Abstract

We show how to apply a truncated eigensystem expansion in the solution of image restoration problems for the case of space invariant point spread functions. The solution is obtained directly from the system of linear equations, which result from the discretization of the Fredholm integral equation of the first kind. Fast Fourier transform techniques are used in obtaining this solution. A procedure is devised to estimate the rank of the coefficient matrix that gives a best or near best solution. It is demonstrated that this algorithm compares favorably with other existing methods. Numerical results using spatially separable point spread functions are given.

© 1980 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. A. Albert, Regression and the Moore-Penrose Pseudo-Inverse (Academic, New York, 1972).
  12. G. Peters, J. H. Wilkinson, Comput. J. 13, 309 (1970).
    [CrossRef]
  13. W. K. Pratt, Digital Image Processing. (Wiley, New York, 1978).
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    [CrossRef] [PubMed]
  15. S. S. Reddi, Appl. Opt. 17, 2340 (1978).
    [CrossRef] [PubMed]

1979

1978

1975

1973

J. M. Varah, SIAM J. Numer. Anal. 10, 257 (1973).
[CrossRef]

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

1971

B. R. Hunt, IEEE Trans. Audio Electroacoust. AU-19, 285 (1971).
[CrossRef]

R. J. Hanson, SIAM J. Numer. Anal. 8, 616 (1971).
[CrossRef]

1970

G. Peters, J. H. Wilkinson, Comput. J. 13, 309 (1970).
[CrossRef]

1964

C. T. H. Baker, L. Fox, D. F. Mayers, K. Wright, Comput. J. 7, 141 (1964).
[CrossRef]

1963

S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).
[CrossRef]

1962

D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).
[CrossRef]

Abdelmalek, N. N.

Albert, A.

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

Andrews, H. C.

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

Baker, C. T. H.

C. T. H. Baker, L. Fox, D. F. Mayers, K. Wright, Comput. J. 7, 141 (1964).
[CrossRef]

Bellman, R.

R. Bellman, Introduction to Matrix Analysis. (McGraw-Hill, New York, 1970).

Fox, L.

C. T. H. Baker, L. Fox, D. F. Mayers, K. Wright, Comput. J. 7, 141 (1964).
[CrossRef]

Hanson, R. J.

R. J. Hanson, SIAM J. Numer. Anal. 8, 616 (1971).
[CrossRef]

Huang, T. S.

Hunt, B. R.

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

B. R. Hunt, IEEE Trans. Audio Electroacoust. AU-19, 285 (1971).
[CrossRef]

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

Kasvand, T.

Mayers, D. F.

C. T. H. Baker, L. Fox, D. F. Mayers, K. Wright, Comput. J. 7, 141 (1964).
[CrossRef]

Narendra, P. M.

Peters, G.

G. Peters, J. H. Wilkinson, Comput. J. 13, 309 (1970).
[CrossRef]

Phillips, D. L.

D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing. (Wiley, New York, 1978).

Reddi, S. S.

Twomey, S.

S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).
[CrossRef]

Varah, J. M.

J. M. Varah, SIAM J. Numer. Anal. 10, 257 (1973).
[CrossRef]

Wilkinson, J. H.

G. Peters, J. H. Wilkinson, Comput. J. 13, 309 (1970).
[CrossRef]

Wright, K.

C. T. H. Baker, L. Fox, D. F. Mayers, K. Wright, Comput. J. 7, 141 (1964).
[CrossRef]

Appl. Opt.

Comput. J.

G. Peters, J. H. Wilkinson, Comput. J. 13, 309 (1970).
[CrossRef]

C. T. H. Baker, L. Fox, D. F. Mayers, K. Wright, Comput. J. 7, 141 (1964).
[CrossRef]

IEEE Trans. Audio Electroacoust.

B. R. Hunt, IEEE Trans. Audio Electroacoust. AU-19, 285 (1971).
[CrossRef]

IEEE Trans. Comput.

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

J. Assoc. Comp. Mach.

D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).
[CrossRef]

S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).
[CrossRef]

SIAM J. Numer. Anal.

R. J. Hanson, SIAM J. Numer. Anal. 8, 616 (1971).
[CrossRef]

J. M. Varah, SIAM J. Numer. Anal. 10, 257 (1973).
[CrossRef]

Other

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

R. Bellman, Introduction to Matrix Analysis. (McGraw-Hill, New York, 1970).

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

W. K. Pratt, Digital Image Processing. (Wiley, New York, 1978).

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Figures (6)

Fig. 1
Fig. 1

The 16 × 16 matrix representing character 5. Each point inside the character is given pixel value 7, and each point outside is given value 0.

