Abstract

We present a new image-restoration algorithm for binary-valued imagery. A trellis-based search method is described that exploits the finite alphabet of the target imagery. This algorithm seeks the maximum-likelihood solution to the image-restoration problem and is motivated by the Viterbi algorithm for traditional binary data detection in the presence of intersymbol interference and noise. We describe a blockwise method to restore two-dimensional imagery on a row-by-row basis and in which a priori knowledge of image pixel correlation structure can be included through a modification to the trellis transition probabilities. The performance of the new Viterbi-based algorithm is shown to be superior to Wiener filtering in terms of both bit error rate and visual quality. Algorithmic choices related to trellis state configuration, complexity reduction, and transition probability selection are investigated, and various trade-offs are discussed.

© 2000 Optical Society of America

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References

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  1. M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Processing Mag. 14, 24–41 (1997).
    [CrossRef]
  2. M. I. Sezan, A. M. Tekalp, “Tutorial review of recent developments in digital image restoration,” in Visual Communications and Image Processing ’90: Fifth in a Series, M. Kunt, ed., Proc. SPIE1360, 1346–1359 (1990).
  3. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, New Jersey, 1977).
  4. P. J. Verveer, T. M. Jovin, “Improved restoration from multiple images of a single object: application to fluorescence microscopy,” Appl. Opt. 37, 6240–6246 (1998).
    [CrossRef]
  5. M. Bertero, P. Boccacci, F. Maggio, “Regularization methods in image restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).
    [CrossRef]
  6. A. Herzog, G. Krell, B. Michaelis, J. Wang, W. Zuschratter, A. K. Brain, “Restoration of three-dimensional quasi-binary images from confocal microscopy and its application to dendritic trees,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 146–157 (1997).
  7. J. C. Brailean, D. Little, M. L. Giger, C.-T. Chen, B. J. Sullivan, “Application of the EM algorithms to radiographic images,” Med. Phys. 19, 1175–1182 (1992).
    [CrossRef] [PubMed]
  8. K. Arzner, A. Magun, “Fast maximum entropy restoration of low-noise solar images,” Astron. Astrophys. 324, 735–742 (1997).
  9. E. Nezry, F. Yakam-Simen, F. Zagolski, I. Supit, “Control systems principals applied to speckle filtering and geophysical information extraction in multi-channel SAR images,” in Image Processing, Signal Processing, and Synthetic Aperture Radar for Remote Sensing, J. Desachy, S. Tajbakhsh, eds., Proc. SPIE3217, 48–57 (1997).
    [CrossRef]
  10. L. D. Marks, “Wiener-filter enhancement of noisy HREM images,” Ultramicroscopy 62, 43–52 (1996).
    [CrossRef] [PubMed]
  11. G. W. Burr, J. Ashley, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, B. Marcus, “Modulation coding for pixel-matched holographic data storage,” Opt. Lett. 22, 639–641 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. J. F. Heanue, M. C. Bashaw, L. Hesselink, “Channel codes for digital holographic data storage,” J. Opt. Soc. Am. A 12, 2432–2439 (1995).
    [CrossRef]
  14. K. M. Chugg, X. Chen, M. A. Neifeld, “Two-dimensional equalization in coherent and incoherent page-oriented optical memory,” J. Opt. Soc. Am. A 16, 549–562 (1999).
    [CrossRef]
  15. B. King, M. A. Neifeld, “Parallel detection algorithm for page-oriented optical memories,” Appl. Opt. 37, 6275–6298 (1998).
    [CrossRef]
  16. J. Heanue, K. Gurkan, L. Hesselink, “Signal detection for page-access optical memories with intersymbol interference,” Appl. Opt. 35, 2431–2438 (1996).
    [CrossRef] [PubMed]
  17. X. Chen, K. M. Chugg, M. A. Neifeld, “Near-optimal parallel distributed data detection for page-oriented optical memories,” IEEE J. Sel. Top. Quantum Electron. 4, 866–879 (1998).
    [CrossRef]
  18. C. Miller, B. R. Hunt, M. A. Neifeld, M. W. Marcellin, “Binary image reconstruction via 2D Viterbi search,” Proceedings of the IEEE International Conference on Image Processing—ICIP97 (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 181–184.
  19. K. M. Chugg, X. Chen, A. Ortega, C.-W. Cheng, “An iterative algorithm for two-dimensional digital least metric problems with applications to digital image compression,” Proceedings of the IEEE International Conference on Image Processing—ICIP98 (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 722–726.
  20. P. W. Wong, “Entropy constrained halftoning using multipath tree coding,” IEEE Trans. Image Process. 6, 1567–1579 (1997).
    [CrossRef]
  21. H. L. van Trees, “Linear estimation,” in Detection, Estimation, and Modulation Theory Pt. I (Wiley, New York, 1968), pp. 481–493.
  22. G. Forney, “Maximum likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inf. Theory IT-18, 363–378 (1972).
    [CrossRef]
  23. G. Forney, “The Viterbi algorithm,” Proc. IEEE 61, 268–278 (1973).
    [CrossRef]
  24. J. B. Anderson, S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun. COM-32, 169–176 (1984).
    [CrossRef]

