Abstract

We describe a pixelwise parallel algorithm for the restoration of images that have been corrupted by a low-pass optical channel and additive noise. This new algorithm is based on an iterative soft-decision method of error correction (i.e., turbo decoding) and offers performance on binary-valued imagery that is comparable to the Viterbi algorithm. We quantify the restoration performance of this new algorithm on random binary imagery for which it is superior to both the Wiener filter and the projection onto convex sets algorithms over a wide range of channels. For typical optical channels, the new algorithm is within 0.5 dB of the two-dimensional Viterbi restoration method [J. Opt. Soc. Am. A 17, 265 (2000)]. We also demonstrate the extension of our new algorithm to correlated and gray-scale images using vector quantization to mitigate the associated complexity burden. A highly parallel focal-plane implementation is also discussed, and a design study is presented to quantify the capabilities of such a VLSI hardware solution. We find that video-rate restoration on 252 × 252 pixel images is possible using current technology.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. C. Andrew, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  2. M. I. Sezan, A. M. Tekalp, “Tutorial review of recent development in digital image restoration,” in Visual Communications and Image Processing 1990, M. Kunt, ed., Proc. SPIE1360, 1346–1359 (1990).
  3. C. Miller, B. R. Hunt, M. W. Marcellin, M. A. Neifeld, “Image restoration with the Viterbi algorithm,” J. Opt. Soc. Am. A 17, 265–275 (2000).
    [Crossref]
  4. M. A. Neifeld, R. Xuan, M. W. Marcellin, “Communication theoretic image restoration for binary-valued imagery,” Appl. Opt. 39, 269–276 (2000).
    [Crossref]
  5. J. F. Heanue, K. Gurkan, L. Hesselink, “Signal detection for page-access optical memories with intersymbol interference,” Appl. Opt. 35, 2431–2438 (1996).
    [Crossref] [PubMed]
  6. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).
  7. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [Crossref]
  8. C. Berrou, A. Glavieux, P. Thitimajishima, “Near Shannon limit error-correction coding and decoding: turbo-codes (1),” in Proceedings of the IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1064–1070.
  9. J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 409–445 (1996).
    [Crossref]
  10. 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]
  11. W.-C. Chou, M. A. Neifeld, “Soft-decision array decoding for volume holographic memory systems,” J. Opt. Soc. Am. A 18, 185–194 (2001).
    [Crossref]
  12. Ref. 6, pp. 359–361.
  13. R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [Crossref]
  14. Y. Linde, A. Buzo, R. M. Gray, “An algorithm for vector quantization design,” IEEE Trans. Commun. T-COM-28, 84–95 (1980).
    [Crossref]
  15. G. Cauwenberghs, “A micropower CMOS algorithm A/D/A converter,” IEEE Trans. Circuits Syst. I 42, 913–919 (1995).
    [Crossref]

2001 (1)

2000 (2)

1998 (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]

1996 (2)

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 409–445 (1996).
[Crossref]

J. F. Heanue, K. Gurkan, L. Hesselink, “Signal detection for page-access optical memories with intersymbol interference,” Appl. Opt. 35, 2431–2438 (1996).
[Crossref] [PubMed]

1995 (1)

G. Cauwenberghs, “A micropower CMOS algorithm A/D/A converter,” IEEE Trans. Circuits Syst. I 42, 913–919 (1995).
[Crossref]

1982 (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[Crossref]

1980 (1)

Y. Linde, A. Buzo, R. M. Gray, “An algorithm for vector quantization design,” IEEE Trans. Commun. T-COM-28, 84–95 (1980).
[Crossref]

1974 (1)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Andrew, H. C.

H. C. Andrew, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Berrou, C.

C. Berrou, A. Glavieux, P. Thitimajishima, “Near Shannon limit error-correction coding and decoding: turbo-codes (1),” in Proceedings of the IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1064–1070.

Buzo, A.

