Abstract

Deconvolution of images of the same object from multiple sensors with different point spread functions as suggested by Berenstein [Proc. IEEE 78, 723 (1990); Stochastic and Neural Methods in Signal Processing, Image Processing, and Computer Vision, S. Chen, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1569, 35 (1991)], opens new opportunities in solving the image-deconvolution problem, which has challenged researchers for years. We attack this problem in a more realistic formulation than that used by Berenstein; it explicitly takes into account image sensor noise and the necessity for adaptive restoration with estimation of all required signal and noise parameters directly from the observed noisy signals. We show that arbitrary restoration accuracy can be achieved by the appropriate choice of the number of sensor channels and the signal-to-noise ratio in each channel. The results are then extended to the practically important situation when true images in different sensor channels are not identical.

© 1994 Optical Society of America

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References

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  1. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  2. A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, New York, 1982), Vol. 1.
  3. B. R. Frieden, introduction in The Computer in Optical Research: Methods and Applications, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  4. C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operator: an overview,” Proc. IEEE 78, 723–734 (1990).
    [CrossRef]
  5. N. Sidoropoulos, J. Baras, C. Berenstein, “Two-dimensional signal deconvolution: design issues related to a novel multiple sensors based approach,” in Stochastic and Neural Methods in Signal Processing, Image Processing, and Computer Vision, S. Chen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1569, 356–366 (1991).
  6. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
    [CrossRef]
  7. L. P. Yaroslavsky, Digital Imaging Processing: An Introduction, Vol. 9 of Springer Series in Information Sciences (Springer-Verlag, New York, 1985).
  8. R. Yu. Vitkus, L. P. Yaroslavsky, “Recursive algorithms for local adaptive linear filtration,” in Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, eds., Vol. 40 of Mathematical Research (Academie Verlag, Berlin, 1987), pp. 34–39.

1990 (1)

C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operator: an overview,” Proc. IEEE 78, 723–734 (1990).
[CrossRef]

1967 (1)

Baras, J.

N. Sidoropoulos, J. Baras, C. Berenstein, “Two-dimensional signal deconvolution: design issues related to a novel multiple sensors based approach,” in Stochastic and Neural Methods in Signal Processing, Image Processing, and Computer Vision, S. Chen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1569, 356–366 (1991).

Berenstein, C.

N. Sidoropoulos, J. Baras, C. Berenstein, “Two-dimensional signal deconvolution: design issues related to a novel multiple sensors based approach,” in Stochastic and Neural Methods in Signal Processing, Image Processing, and Computer Vision, S. Chen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1569, 356–366 (1991).

Berenstein, C. A.

C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operator: an overview,” Proc. IEEE 78, 723–734 (1990).
[CrossRef]

Frieden, B. R.

B. R. Frieden, introduction in The Computer in Optical Research: Methods and Applications, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Helstrom, C. W.

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, New York, 1982), Vol. 1.

Patrick, E. V.

C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operator: an overview,” Proc. IEEE 78, 723–734 (1990).
[CrossRef]

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, New York, 1982), Vol. 1.

Sidoropoulos, N.

N. Sidoropoulos, J. Baras, C. Berenstein, “Two-dimensional signal deconvolution: design issues related to a novel multiple sensors based approach,” in Stochastic and Neural Methods in Signal Processing, Image Processing, and Computer Vision, S. Chen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1569, 356–366 (1991).

Vitkus, R. Yu.

R. Yu. Vitkus, L. P. Yaroslavsky, “Recursive algorithms for local adaptive linear filtration,” in Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, eds., Vol. 40 of Mathematical Research (Academie Verlag, Berlin, 1987), pp. 34–39.

Yaroslavsky, L. P.

R. Yu. Vitkus, L. P. Yaroslavsky, “Recursive algorithms for local adaptive linear filtration,” in Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, eds., Vol. 40 of Mathematical Research (Academie Verlag, Berlin, 1987), pp. 34–39.

