Abstract

A method for restoring an optical image which is subjected to low-pass frequency filtering is presented. It is assumed that the object whose image is restored is of finite spatial extent. The problem is treated as an algebraic image-restoration problem which is then solved as a quadratic programming problem with bounded variables. The regularization technique for the ill-posed system is to replace the consistent system of the quadratic programming problem by an approximate system of smaller rank. The rank which gives a best or near-best solution is estimated. This method is a novel one, and it compares favorably with other known methods. Computer-simulated examples are presented. Comments and conclusions are given.

© 1983 Optical Society of America

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References

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  1. S. J. Howard, J. Opt. Soc. Am. 71, 95 (1981).
    [CrossRef]
  2. S. J. Howard, J. Opt. Soc. Am. 71, 819 (1981).
    [CrossRef]
  3. R. Mammone, G. Eichmann, J. Opt. Soc. Am. 72, 987 (1982).
    [CrossRef]
  4. M. Severcan, Appl. Opt. 21, 1073 (1982).
    [CrossRef] [PubMed]
  5. C. K. Rushforth, A. E. Crawford, Y. Zhou, J. Opt. Soc. Am. 72, 204 (1982).
    [CrossRef]
  6. Y. Zhou, C. K. Rushforth, Appl. Opt. 21, 1249 (1982).
    [CrossRef] [PubMed]
  7. N. N. Abdelmalek, “An Algorithm for the Solution of Ill-Posed Linear Systems Arising From the Discretization of Fredholm Integral Equation of the First Kind,” J. Math. Anal. Appl., 97, No. 1 (Nov.1983); accepted for publication.
  8. N. N. Abdelmalek, T. Kasvand, Appl. Opt. 19, 3407 (1980).
    [CrossRef] [PubMed]
  9. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  10. J. M. Varah, SIAM J. Numer. Anal. 10, 257 (1973).
    [CrossRef]
  11. G. Peters, J. H. Wilkinson, Computer J. 13, 309 (1970).
    [CrossRef]
  12. N. N. Abdelmalek, Int. J. Syst. Sci. 10, 77 (1979).
    [CrossRef]
  13. N. N. Abdelmalek, “A fortran Program for the Solution of the Minimum Energy Problem for Discrete Linear Admissible Control systems,” NRC Tech. Rep. ERB-916 (1979).
  14. J. J. Dongarra, J. B. Bunch, C. B. Moler, G. W. Stewart, linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa.1979.

1983 (1)

N. N. Abdelmalek, “An Algorithm for the Solution of Ill-Posed Linear Systems Arising From the Discretization of Fredholm Integral Equation of the First Kind,” J. Math. Anal. Appl., 97, No. 1 (Nov.1983); accepted for publication.

1982 (4)

1981 (2)

1980 (1)

1979 (1)

N. N. Abdelmalek, Int. J. Syst. Sci. 10, 77 (1979).
[CrossRef]

1973 (1)

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

1970 (1)

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

Abdelmalek, N. N.

N. N. Abdelmalek, “An Algorithm for the Solution of Ill-Posed Linear Systems Arising From the Discretization of Fredholm Integral Equation of the First Kind,” J. Math. Anal. Appl., 97, No. 1 (Nov.1983); accepted for publication.

N. N. Abdelmalek, T. Kasvand, Appl. Opt. 19, 3407 (1980).
[CrossRef] [PubMed]

N. N. Abdelmalek, Int. J. Syst. Sci. 10, 77 (1979).
[CrossRef]

N. N. Abdelmalek, “A fortran Program for the Solution of the Minimum Energy Problem for Discrete Linear Admissible Control systems,” NRC Tech. Rep. ERB-916 (1979).

Andrews, H. C.

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

Bunch, J. B.

J. J. Dongarra, J. B. Bunch, C. B. Moler, G. W. Stewart, linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa.1979.

Crawford, A. E.

Dongarra, J. J.

J. J. Dongarra, J. B. Bunch, C. B. Moler, G. W. Stewart, linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa.1979.

Eichmann, G.

Howard, S. J.

Hunt, B. R.

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

Kasvand, T.

Mammone, R.

Moler, C. B.

J. J. Dongarra, J. B. Bunch, C. B. Moler, G. W. Stewart, linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa.1979.

Peters, G.

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

Rushforth, C. K.

Severcan, M.

Stewart, G. W.

J. J. Dongarra, J. B. Bunch, C. B. Moler, G. W. Stewart, linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa.1979.

Varah, J. M.

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

Wilkinson, J. H.

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

Zhou, Y.