1999 (2)

1998 (3)

1997 (4)

K. Arzner, A. Magun, “Fast maximum entropy restoration of low-noise solar images,” Astron. Astrophys. 324, 735–742 (1997).

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Processing Mag. 14, 24–41 (1997).
[CrossRef]

G. W. Burr, J. Ashley, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, B. Marcus, “Modulation coding for pixel-matched holographic data storage,” Opt. Lett. 22, 639–641 (1997).
[CrossRef] [PubMed]

P. W. Wong, “Entropy constrained halftoning using multipath tree coding,” IEEE Trans. Image Process. 6, 1567–1579 (1997).
[CrossRef]

1996 (2)

1995 (2)

M. Bertero, P. Boccacci, F. Maggio, “Regularization methods in image restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).
[CrossRef]

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Channel codes for digital holographic data storage,” J. Opt. Soc. Am. A 12, 2432–2439 (1995).
[CrossRef]

1992 (1)

J. C. Brailean, D. Little, M. L. Giger, C.-T. Chen, B. J. Sullivan, “Application of the EM algorithms to radiographic images,” Med. Phys. 19, 1175–1182 (1992).
[CrossRef] [PubMed]

1984 (1)

J. B. Anderson, S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun. COM-32, 169–176 (1984).
[CrossRef]

1973 (1)

G. Forney, “The Viterbi algorithm,” Proc. IEEE 61, 268–278 (1973).
[CrossRef]

1972 (1)

G. Forney, “Maximum likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inf. Theory IT-18, 363–378 (1972).
[CrossRef]

Anderson, J. B.

J. B. Anderson, S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun. COM-32, 169–176 (1984).
[CrossRef]

Andrews, H. C.

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

Arzner, K.

K. Arzner, A. Magun, “Fast maximum entropy restoration of low-noise solar images,” Astron. Astrophys. 324, 735–742 (1997).

Ashley, J.

Banham, M. R.

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Processing Mag. 14, 24–41 (1997).
[CrossRef]

Bashaw, M. C.

Bertero, M.

M. Bertero, P. Boccacci, F. Maggio, “Regularization methods in image restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, F. Maggio, “Regularization methods in image restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).
[CrossRef]

Brailean, J. C.

J. C. Brailean, D. Little, M. L. Giger, C.-T. Chen, B. J. Sullivan, “Application of the EM algorithms to radiographic images,” Med. Phys. 19, 1175–1182 (1992).
[CrossRef] [PubMed]

Brain, A. K.

A. Herzog, G. Krell, B. Michaelis, J. Wang, W. Zuschratter, A. K. Brain, “Restoration of three-dimensional quasi-binary images from confocal microscopy and its application to dendritic trees,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 146–157 (1997).

Burr, G. W.

Chen, C.-T.

J. C. Brailean, D. Little, M. L. Giger, C.-T. Chen, B. J. Sullivan, “Application of the EM algorithms to radiographic images,” Med. Phys. 19, 1175–1182 (1992).
[CrossRef] [PubMed]

Chen, X.