Y. Linde, A. Buzo, R. M. Gray, “An algorithm for vector quantization design,” IEEE Trans. Commun. T-COM-28, 84–95 (1980).
[Crossref]

Cauwenberghs, G.

G. Cauwenberghs, “A micropower CMOS algorithm A/D/A converter,” IEEE Trans. Circuits Syst. I 42, 913–919 (1995).
[Crossref]

Chen, X.

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]

Chou, W.-C.

Chugg, K. M.

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]

Gerchberg, R. W.

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Glavieux, A.

C. Berrou, A. Glavieux, P. Thitimajishima, “Near Shannon limit error-correction coding and decoding: turbo-codes (1),” in Proceedings of the IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1064–1070.

Gray, R. M.

Y. Linde, A. Buzo, R. M. Gray, “An algorithm for vector quantization design,” IEEE Trans. Commun. T-COM-28, 84–95 (1980).
[Crossref]

Gurkan, K.

Hagenauer, J.

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 409–445 (1996).
[Crossref]

Heanue, J. F.

Hesselink, L.

Hunt, B. R.

Lim, J. S.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).

Linde, Y.

Y. Linde, A. Buzo, R. M. Gray, “An algorithm for vector quantization design,” IEEE Trans. Commun. T-COM-28, 84–95 (1980).
[Crossref]

Marcellin, M. W.

Miller, C.

Neifeld, M. A.

Offer, E.

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 409–445 (1996).
[Crossref]

Papke, L.

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 409–445 (1996).
[Crossref]

Sezan, M. I.

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

Tekalp, A. M.

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

Thitimajishima, P.

C. Berrou, A. Glavieux, P. Thitimajishima, “Near Shannon limit error-correction coding and decoding: turbo-codes (1),” in Proceedings of the IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1064–1070.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[Crossref]

Xuan, R.

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[Crossref]

Appl. Opt. (2)

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 Trans. Circuits Syst. I (1)

G. Cauwenberghs, “A micropower CMOS algorithm A/D/A converter,” IEEE Trans. Circuits Syst. I 42, 913–919 (1995).
[Crossref]

IEEE Trans. Commun. (1)

Y. Linde, A. Buzo, R. M. Gray, “An algorithm for vector quantization design,” IEEE Trans. Commun. T-COM-28, 84–95 (1980).
[Crossref]

IEEE Trans. Inf. Theory (1)

J. Hagenauer, E. Offer, L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory 42, 409–445 (1996).
[Crossref]

IEEE Trans. Med. Imaging (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[Crossref]

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

Opt. Acta (1)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Other (5)

Ref. 6, pp. 359–361.

C. Berrou, A. Glavieux, P. Thitimajishima, “Near Shannon limit error-correction coding and decoding: turbo-codes (1),” in Proceedings of the IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1064–1070.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).

H. C. Andrew, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Canonical incoherent optical imaging system.

Fig. 2
Fig. 2

Benchmark binary-valued object.

Fig. 3
Fig. 3

L = 3 example illustrating the definitions of Ω, N l , and h′ along with the computation of g 0 and g 1.

Fig. 4
Fig. 4

Bit error rate (BER) versus number of LPDIR iterations k for several values of the relaxation parameter β.

Fig. 5
Fig. 5

LPDIR performance comparison with three baseline restoration methods for class A channels: WF, POCS, and VA. (a) BER versus SNR. (b) Minimum acceptable SNR required to achieve a BER of 10-3 versus channel blur W/Δ.

Fig. 6
Fig. 6

Corrupted received images by use of class A channels with (a) W = 1.5Δ and (b) W = 2.0Δ; restored images by WF for (c) W = 1.5Δ and (d) W = 2.0Δ; restored images by POCS for (e) W = 1.5Δ and (f) W = 2.0Δ; restored images by LPDIR for (g) W = 1.5Δ and (h) W = 2.0Δ.

