Abstract

We demonstrate by a computer simulation example that singular value decomposition is a powerful tool for restoring noisy linearly degraded images. We also discuss a way of reducing the computation time requirement.

© 1975 Optical Society of America

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References

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  1. S. Treitel, J. L. Shanks, IEEE Trans. Geosci. Electron. GE-9, 10 (1971).
    [CrossRef]
  2. T. S. Huang, W. F. Schreiber, O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
    [CrossRef]
  3. G. Sherman, IEEE Trans. Computers, in press.
  4. H. Andrews, Computer 7, (5), 36 (May1974).
    [CrossRef]
  5. A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).
  6. W. Pratt, Semi-annual Reports under an ARPA Research Contract, Department of Electrical Engineering, University of Southern California, Los Angeles (1974).
  7. C. Lanczos, Linear Differential Operators (Van Nostrand, New York, 1961), Chap. 3.
  8. H. Andrews, “Two-Dimensional Transforms,” in Digital Picture Processing, T. S. Huang, Ed. (Springer-Verlag, Berlin, 1975), Chap. 2.
  9. G. M. Robbins, T. S. Huang, “Inverse Filtering for Linearly Shift-Variant Imaging Systems,” in Proc. Symp. Bildverar-beitung und Interaktive Systeme, DLR-MITT, 73-11, DFVLR-Forschungszentrum, Oberpfaffenhofen, West Germany (Dec. 1971), pp. 40–49.
  10. H. C. Andrews, C. L. Patterson, Amer. Math. Monthly, 82, 1 (Jan.1975).
    [CrossRef]

1975 (1)

H. C. Andrews, C. L. Patterson, Amer. Math. Monthly, 82, 1 (Jan.1975).
[CrossRef]

1974 (1)

H. Andrews, Computer 7, (5), 36 (May1974).
[CrossRef]

1971 (2)

S. Treitel, J. L. Shanks, IEEE Trans. Geosci. Electron. GE-9, 10 (1971).
[CrossRef]

T. S. Huang, W. F. Schreiber, O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[CrossRef]

Albert, A.

A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).

Andrews, H.

H. Andrews, Computer 7, (5), 36 (May1974).
[CrossRef]

H. Andrews, “Two-Dimensional Transforms,” in Digital Picture Processing, T. S. Huang, Ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

Andrews, H. C.

H. C. Andrews, C. L. Patterson, Amer. Math. Monthly, 82, 1 (Jan.1975).
[CrossRef]

Huang, T. S.

T. S. Huang, W. F. Schreiber, O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[CrossRef]

G. M. Robbins, T. S. Huang, “Inverse Filtering for Linearly Shift-Variant Imaging Systems,” in Proc. Symp. Bildverar-beitung und Interaktive Systeme, DLR-MITT, 73-11, DFVLR-Forschungszentrum, Oberpfaffenhofen, West Germany (Dec. 1971), pp. 40–49.

Lanczos, C.

C. Lanczos, Linear Differential Operators (Van Nostrand, New York, 1961), Chap. 3.

Patterson, C. L.

H. C. Andrews, C. L. Patterson, Amer. Math. Monthly, 82, 1 (Jan.1975).
[CrossRef]

Pratt, W.

W. Pratt, Semi-annual Reports under an ARPA Research Contract, Department of Electrical Engineering, University of Southern California, Los Angeles (1974).

Robbins, G. M.

G. M. Robbins, T. S. Huang, “Inverse Filtering for Linearly Shift-Variant Imaging Systems,” in Proc. Symp. Bildverar-beitung und Interaktive Systeme, DLR-MITT, 73-11, DFVLR-Forschungszentrum, Oberpfaffenhofen, West Germany (Dec. 1971), pp. 40–49.

Schreiber, W. F.

T. S. Huang, W. F. Schreiber, O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[CrossRef]

Shanks, J. L.

S. Treitel, J. L. Shanks, IEEE Trans. Geosci. Electron. GE-9, 10 (1971).
[CrossRef]

Sherman, G.

G. Sherman, IEEE Trans. Computers, in press.

Treitel, S.

S. Treitel, J. L. Shanks, IEEE Trans. Geosci. Electron. GE-9, 10 (1971).
[CrossRef]

Tretiak, O. J.

T. S. Huang, W. F. Schreiber, O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[CrossRef]

Amer. Math. Monthly (1)

H. C. Andrews, C. L. Patterson, Amer. Math. Monthly, 82, 1 (Jan.1975).
[CrossRef]

Computer (1)

H. Andrews, Computer 7, (5), 36 (May1974).
[CrossRef]

IEEE Trans. Geosci. Electron. (1)

S. Treitel, J. L. Shanks, IEEE Trans. Geosci. Electron. GE-9, 10 (1971).
[CrossRef]

Proc. IEEE (1)

T. S. Huang, W. F. Schreiber, O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[CrossRef]

Other (6)

G. Sherman, IEEE Trans. Computers, in press.

A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).

W. Pratt, Semi-annual Reports under an ARPA Research Contract, Department of Electrical Engineering, University of Southern California, Los Angeles (1974).

C. Lanczos, Linear Differential Operators (Van Nostrand, New York, 1961), Chap. 3.

H. Andrews, “Two-Dimensional Transforms,” in Digital Picture Processing, T. S. Huang, Ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

G. M. Robbins, T. S. Huang, “Inverse Filtering for Linearly Shift-Variant Imaging Systems,” in Proc. Symp. Bildverar-beitung und Interaktive Systeme, DLR-MITT, 73-11, DFVLR-Forschungszentrum, Oberpfaffenhofen, West Germany (Dec. 1971), pp. 40–49.

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Figures (3)

Fig. 1
Fig. 1

(a) Original; (b) smeared image with additive Gaussian noise (mean = 0, standard deviation = 0.1); (c) smeared image with additive Gaussian noise (mean = 0, standard deviation = 0.5).

Fig. 2
Fig. 2

Restored image from Fig. 1(b) using SVD. The number of terms used are (a) 50; (b) 48; (c) 42; (d) 36.

Fig. 3
Fig. 3

Restored image from Fig. 1(c) using SVD. The number of terms used are (a) 44; (b) 38; (c) 32.

Equations (8)

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

g = [ H ] f + n ,
f = [ f 1 f 2 f p ] ,
f ̂ = [ H ] + g ,
f ̂ = [ H ] + [ H ] f + [ H ] + n ,
[ H ] = i = 1 R ( λ i ) 1 / 2 U i V i t ,
[ H ] + = i = 1 R 1 ( λ i ) 1 / 2 V i U i t .
f = i = 1 R λ i 1 / 2 V i U i t { [ H ] f } + i = 1 R λ 1 / 2 V i U i t n .
f ̂ = i = 1 P λ i 1 / 2 V i U i t g .

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