K. M. Chugg, X. Chen, M. A. Neifeld, “Two-dimensional equalization in coherent and incoherent page-oriented optical memory,” J. Opt. Soc. Am. A 16, 549–562 (1999).
[CrossRef]

X. Chen, K. M. Chugg, M. A. Neifeld, “Near-optimal parallel distributed data detection for page-oriented optical memories,” IEEE J. Sel. Top. Quantum Electron. 4, 866–879 (1998).
[CrossRef]

K. M. Chugg, X. Chen, A. Ortega, C.-W. Cheng, “An iterative algorithm for two-dimensional digital least metric problems with applications to digital image compression,” Proceedings of the IEEE International Conference on Image Processing—ICIP98 (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 722–726.

Cheng, C.-W.

K. M. Chugg, X. Chen, A. Ortega, C.-W. Cheng, “An iterative algorithm for two-dimensional digital least metric problems with applications to digital image compression,” Proceedings of the IEEE International Conference on Image Processing—ICIP98 (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 722–726.

Chugg, K. M.

K. M. Chugg, X. Chen, M. A. Neifeld, “Two-dimensional equalization in coherent and incoherent page-oriented optical memory,” J. Opt. Soc. Am. A 16, 549–562 (1999).
[CrossRef]

X. Chen, K. M. Chugg, M. A. Neifeld, “Near-optimal parallel distributed data detection for page-oriented optical memories,” IEEE J. Sel. Top. Quantum Electron. 4, 866–879 (1998).
[CrossRef]

K. M. Chugg, X. Chen, A. Ortega, C.-W. Cheng, “An iterative algorithm for two-dimensional digital least metric problems with applications to digital image compression,” Proceedings of the IEEE International Conference on Image Processing—ICIP98 (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 722–726.

Coufal, H.

Forney, G.

G. Forney, “The Viterbi algorithm,” Proc. IEEE 61, 268–278 (1973).
[CrossRef]

G. Forney, “Maximum likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inf. Theory IT-18, 363–378 (1972).
[CrossRef]

Giger, M. L.

J. C. Brailean, D. Little, M. L. Giger, C.-T. Chen, B. J. Sullivan, “Application of the EM algorithms to radiographic images,” Med. Phys. 19, 1175–1182 (1992).
[CrossRef] [PubMed]

Grygier, R. K.

Gurkan, K.

Heanue, J.

Heanue, J. F.

Herzog, A.

A. Herzog, G. Krell, B. Michaelis, J. Wang, W. Zuschratter, A. K. Brain, “Restoration of three-dimensional quasi-binary images from confocal microscopy and its application to dendritic trees,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 146–157 (1997).

Hesselink, L.

Hoffnagle, J. A.

Hunt, B. R.

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

C. Miller, B. R. Hunt, M. A. Neifeld, M. W. Marcellin, “Binary image reconstruction via 2D Viterbi search,” Proceedings of the IEEE International Conference on Image Processing—ICIP97 (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 181–184.

Jefferson, C. M.

Jovin, T. M.

Katsaggelos, A. K.

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Processing Mag. 14, 24–41 (1997).
[CrossRef]

Keskinoz, M.

King, B.

Krell, G.

A. Herzog, G. Krell, B. Michaelis, J. Wang, W. Zuschratter, A. K. Brain, “Restoration of three-dimensional quasi-binary images from confocal microscopy and its application to dendritic trees,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 146–157 (1997).

Little, D.

J. C. Brailean, D. Little, M. L. Giger, C.-T. Chen, B. J. Sullivan, “Application of the EM algorithms to radiographic images,” Med. Phys. 19, 1175–1182 (1992).
[CrossRef] [PubMed]

Maggio, F.

M. Bertero, P. Boccacci, F. Maggio, “Regularization methods in image restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).
[CrossRef]

Magun, A.

K. Arzner, A. Magun, “Fast maximum entropy restoration of low-noise solar images,” Astron. Astrophys. 324, 735–742 (1997).

