W. H.

W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. A 62, 55–59 (1972).
[CrossRef]

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–121 (1982).
[CrossRef]

Snyder, D. L.

D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
[CrossRef] [PubMed]

D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imaging 10, 82–89 (1991).
[CrossRef] [PubMed]

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–121 (1982).
[CrossRef]

White, R. L.

Astron. J. (1)

L. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

C. Helstrom, “Approximate evaluation of detection probabilities in radar and optical communications,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 630–640 (1978).
[CrossRef]

IEEE Trans. Med. Imaging (2)

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–121 (1982).
[CrossRef]

D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imaging 10, 82–89 (1991).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

Other (3)

J. Llacer, J. Nunez, “Iterative maximum-likelihood and Bayesian algorithms for image reconstruction in astronomy,” in Restoration of Hubble Space Telescope Images, R. L. White, R. J. Allen, eds. (Space Telescope Science Institute, Baltimore, Md., 1990), pp. 62–69.

W. Feller, Introduction to Probability Theory and Its Applications (Wiley, New York, 1968), pp. 190, 245.

G. C. Newton, L. A. Gould, J. F. Kaiser, Analytical Design of Linear Feedback Controls (Wiley, New York, 1961), p. 239.

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Figures (11)

Fig. 1
Fig. 1

(Left) Simulated star cluster and (right) simulated HST image of the star cluster.

Fig. 2
Fig. 2

Restorations of simulated HST star-cluster data with (left) the Poisson and (right) the saddle-point approximations and 50 iterations of Eq. (3).

Fig. 3
Fig. 3

Ratio of estimated to true brightness versus true brightness for simulated star-cluster data obtained with 50 iterations of Eq. (3) and (a) the Poisson and (b) the saddle-point approximations. EM refers to the expectation-maximization algorithm.

Fig. 4
Fig. 4

Smoothed ratio of estimated to true brightness versus true brightness for star-cluster data obtained from one restoration of a simulated HST star-cluster image with 50 iterations of Eq. (3) and the Poisson (dashed curve) and the saddle-point (solid curve) approximations.

Fig. 5
Fig. 5

Smoothed ratio of estimated to true brightness versus true brightness for star-cluster data obtained by the averaging of 25 restorations of the simulated HST star-cluster images with 50 iterations of Eq. (3) each and the Poisson (dashed curve) and the saddle-point (solid curve) approximations.

Fig. 6
Fig. 6

Relative bias in estimating brightness with (a) the Poisson and (b) the saddle-point approximations for a simulated star-cluster field obtained by the averaging of errors in 25 simulations with restorations performed with 50 iterations of Eq. (3) each.

Fig. 7
Fig. 7

Relative standard deviation of error in estimating brightness with (a) the Poisson and (b) the saddle-point approximations for a simulated star-cluster field obtained by the averaging of errors in 25 simulations with restorations performed with 50 iterations of Eq. (3) each.

Fig. 8
Fig. 8

(a) Mean and (b) standard deviation of error for the saddle-point (solid curves) and the Poisson (dashed curves) approximations obtained by the averaging of restorations of 25 simulated HST images of the star cluster, each restored with 50 iterations of Eq. (3).

Fig. 9
Fig. 9

Ratio of estimated to true brightness versus true brightness for simulated star-cluster data obtained with 500 iterations of Eq. (3) and (a) the Poisson and (b) the saddle-point approximations.

Fig. 10
Fig. 10

Restorations of one simulated HST star-cluster image obtained with 500 iterations of Eq. (3) and (left) the Poisson and (right) the saddle-point approximations.

Fig. 11
Fig. 11

Average of restorations of 25 simulated HST star-cluster images obtained with 500 iterations of Eq. (3) each and (left) the Poisson and (right) the saddle-point approximations.

