Abstract

An iterative method of reconstructing degraded images is developed from consideration of a mixed-noise imaging situation. Both photon noise in the image itself and postdetection Gaussian noise are combined by use of the standard maximum-likelihood method to produce a mixed-expectation reconstruction technique that demonstrates good performance in the presence of both noise sources. The new algorithm is evaluated through computer simulations.

© 1999 Optical Society of America

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References

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  1. M. R. Descour, C. E. Volin, T. M. Gleeson, E. L. Dereniak, M. F. Hopkins, D. W. Wilson, P. D. Maker, “Demonstration of a computed-tomography imaging spectrometer using a computer-generated hologram disperser,” Appl. Opt. 36, 3694–3698 (1997).
    [CrossRef] [PubMed]
  2. T. R. Miller, J. W. Wallis, “Clinically important characteristics of maximum-likelihood reconstruction,” J. Nucl. Med. 33, 1678–1684 (1992).
    [PubMed]
  3. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
    [CrossRef]
  4. A. Lent, “A convergent algorithm for maximum entropy image restoration,” in Image Analysis and Evaluation, Proceedings of the SPSE Conference, 1976, R. Shaw, ed. (Society for Photographic Scientists and Engineers, Washington, D.C., 1976), pp. 249–257.
  5. B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer-Verlag, New York, 1983), p. 51.
  6. H. H. Barrett, “Image reconstruction and the solution of inverse problems in medical imaging,” in The Formation, Handling, and Evaluation of Medical Images, A. Todd-Pokropek, M. A. Viergever, eds. (Springer-Verlag, Berlin, 1991), pp. 33–39.
  7. M. R. Descour, “Non-scanning imaging spectrometry,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1994), pp. 65–68.

1997

1992

T. R. Miller, J. W. Wallis, “Clinically important characteristics of maximum-likelihood reconstruction,” J. Nucl. Med. 33, 1678–1684 (1992).
[PubMed]

1982

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Barrett, H. H.

H. H. Barrett, “Image reconstruction and the solution of inverse problems in medical imaging,” in The Formation, Handling, and Evaluation of Medical Images, A. Todd-Pokropek, M. A. Viergever, eds. (Springer-Verlag, Berlin, 1991), pp. 33–39.

Dereniak, E. L.

Descour, M. R.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer-Verlag, New York, 1983), p. 51.

Gleeson, T. M.

Hopkins, M. F.

Lent, A.

A. Lent, “A convergent algorithm for maximum entropy image restoration,” in Image Analysis and Evaluation, Proceedings of the SPSE Conference, 1976, R. Shaw, ed. (Society for Photographic Scientists and Engineers, Washington, D.C., 1976), pp. 249–257.

Maker, P. D.

Miller, T. R.

T. R. Miller, J. W. Wallis, “Clinically important characteristics of maximum-likelihood reconstruction,” J. Nucl. Med. 33, 1678–1684 (1992).
[PubMed]

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Volin, C. E.

Wallis, J. W.

T. R. Miller, J. W. Wallis, “Clinically important characteristics of maximum-likelihood reconstruction,” J. Nucl. Med. 33, 1678–1684 (1992).
[PubMed]

Wilson, D. W.

Appl. Opt.

IEEE Trans. Med. Imaging

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

J. Nucl. Med.

T. R. Miller, J. W. Wallis, “Clinically important characteristics of maximum-likelihood reconstruction,” J. Nucl. Med. 33, 1678–1684 (1992).
[PubMed]

Other

A. Lent, “A convergent algorithm for maximum entropy image restoration,” in Image Analysis and Evaluation, Proceedings of the SPSE Conference, 1976, R. Shaw, ed. (Society for Photographic Scientists and Engineers, Washington, D.C., 1976), pp. 249–257.

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer-Verlag, New York, 1983), p. 51.

H. H. Barrett, “Image reconstruction and the solution of inverse problems in medical imaging,” in The Formation, Handling, and Evaluation of Medical Images, A. Todd-Pokropek, M. A. Viergever, eds. (Springer-Verlag, Berlin, 1991), pp. 33–39.

M. R. Descour, “Non-scanning imaging spectrometry,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1994), pp. 65–68.

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Figures (6)

Fig. 1
Fig. 1

Block diagram of an imaging system. We assume that the noiseless object vector f acquires Poisson statistics by means of n1 to become a vector of random variables after passing through the system represented by the matrix H. Postdetection noise n2 is then added to create the measurement g.

Fig. 2
Fig. 2

Noiseless object corresponding to the vector f.

Fig. 3
Fig. 3

Noisy and blurred image corresponding to the measurement vector g.

Fig. 4
Fig. 4

Second, fourth, sixth, and eighth iterations (top to bottom) of MERT. The first iteration is the uniform initial estimate.

Fig. 5
Fig. 5

Error vector f̂ - f plotted versus the element index (1–289) for the second, fourth, sixth, and eighth iterations of MERT. The maximum level in object f is 50.

Fig. 6
Fig. 6

Average error per pixel for the MERT plotted versus the iteration number. Iteration 1 is the uniform initial estimate of the object.

Equations (13)

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g=Hf+n1+n2.
Pgm|f=12πσm21/2exp-gm-Hfm22σm2,
σm2=σs2+Hfm.
Pg|f=m=1M12πσm21/2exp-gm-Hfm22σm2,
lnPg|fˆ=m=1M-12ln2πHfˆm+σs2-gm-Hfˆm22Hfˆm+σs2,
Pg|fˆ=maximum,  at fˆ=fˆML,
lnPg|fˆ=maximum,  at fˆ=fˆML.
lnPg|fˆm=1M-gm-Hfˆm22Hfˆm+σs2
lnPg|fˆfˆn=0  for all n.
Hfˆmfˆn=i Hmifˆifˆn=Hmn,
m=1MHfˆm2+2Hfˆmσs2HmnHfˆm+σs22=m=1Mgm2+2gmσs2HmnHfˆm+σs22.
fˆnk+1=fˆnkm=1Mgm2+2gmσs2HmnHfˆkm+σs22m=1MHfˆkm2+2Hfˆkmσs2HmnHfˆkm+σs22.
Error=1289j=1289 |fˆj-fj|.

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