Marcellin, M. W.

C. Miller, B. R. Hunt, M. A. Neifeld, M. W. Marcellin, “Binary image reconstruction via 2D Viterbi search,” Proceedings of the IEEE International Conference on Image Processing—ICIP97 (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 181–184.

Marcus, B.

Marks, L. D.

L. D. Marks, “Wiener-filter enhancement of noisy HREM images,” Ultramicroscopy 62, 43–52 (1996).
[CrossRef] [PubMed]

Michaelis, B.

A. Herzog, G. Krell, B. Michaelis, J. Wang, W. Zuschratter, A. K. Brain, “Restoration of three-dimensional quasi-binary images from confocal microscopy and its application to dendritic trees,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 146–157 (1997).

Miller, C.

C. Miller, B. R. Hunt, M. A. Neifeld, M. W. Marcellin, “Binary image reconstruction via 2D Viterbi search,” Proceedings of the IEEE International Conference on Image Processing—ICIP97 (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 181–184.

Mohan, S.

J. B. Anderson, S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun. COM-32, 169–176 (1984).
[CrossRef]

Neifeld, M. A.

K. M. Chugg, X. Chen, M. A. Neifeld, “Two-dimensional equalization in coherent and incoherent page-oriented optical memory,” J. Opt. Soc. Am. A 16, 549–562 (1999).
[CrossRef]

B. King, M. A. Neifeld, “Parallel detection algorithm for page-oriented optical memories,” Appl. Opt. 37, 6275–6298 (1998).
[CrossRef]

X. Chen, K. M. Chugg, M. A. Neifeld, “Near-optimal parallel distributed data detection for page-oriented optical memories,” IEEE J. Sel. Top. Quantum Electron. 4, 866–879 (1998).
[CrossRef]

C. Miller, B. R. Hunt, M. A. Neifeld, M. W. Marcellin, “Binary image reconstruction via 2D Viterbi search,” Proceedings of the IEEE International Conference on Image Processing—ICIP97 (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 181–184.

Nezry, E.

E. Nezry, F. Yakam-Simen, F. Zagolski, I. Supit, “Control systems principals applied to speckle filtering and geophysical information extraction in multi-channel SAR images,” in Image Processing, Signal Processing, and Synthetic Aperture Radar for Remote Sensing, J. Desachy, S. Tajbakhsh, eds., Proc. SPIE3217, 48–57 (1997).
[CrossRef]

Ortega, A.

K. M. Chugg, X. Chen, A. Ortega, C.-W. Cheng, “An iterative algorithm for two-dimensional digital least metric problems with applications to digital image compression,” Proceedings of the IEEE International Conference on Image Processing—ICIP98 (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 722–726.

Sezan, M. I.

M. I. Sezan, A. M. Tekalp, “Tutorial review of recent developments in digital image restoration,” in Visual Communications and Image Processing ’90: Fifth in a Series, M. Kunt, ed., Proc. SPIE1360, 1346–1359 (1990).

Sullivan, B. J.

J. C. Brailean, D. Little, M. L. Giger, C.-T. Chen, B. J. Sullivan, “Application of the EM algorithms to radiographic images,” Med. Phys. 19, 1175–1182 (1992).
[CrossRef] [PubMed]

Supit, I.

E. Nezry, F. Yakam-Simen, F. Zagolski, I. Supit, “Control systems principals applied to speckle filtering and geophysical information extraction in multi-channel SAR images,” in Image Processing, Signal Processing, and Synthetic Aperture Radar for Remote Sensing, J. Desachy, S. Tajbakhsh, eds., Proc. SPIE3217, 48–57 (1997).
[CrossRef]

Tekalp, A. M.

M. I. Sezan, A. M. Tekalp, “Tutorial review of recent developments in digital image restoration,” in Visual Communications and Image Processing ’90: Fifth in a Series, M. Kunt, ed., Proc. SPIE1360, 1346–1359 (1990).

van Trees, H. L.

H. L. van Trees, “Linear estimation,” in Detection, Estimation, and Modulation Theory Pt. I (Wiley, New York, 1968), pp. 481–493.