L. P. Yaroslavsky, Digital Imaging Processing: An Introduction, Vol. 9 of Springer Series in Information Sciences (Springer-Verlag, New York, 1985).

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operator: an overview,” Proc. IEEE 78, 723–734 (1990).
[CrossRef]

Other (6)

N. Sidoropoulos, J. Baras, C. Berenstein, “Two-dimensional signal deconvolution: design issues related to a novel multiple sensors based approach,” in Stochastic and Neural Methods in Signal Processing, Image Processing, and Computer Vision, S. Chen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1569, 356–366 (1991).

L. P. Yaroslavsky, Digital Imaging Processing: An Introduction, Vol. 9 of Springer Series in Information Sciences (Springer-Verlag, New York, 1985).

R. Yu. Vitkus, L. P. Yaroslavsky, “Recursive algorithms for local adaptive linear filtration,” in Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, eds., Vol. 40 of Mathematical Research (Academie Verlag, Berlin, 1987), pp. 34–39.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, New York, 1982), Vol. 1.

B. R. Frieden, introduction in The Computer in Optical Research: Methods and Applications, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Comparison of the image restoration from two blurred images by an optimum filter and by two independent Wiener filters in each of two channels.

Fig. 2
Fig. 2

Frequency responses of the two distorting channels.

Fig. 3
Fig. 3

Frequency responses of the two restoration filters.

Equations (31)

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k = 1 m S k F k = δ ,
i ( x , y ) = s ( x , y ) o ( x , y ) .
i = s o + n ,
I = S O + N .
F inv = 1 / S = S * / S 2
F W = 1 S SNR 1 + SNR ,
SNR = S 2 O 2 0 / N 2 n ,
F w F inv = 1 / S ,
F w 0 ,
F W = 1 S SNR ¯ 1 + SNR ¯ ,
SNR ¯ = S 2 O 2 ¯ / N 2 n .
S 2 O 2 ¯ { I 2 - N 2 n if this difference is nonnegative 0 otherwise ,
i 1 = s 1 * o + n 1 , i 2 = s 2 * o + n 2 .
O ^ = F 1 I 1 + F 2 I 2 .
= o - o ^ n 1 n 2 = O - O ^ n 1 n 2 = O - F 1 I 1 + F 2 I 2 n 1 n 2 ,
N 1 * N 2 n 1 n 2 = N 1 N 2 * n 1 n 2 = 0.
F 1 ( 2 ) = 1 S 1 SNR 1 ( 2 ) ¯ 1 + SNR 1 ¯ + SNR 2 ¯ ,
SNR 1 ( 2 ) ¯ = S 1 ( 2 ) 2 O 2 / N 1 ( 2 ) 2 n 1 ( 2 )
O ( f ) = exp ( - f 2 / 3500 ) ,
S 1 ( f ) = sin ( 2 π f / 33 ) 2 π f / 33 , S 2 ( f ) = sin ( 2 π f / 47 ) 2 π f / 47 ,
N 1 ( f ) 2 n 1 = 0.001 ,     N 2 ( f ) 2 n 2 = 0.002.
{ I k = S k O + N k ;             k = 1 , 2 , , K }
O ^ = k = 1 K F k I k .
{ F k S k + SNR k ¯ l = 1 K F 1 S 1 = SNR k ¯ } ,
SNR k ¯ = S k 2 O 2 / N k 2 n k
{ F k = 1 S k SNR k ¯ 1 + k = 1 K SNR k ¯ ;             k = 1 , 2 , , K } .
O ^ = k = 1 K SNR k ¯ 1 + k = 1 K SNR k ¯ O + k = 1 K N k SNR k ¯ / S k 1 + k = 1 K SNR k ¯ .
image energy restoration noise energy = k = 1 K SNR k ¯ .
{ I k = S k O k + N k } ,
O ^ k = l = 1 K F k , l I l .
F k , l = 1 S 1 O k O 1 * O 1 2 SNR 1 ¯ 1 + m = 1 K SNR m ¯ ,

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