Abstract

The problem of 2-D digital restoration of images degraded by spatially varying pointspread functions is considered. It is shown that the conjugate gradient method can be applied to this problem. This algorithm preserves the sparse matrix properties of other iterative approaches to least squares restoration but has improved convergence properties. The approximation of pointspread matrices by banded matrices is shown to improve the speed of the algorithm without significantly affecting the restoration.

© 1978 Optical Society of America

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References

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  1. A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
    [CrossRef]
  2. B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
    [CrossRef] [PubMed]
  3. H. C. Andrews, C. L. Patterson, IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
    [CrossRef]
  4. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N. J., 1977).
  5. E. S. Angel, A. K. Jain, J. Opt. Soc. Am. 65, 1203 (1975).
  6. T. S. Huang, D. A. Barker, S. P. Berger, Appl. Opt. 14, 1165 (1975).
    [CrossRef] [PubMed]
  7. P. H. Van Cittert, Z. Phys. 69, 298 (1931).
    [CrossRef]
  8. P. H. Jansson, R. H. Hunt, E. K. Plyler, J. Opt. Soc. Am. 58, 1665 (1968).
    [CrossRef]
  9. D. G. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1973).
  10. E. S. Angel, R. Bellman, Dynamic Programming and Partial Differential Equations (Academic, New York, 1972).

1976 (1)

H. C. Andrews, C. L. Patterson, IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
[CrossRef]

1975 (2)

E. S. Angel, A. K. Jain, J. Opt. Soc. Am. 65, 1203 (1975).

T. S. Huang, D. A. Barker, S. P. Berger, Appl. Opt. 14, 1165 (1975).
[CrossRef] [PubMed]

1972 (2)

1968 (1)

1931 (1)

P. H. Van Cittert, Z. Phys. 69, 298 (1931).
[CrossRef]

Andrews, H. C.

H. C. Andrews, C. L. Patterson, IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
[CrossRef]

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

Angel, E. S.

E. S. Angel, A. K. Jain, J. Opt. Soc. Am. 65, 1203 (1975).

E. S. Angel, R. Bellman, Dynamic Programming and Partial Differential Equations (Academic, New York, 1972).

Barker, D. A.

Bellman, R.

E. S. Angel, R. Bellman, Dynamic Programming and Partial Differential Equations (Academic, New York, 1972).

Berger, S. P.

Frieden, B. R.

Huang, T. S.

Hunt, B. R.

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

Hunt, R. H.

Jain, A. K.

E. S. Angel, A. K. Jain, J. Opt. Soc. Am. 65, 1203 (1975).

Jansson, P. H.

Luenberger, D. G.

D. G. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1973).

Patterson, C. L.

H. C. Andrews, C. L. Patterson, IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
[CrossRef]

Plyler, E. K.

Sawchuk, A. A.

A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
[CrossRef]

Van Cittert, P. H.

P. H. Van Cittert, Z. Phys. 69, 298 (1931).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

H. C. Andrews, C. L. Patterson, IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. IEEE (1)

A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
[CrossRef]

Z. Phys. (1)

P. H. Van Cittert, Z. Phys. 69, 298 (1931).
[CrossRef]

Other (3)

D. G. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1973).

E. S. Angel, R. Bellman, Dynamic Programming and Partial Differential Equations (Academic, New York, 1972).

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

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Figures (5)

Fig. 1
Fig. 1

Degrading pointspread function.

Fig. 2
Fig. 2

Original S image.

Fig. 3
Fig. 3

Blurred S image.

Fig. 4
Fig. 4

(a) Iteration 1; (b) iteration 2; (c) iteration 5; (d) iteration 10.

Fig. 5
Fig. 5

(a) Blurred S + noise; (b) iteration 1; (c) iteration 3; (d) iteration 5.

Equations (40)

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υ ( x , y ) = h ( x , y , x , y ) u ( x , y ) d x d y .
g = H f ,
h ( x , y , x , y ) = 0 ,
min f J = min f  ∥ g H f 2 ,
f 2 = f T f .
H T H f ˆ = H T g .
f ˆ = H 1 g  ,
f ( k ) = f ( k ) σ H T ( g H T f ( k ) ) , = ( I σ H T H ) f ( k ) + σ H T g .
e ( k ) = f ˆ f ( k ) ,
e ( k ) = ( I σ H T H ) K e ( o ) ,
d i T Q d j = 0 ,  i j .
f = i   = 1 n α i d i .
f ( k + 1 ) = f ( k ) + α k d k  ,
α k = c k T d k d k T Q d k ,
d k + 1 = c k + 1 + β k d k  ,
β k = c k + 1 T Q d k d k T Q d k .
c k = H g T Q f ( k ) = c k 1 α k 1 Q d k 1  ,
f ( 1 ) = g  ,
d 1 = c 1 ,
h ( x , y , x , y ) = h 1 ( x , x ) h 2 ( y , y ) ,
G = A F B ,
X , Y = i = 1 N j = 1 M x i j y i j  ,
F ( k + 1 ) = F ( k ) + α k D k ,
α k = C k , Β k D k Α T Α D k Β Β T ,
D k + 1 = C k + 1 + β k D k ,
β k = C k + 1 , Α T Α D k Β Β T D k , Α T Α D k Β Β T ,
C k = A T G B T A T A F ( k ) B B T ,
F ( 1 ) = G ,
D 1 = C 1 .
h ( x , y , x , y ) = exp [ ρ x ( x x ) 2 ρ y ( y y ) 2 ]  ,
A = B = ( a i j ) ,
a i j = exp [ γ i ( i j ) 2 ]  ,
γ i = 2 19 | 16 i | 150 .
a i j < 0.1 ,  | i j | 4.
H = I  ⊗  A + B  ⊗  I ,
F = [ F 11 , F 12 …… F 1 r F s 1 , …….. F s r ]  ,
G = [ G 11 , G 12 …… G 1 r G s 1 , …….. G s r ]  ,
[ A ^ = A 1 A 2 . . A r ] ,
[ B ^ = B 1 B 2 . . B s ] .
G i j = A i F i j B j ,

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