Fig. 2
Fig. 2

(a) Smeared image with additive Gaussian noise, mean = 0, standard deviation = 0.1, SNR ≈ 920; (b) smeared image with additive Gaussian noise, mean = 0, standard deviation = 0.5, SNR ≈ 37; (c) restored for (a); (d) restored for (b).

Fig. 3
Fig. 3

The 103 × 64 matrix representing a portion of the image of the GIRL decimated by taking every second pixel every second line.

Fig. 4
Fig. 4

Restoration from moderate SIPSF blur (K = 1). (a) Blurred noisy image,K = 1,L = 5, with additive Gaussian noise, mean = 0, standard deviation = 1, SNR ≈ 1750; (b) blurred noisy image,K = 1, L = 5, with additive Gaussian noise, mean = 0, standard deviation = 3, SNR ≈ 195; (c) restored for (a); (d) restored for (b).

Fig. 5
Fig. 5

Restoration from severe SIPSF blur (K = 4). (a) Blurred image with no noise,K = 4, L = 15. (b) Restored for (a), rank of [Āc] = 118, and rank of [ B ̅ c ] = 60.

Fig. 6
Fig. 6

Restoration from severe SIPSF blur (K = 4). (a) Blurred noisy image, K = 4, L = 15, with additive Gaussian noise, mean = 0, standard deviation = 0.5, SNR ≈ 7000; (b) blurred noisy image, K = 4, L = 15, with additive Gaussian noise, mean = 0, standard deviation = 1, SNR ≈ 1750; (c) restored for (a), rank of [Āc] = 35, and rank of; [ B ̅ c ] = 17 (d) restored for (b), rank of [Āc] = 31, and rank of [ B ̅ c ] = 15.

Equations (27)

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

g = [ H ] f + e ,
[ H ] T g = [ H ] T [ H ] f + [ H ] T e .
[ H c ] = [ h 0 h 1 h 2 . . . h N 1 h N 1 h 0 h 1 . . . h N 2 · · · . . . · h 1 h 2 h 3 . . . h 0 ] .
[ H c ] = [ W ] [ D ] [ W ] 1 ,
{ W } kj = exp ( 2 π i N kj ) , k , j = 0 , 1 , . . . , N 1 ,
{ W 1 } kj = 1 N exp ( 2 π i N kj ) , k , j = 0 , 1 , . . . , N 1 ,
d k = h 0 + h 1 r k 1 + h 2 r k 2 + . . . + h N 1 r k N 1 , k = 0 , 1 , . . . , N 1 ,
[ W ] 1 = ( 1 / N ) [ W ] t ,
[ H c ] = ( 1 / N ) [ W ] [ D ] [ W ] t .
d k = h 0 + h 1 ( r k 1 + r k N 1 ) + h 2 ( r k 2 + r k N 2 ) + . . . + L , k = 0 , 1 , . . . , N 1 ,
d k = h 0 + 2 h 1 cos ( 2 π k N ) + 2 h 2 cos ( 4 π k N ) + . . . + L , k = 0 , 1 , . . . , N 1 ,
g = [ H c ] f + e ,
g = [ H c ] f + ρ ,
[ H ̅ c ] = ( 1 / N ) [ W ] [ D ̅ ] [ W ] t .
f ̂ = [ H ̅ c ] + g ,
[ H ̅ c ] + = ( 1 / N ) [ W ] [ S ] [ W ] t ,
f ̂ = [ W ] [ S ] [ W ] 1 g .
e 2 = ( N 1 ) S 2 ( e ) + μ 2 ( e ) .
[ H ̅ c ] f ̂ = [ W ] [ Ī ] [ W ] 1 g ,
[ H ] = [ A ] [ B ] ,
[ G ] = [ A ] [ F ] [ B ] T + [ E ] ,
[ G ] = [ A c ] [ F ] [ B c ] T + [ E ] .
[ G ] = [ A c ] [ X ] + [ E ] .
g j = [ Ā c ] x j + ρ j , j = 1 , . . . , J .
[ B ̅ c ] [ F ] T = [ X ] T + [ R ]
SNR = variance of ideal image / variance of noise .
a i j = C exp [ ( i j ) 2 / K 2 ] ,

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