Abstract

A novel technique for the blind deconvolution of an image from an unknown blurring function is presented. This iterative technique consists of two steps. We first estimate the blurring function and then use this result to estimate the original image. The problem is posed in terms of successive interlinked energy minimizations that are mapped onto two Hopfield neural networks.

© 1994 Optical Society of America

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References

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  1. G. R. Ayers, J. C. Dainty, Opt. Lett. 13, 547 (1988).
    [CrossRef] [PubMed]
  2. N. Nakajima, Opt. Commun. 100, 59 (1993).
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    [CrossRef]
  4. O. Shalvi, E. Weinstein, IEEE Trans. Inf. Theory 39, 509 (1993).
    [CrossRef]
  5. B. C. McCallum, Opt. Commun. 75, 101 (1990).
    [CrossRef]
  6. J. J. Hopfield, Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
    [CrossRef] [PubMed]
  7. J. J. Hopfield, Proc. Natl. Acad. Sci. USA 81, 3088 (1984).
    [CrossRef] [PubMed]
  8. J. B. Abbiss, B. J. Brames, M. A. Fiddy, Proc. Soc. Photo-Opt. Instrum. Eng. 1058, 138 (1989).
  9. J. B. Abbiss, M. A. Fiddy, R. Steriti, Advances in Electronics and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 1994), Vol. 87, pp. 1–48.
    [CrossRef]
  10. C. K. Rushforth, in Image Recovery Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 1–27.
  11. M. E. Hayes, J. H. McClellan, Proc. IEEE 70, 197 (1982).
    [CrossRef]

1993

N. Nakajima, Opt. Commun. 100, 59 (1993).
[CrossRef]

O. Shalvi, E. Weinstein, IEEE Trans. Inf. Theory 39, 509 (1993).
[CrossRef]

1990

1989

J. B. Abbiss, B. J. Brames, M. A. Fiddy, Proc. Soc. Photo-Opt. Instrum. Eng. 1058, 138 (1989).

1988

1984

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 81, 3088 (1984).
[CrossRef] [PubMed]

1982

M. E. Hayes, J. H. McClellan, Proc. IEEE 70, 197 (1982).
[CrossRef]

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

Abbiss, J. B.

J. B. Abbiss, B. J. Brames, M. A. Fiddy, Proc. Soc. Photo-Opt. Instrum. Eng. 1058, 138 (1989).

J. B. Abbiss, M. A. Fiddy, R. Steriti, Advances in Electronics and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 1994), Vol. 87, pp. 1–48.
[CrossRef]

Ayers, G. R.

Bates, R. H. T.

Brames, B. J.

J. B. Abbiss, B. J. Brames, M. A. Fiddy, Proc. Soc. Photo-Opt. Instrum. Eng. 1058, 138 (1989).

Dainty, J. C.

Fiddy, M. A.

J. B. Abbiss, B. J. Brames, M. A. Fiddy, Proc. Soc. Photo-Opt. Instrum. Eng. 1058, 138 (1989).

J. B. Abbiss, M. A. Fiddy, R. Steriti, Advances in Electronics and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 1994), Vol. 87, pp. 1–48.
[CrossRef]

Hayes, M. E.

M. E. Hayes, J. H. McClellan, Proc. IEEE 70, 197 (1982).
[CrossRef]

Hopfield, J. J.

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 81, 3088 (1984).
[CrossRef] [PubMed]

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

McCallum, B. C.

B. C. McCallum, Opt. Commun. 75, 101 (1990).
[CrossRef]

McClellan, J. H.

M. E. Hayes, J. H. McClellan, Proc. IEEE 70, 197 (1982).
[CrossRef]

Nakajima, N.

N. Nakajima, Opt. Commun. 100, 59 (1993).
[CrossRef]

Parker, C. R.

Quek, B. K.

Rushforth, C. K.

C. K. Rushforth, in Image Recovery Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 1–27.

Shalvi, O.

O. Shalvi, E. Weinstein, IEEE Trans. Inf. Theory 39, 509 (1993).
[CrossRef]

Steriti, R.

J. B. Abbiss, M. A. Fiddy, R. Steriti, Advances in Electronics and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 1994), Vol. 87, pp. 1–48.
[CrossRef]

Weinstein, E.

O. Shalvi, E. Weinstein, IEEE Trans. Inf. Theory 39, 509 (1993).
[CrossRef]

IEEE Trans. Inf. Theory

O. Shalvi, E. Weinstein, IEEE Trans. Inf. Theory 39, 509 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

N. Nakajima, Opt. Commun. 100, 59 (1993).
[CrossRef]

B. C. McCallum, Opt. Commun. 75, 101 (1990).
[CrossRef]

Opt. Lett.

Proc. IEEE

M. E. Hayes, J. H. McClellan, Proc. IEEE 70, 197 (1982).
[CrossRef]

Proc. Natl. Acad. Sci. USA

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 81, 3088 (1984).
[CrossRef] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng.

J. B. Abbiss, B. J. Brames, M. A. Fiddy, Proc. Soc. Photo-Opt. Instrum. Eng. 1058, 138 (1989).

Other

J. B. Abbiss, M. A. Fiddy, R. Steriti, Advances in Electronics and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 1994), Vol. 87, pp. 1–48.
[CrossRef]

C. K. Rushforth, in Image Recovery Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 1–27.

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Figures (3)

Fig. 1
Fig. 1

(a) Original image, (b) the blurring function, (c) the data [(a) convolved with (b)].

Fig. 2
Fig. 2

Initial estimate of the image.

Fig. 3
Fig. 3

(a) Estimate of the original image, (b) estimate of the blurring function, (c) convolution of (a) and (b).

Equations (15)

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g ( x ) = f ( x 1 ) h ( x - x 1 ) d x 1 + noise ,
g i j = k = 1 N l = 1 N f k , l h i - k + N / 2 , j - l + N / 2 = k = 1 N l = 1 N h k , l f i - k + N / 2 , j - 1 + N / 2 ,             i , j = { 1 , 2 , , N } ,
g i j = k = 1 N l = 1 N A i j , k l f k l = k = 1 N l = 1 N C i j , k l h k l ,
A i j , k l = h i - k + N / 2 , j - l + N / 2 ,             C i j , k l = f i - k + N / 2 , j - l + N / 2 .
E = A f - g 2 ,
E = C h - g 2 .
E = - 1 2 T v + b 2 ,
T = - 2 A * A ,             b = 2 A * g ,
T = - 2 C * C ,             b = 2 C * g .
v i j = s ( u i j ) ,
d u i j d t = k = 1 N l = 1 N T i j , k l v k l + b i j ,
u i j t + 1 = u i j t + k = 1 N l = 1 N T i j , k l v k l t + b i j ,
g = f * h = g * δ ,
g = f * h = f 1 * f 2 * h = f 1 * a
g = f * h = f * h 1 * h 2 = b * h 2 .

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