Verveer, P. J.

Vijaya Kumar, B. V. K.

Wang, J.

A. Herzog, G. Krell, B. Michaelis, J. Wang, W. Zuschratter, A. K. Brain, “Restoration of three-dimensional quasi-binary images from confocal microscopy and its application to dendritic trees,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 146–157 (1997).

Wong, P. W.

P. W. Wong, “Entropy constrained halftoning using multipath tree coding,” IEEE Trans. Image Process. 6, 1567–1579 (1997).
[CrossRef]

Yakam-Simen, F.

E. Nezry, F. Yakam-Simen, F. Zagolski, I. Supit, “Control systems principals applied to speckle filtering and geophysical information extraction in multi-channel SAR images,” in Image Processing, Signal Processing, and Synthetic Aperture Radar for Remote Sensing, J. Desachy, S. Tajbakhsh, eds., Proc. SPIE3217, 48–57 (1997).
[CrossRef]

Zagolski, F.

E. Nezry, F. Yakam-Simen, F. Zagolski, I. Supit, “Control systems principals applied to speckle filtering and geophysical information extraction in multi-channel SAR images,” in Image Processing, Signal Processing, and Synthetic Aperture Radar for Remote Sensing, J. Desachy, S. Tajbakhsh, eds., Proc. SPIE3217, 48–57 (1997).
[CrossRef]

Zuschratter, W.

A. Herzog, G. Krell, B. Michaelis, J. Wang, W. Zuschratter, A. K. Brain, “Restoration of three-dimensional quasi-binary images from confocal microscopy and its application to dendritic trees,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 146–157 (1997).

Appl. Opt. (4)

Astron. Astrophys. (1)

K. Arzner, A. Magun, “Fast maximum entropy restoration of low-noise solar images,” Astron. Astrophys. 324, 735–742 (1997).

IEEE J. Sel. Top. Quantum Electron. (1)

X. Chen, K. M. Chugg, M. A. Neifeld, “Near-optimal parallel distributed data detection for page-oriented optical memories,” IEEE J. Sel. Top. Quantum Electron. 4, 866–879 (1998).
[CrossRef]

IEEE Signal Processing Mag. (1)

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Processing Mag. 14, 24–41 (1997).
[CrossRef]

IEEE Trans. Commun. (1)

J. B. Anderson, S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun. COM-32, 169–176 (1984).
[CrossRef]

IEEE Trans. Image Process. (1)

P. W. Wong, “Entropy constrained halftoning using multipath tree coding,” IEEE Trans. Image Process. 6, 1567–1579 (1997).
[CrossRef]

IEEE Trans. Inf. Theory (1)

G. Forney, “Maximum likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inf. Theory IT-18, 363–378 (1972).
[CrossRef]

Int. J. Imaging Syst. Technol. (1)

M. Bertero, P. Boccacci, F. Maggio, “Regularization methods in image restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).
[CrossRef]

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

Med. Phys. (1)

J. C. Brailean, D. Little, M. L. Giger, C.-T. Chen, B. J. Sullivan, “Application of the EM algorithms to radiographic images,” Med. Phys. 19, 1175–1182 (1992).
[CrossRef] [PubMed]

Opt. Lett. (1)

Proc. IEEE (1)

G. Forney, “The Viterbi algorithm,” Proc. IEEE 61, 268–278 (1973).
[CrossRef]

Ultramicroscopy (1)

L. D. Marks, “Wiener-filter enhancement of noisy HREM images,” Ultramicroscopy 62, 43–52 (1996).
[CrossRef] [PubMed]

Other (7)

E. Nezry, F. Yakam-Simen, F. Zagolski, I. Supit, “Control systems principals applied to speckle filtering and geophysical information extraction in multi-channel SAR images,” in Image Processing, Signal Processing, and Synthetic Aperture Radar for Remote Sensing, J. Desachy, S. Tajbakhsh, eds., Proc. SPIE3217, 48–57 (1997).
[CrossRef]

A. Herzog, G. Krell, B. Michaelis, J. Wang, W. Zuschratter, A. K. Brain, “Restoration of three-dimensional quasi-binary images from confocal microscopy and its application to dendritic trees,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 146–157 (1997).