Tables (3)

Tables Icon

Table 1 Comparison of Starting Values for Newton Iterations

Tables Icon

Table 2 Evaluations of F(·)

Tables Icon

Table 3 More Evaluations of F(·)

Equations (26)

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r ( j ) = n obj ( j ) + n 0 ( j ) + g ( j ) ,             j = 0 , 1 , , J - 1 ,
μ obj ( j ) = β ( j ) i = 0 I - 1 p ( j i ) λ ( i ) ,
{ p ( j i ) , i = 0 , 1 , , I - 1 , j = 0 , 1 , , J - 1 }
λ ^ new ( i ) = λ ^ old ( i ) 1 β ¯ ( i ) j = 0 J - 1 [ β ( j ) p ( j i ) μ ^ old ( j ) ] × F [ r ( j ) , μ ^ old ( j ) , σ ] ,
β ¯ ( i ) = j = 0 J - 1 p ( j i ) β ( j ) ,
λ ^ old ( j ) = β ( j ) i = 0 I = 1 p ( j i ) γ ^ old ( i ) + μ 0 ( j ) ,
F [ r ( j ) , μ ^ old ( j ) , σ ] = E [ n obj ( j ) + n 0 ( j ) r , λ ^ old ] = E [ r ( j ) - g ( j ) r , λ ^ old ]
F ( r , μ , σ ) = μ p ( r - m - 1 : μ , σ ) p ( r - m : μ , σ ) ,
p ( x : μ , σ ) = n = 0 1 n ! μ n exp ( - μ ) × 1 ( 2 π ) 1 / 2 σ exp [ - 1 2 σ 2 ( x - n ) 2 ] .
r ( j ) - m + σ 2 = n obj ( j ) + n 0 ( j ) + n readout ( j ) ,             j = 0 , 1 , , J - 1 ,
λ ^ new ( i ) = λ ^ old ( i ) 1 β ¯ ( i ) × j = 0 J - 1 [ β ( j ) p ( j i ) β ( j ) i = 0 I - 1 p ( j i ) λ ^ old ( i ) + μ 0 ( j ) + σ 2 ] × [ r ( j ) - m + σ 2 ] ,
F ( r , μ , σ ) μ μ + σ 2 ( r - m + σ 2 ) .
L [ p ( · : μ , σ ) ] ( s ) = - p ( x : μ , σ ) exp ( - s x ) d x = exp { μ [ exp ( - s ) - 1 ] + ½ σ 2 s 2 } ,
p ( x : μ , σ ) = 1 2 π j c - j c + j exp [ Φ ( s ) ] d s ,
Φ ( s ) = x s + ½ σ 2 s 2 + μ [ exp ( - s ) - 1 ] .
Φ ( s ˜ ) = d Φ ( s ) d s | s = s ˜ = x + σ 2 s ˜ - μ exp ( - s ˜ ) = 0.
s ˜ new = s ˜ old - Φ ( s ˜ o l d ) Φ ( s ˜ old ) ,
p ( x : μ , σ ) 1 [ 2 π Φ ( s ˜ ) ] 1 / 2 exp [ Φ ( s ˜ ) ] .
exp ( - s ˜ ) 1 - s ˜ ,
s ˜ old = μ - x μ + σ 2
exp ( - s ˜ ) 1 - ½ s ˜ 1 + ½ s ˜ .
s ˜ old = - ln { ( x - 2 σ 2 - μ ) + [ ( x - 2 σ 2 - μ ) 2 + 4 μ ( x + 2 σ 2 ) ] 1 / 2 2 μ }
s ˜ new = ln ( μ x + σ 2 s ˜ old )
F ( r , μ , σ ) = μ exp { ln [ p ( r - m - 1 : μ , σ ) ] - ln [ p ( r - m : μ , σ ) ] } ,
ln [ p ( x : μ , σ ) ] Φ ( s ˜ ) - ½ ln [ 2 π Φ ( s ˜ ) ] .
p ( x : μ , σ ) = 1 π exp ( - μ ) 0 Re ( exp { x ( s ˜ + j y ) + ½ ( s ˜ + j y ) 2 + μ exp [ - ( s ˜ + j y ) ] } ) d y .

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