Fig. 7
Fig. 7

LPDIR performance comparison with three baseline restoration methods for class B channels: (a) BER versus SNR; (b) minimum acceptable SNR required to achieve a BER of 10-3 versus channel blur W/Δ.

Fig. 8
Fig. 8

Binary-valued images used to evaluate the VQ LPDIR method. (a) The original text object, (b) the training image, (c) the received image for a SNR of 20 dB, (d) the LPDIR restoration result by use of an estimated 3 × 3 pixel channel, (e) the VQ LPDIR restoration result.

Fig. 9
Fig. 9

BER performance of the VQ LPDIR and LPDIR algorithms for a 5 × 5 channel with W = 2.0Δ.

Fig. 10
Fig. 10

Gray-scale images used to evaluate the VQ LPDIR method. (a) The original image used for training and testing, (b) the distorted received image for W = 2.5Δ and σ2 = 5 dB, (c) the restored image obtained with the WF, (d) the restored image obtained with the POCS algorithm, (e) the restored image obtained with the VQ LPDIR method.

Fig. 11
Fig. 11

Gray-scale images used to evaluate the training sensitivity of the VQ LPDIR method. (a) The new training image, (b) the restored image obtained with the VQ LPDIR method.

Fig. 12
Fig. 12

Focal-plane implementation of the LPDIR algorithm. (a) The overall chip layout, (b) the layout of the acquisition component, (c) the layout of the processing unit, (d) the circuitry required within each shared processing unit (SPU) and each pixel-specific processor (PSP).

Fig. 13
Fig. 13

Results obtained with the reduced complexity LPDIR method. Minimum acceptable SNR required to achieve a BER of 10-3 versus channel blur W/Δ for several values of the integer word length B = 6, 7, and 8 along with the VA baseline.

Tables (2)

Tables Icon

Table 1 Required Support Region for Several Values of W

Tables Icon

Table 2 Maximum Number of Pixels and the Associated Clock Rate for Various Choices of M1, M2, and B

Equations (19)

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

hi, j=i-1/2Δi+1/2Δj-1/2Δj+1/2Δ hcx, ydxdy,
i,j=-1i,j=1 hi, j=1.
PNl=i,jNH Pfp+i, q+j=Nli, j,
pgp, q|fp, q=NlΩ pgp, q|fp, q, Nl×PNl,
pgp, q|fp, q=0, Nl=1σ2πexp×-|g0Nl-gp, q|2/2σ2,
pgp, q|fp, q=1, Nl=1σ2πexp×-|g1Nl-gp, q|2/2σ2,
P0k=1-βP0k-1+βPU0kp, q,
P1k=1-βP1k-1+βPU1kp, q,
PU0kp, q=NlΩ Cl0i,jNH Pk-1×fp+i, q+j=Nli, j
PU1kp, q=NlΩ Cl1i,jNH Pk-1×fp+i, q+j=Nli, j.
hei, j=hi, j+110|l|>1,|m|>1 hl, m.
PU0kp, q=NlΩ Cl0i,jNHNli, jPk-1×fp+i, q+j=1+|1-Nli, j|Pk-1×fp+i, q+j=0,
PU1kp, q=NlΩ Cl1i,jNHNli, jPk-1×fp+i, q+j=1+|1-Nli, j|Pk-1×fp+i, q+j=0,
Pkfp, q=n=1-βPk-1fp, q=n+βPUkp, q, n,
PUkp, q, n=NlΩ Clni,jNHn=0255×|n-Nli, j|Pk-1fp+i, q+j=n,
L0k=0.75×L0k-1+0.25×LU0kp, q,
L1k=0.75×L1k-1+0.25×LU1kp, q,
LU0kp, q=minNlΩCl0i,jNH Lk-1fp+i, q+j=Nli, j,
LU1kp, q=minNlΩCl1i,jNH Lk-1fp+i, q+j=Nli, j.

Metrics