M. I. Sezan, A. M. Tekalp, “Tutorial review of recent developments in digital image restoration,” in Visual Communications and Image Processing ’90: Fifth in a Series, M. Kunt, ed., Proc. SPIE1360, 1346–1359 (1990).

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

H. L. van Trees, “Linear estimation,” in Detection, Estimation, and Modulation Theory Pt. I (Wiley, New York, 1968), pp. 481–493.

C. Miller, B. R. Hunt, M. A. Neifeld, M. W. Marcellin, “Binary image reconstruction via 2D Viterbi search,” Proceedings of the IEEE International Conference on Image Processing—ICIP97 (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 181–184.

K. M. Chugg, X. Chen, A. Ortega, C.-W. Cheng, “An iterative algorithm for two-dimensional digital least metric problems with applications to digital image compression,” Proceedings of the IEEE International Conference on Image Processing—ICIP98 (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 2, pp. 722–726.

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Figures (7)

Fig. 1
Fig. 1

Original binary-valued object used as the benchmark image throughout this paper. Three different font sizes are used to test recovery of image detail for various algorithms.

Fig. 2
Fig. 2

Schematic representation of the Viterbi algorithm applied to image restoration. (a) Columnwise symbols are used to convert a 2D image to a 1D sequence of columns. (b) Viterbi trellis associated with a two-row example of columnwise restoration. Branches in the trellis represent permitted transitions between states in the top row and those in the bottom row. (c) Blockwise states with n 1 × n 2 = 5 × 3 pixels together with decision feedback facilitate reduction in restoration complexity. Boundary rows initialize the decision-feedback algorithm. (d) Blockwise Viterbi configuration for an arbitrary image row. Previously restored rows support the decision-feedback algorithm.

Fig. 3
Fig. 3

BER versus SNR for three restoration algorithms operating on the mild blur channel. This channel uses W = 1.5, indicating that the sinc2 optical system PSF has a main lobe whose half-width is 1.5 times the pixel pitch.

Fig. 4
Fig. 4

Images depicting the visual quality of various image-restoration results. (a) Measured image after corruption by W = 2.5 channel and noise σ = 0.08. (b) Restored image by means of the threshold algorithm. An optimal threshold is used. (c) Restored image by means of the Wiener filter followed by an optimal threshold. (d) Restored image by means of the Viterbi algorithm with n 1 × n 2 = 5 × 2 and M = 50.

Fig. 5
Fig. 5

BER versus Viterbi state configuration n 1 × n 2 for three different values of the complexity reduction parameter M. These data were produced with the W = 1.5 channel with noise σ = 0.08.

Fig. 6
Fig. 6

BER versus SNR for five different methods of obtaining the a priori local pixel correlation structure { ij }. The W = 1.5 channel was used to obtain these data.

Fig. 7
Fig. 7

Images depicting the visual quality of Viterbi restoration for five different methods of obtaining the a priori local pixel correlation structure { ij }. (a) { ij } estimated with font 1 only. (b) { ij } estimated with font 2 only. (c) { ij } estimated with font 3 only. (d) Restoration with the equal priors assumption.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

hi, j=-Δ/2Δ/2-Δ/2Δ/2sinc2x-iΔ/W×sinc2y-jΔ/Wdxdy,
si, j=ni, j+l=-LLm=-LL ai-l, j-mhl, m,
cijl=p,qNsp, q-tjp, q2,
psˆ¯|s¯=fs¯|sˆ¯psˆ¯fs¯,
maxsˆ¯12πσexp-sˆ¯-s¯2/2σ2psˆ¯.
minsˆ¯sˆ¯-s¯2-2σ2 ln psˆ¯,
aˆ¯=arg mina¯a¯*h¯-s¯2-2σ2 ln pa¯,
cijl=p,qNsp, q-tjp, q2-2σ2 ln pij.

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