Appl. Opt. (3)

Computer J. (1)

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

Int. J. Syst. Sci. (1)

N. N. Abdelmalek, Int. J. Syst. Sci. 10, 77 (1979).
[CrossRef]

J. Math. Anal. Appl. (1)

N. N. Abdelmalek, “An Algorithm for the Solution of Ill-Posed Linear Systems Arising From the Discretization of Fredholm Integral Equation of the First Kind,” J. Math. Anal. Appl., 97, No. 1 (Nov.1983); accepted for publication.

J. Opt. Soc. Am. (4)

SIAM J. Numer. Anal. (1)

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

Other (3)

N. N. Abdelmalek, “A fortran Program for the Solution of the Minimum Energy Problem for Discrete Linear Admissible Control systems,” NRC Tech. Rep. ERB-916 (1979).

J. J. Dongarra, J. B. Bunch, C. B. Moler, G. W. Stewart, linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, Pa.1979.

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

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Figures (9)

Fig. 1
Fig. 1

Original and observed images for example 1 for cutoff frequency M = 9. Noise is added with ρ = 10−3.

Fig. 2
Fig. 2

Original and restored images for Fig. 1 with no inequality constraints imposed.

Fig. 3
Fig. 3

Original and restored images for Fig. 1 with the inquality constraints imposed.

Fig. 4
Fig. 4

Original and the observed images for Example 1 for cutoff frequency M = 6. No noise is added.

Fig. 5
Fig. 5

Original and the restored images for Fig. 4 with the inequality constraints imposed.

Fig. 6
Fig. 6

Original and restored images for Fig. 1. Matrix C is an 10 × 10 matrix as formulated by Severcan (instead of being an 64 × 10). Inequality constraints are satisfied.

Fig. 7
Fig. 7

Original and the observed images for example 2 for cutoff frequency M = 12. No noise is added.

Fig. 8
Fig. 8

Original and the restored images for Fig. 7 with the inequality constraints imposed.

Fig. 9
Fig. 9

Same as Fig. 8, except that the parameter EPS is taken as 10−3 instead of 10−4.

Equations (30)

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C y = b .
C T C y = C T b .
0 y j y max ,             j = 1 , 2 , , L ,
X = F x ,
F = exp ( - 2 π i N m n ) ,             m , n = 0 , 1 , , N - 1.
B = diag ( , , 1 ( M + 1 ) ,             0 , , 0 ( N - 2 M - 1 ) ,             1 , , 1 ) ( M ) ,
D = diag ( 1 , , 1 ( L ) , 0 , , 0 ) ( N - L ) .
F - 1 = 1 N exp ( 2 π i N m n ) ,             m , n = 0 , 1 , , N - 1.
b = A x ,
A = F - 1 BFD .
C y = b .
C T C y = C T b .
C m n = { 1 + 2 j = 1 M cos [ 2 π ( m - n ) j / N ] } / N             m = 0 , 1 , , N - 1 , n = 0 , 1 , , L - 1.
C m n = 1 N sin [ π ( m - n ) ( 2 M + 1 ) / N ] sin [ π ( m - n ) / N ] ,             m n C n n = ( 2 M + 1 ) / N             m = 0 , 1 , , N - 1 , n = 0 , 1 , , L - 1.
G y = g ,
G ( k ) y = g ( k ) .
[ G ( k ) ] = [ G ( k ) ] T { G ( k ) [ G ( k ) ] T } - 1 .
G ( k ) [ G ( k ) ] T = LDL ¯ T ,
( s k + 1 { G ( k + 1 ) [ G ( k + 1 ) ] T } ) 1 / 4 s k + 1 ( C ) ( L - k ) 1 / 4 D ¯ k + 1 1 / 4 ,
y j = y max ( z j + 1 ) / 2 ,
minimize z 2
G z = 2 b / y max - G e
- 1 z j 1 ,             j = 1 , , L ,
r ( k ) = C y ( k ) - b ,             k = 1 , 2 , .
r ( k + 1 ) 2 r ( k ) 2 .
δ b = ρ ( 0.5 - random number ) ,
x ( n ) = { 0.5 exp [ - 2 ( n - 3 ) 3 ] + exp [ - 2 ( n - 7 ) 2 ] , n = 0 , 1 , , 9 0 , otherwise .
B ¯ j = { 1 , 0 j M η ( j ) , M < j N / 2.
η ( j ) = 1 - exp { - [ j - ( M + 1 ) ] 2 } , M < j N / 2 ,
C ¯ m n = { 1 + ( - 1 ) m + n η ( N / 2 ) + 2 j = 1 M cos [ 2 π ( m - n ) j / N ] + 2 j = M + 1 N / 2 - 1 η ( j ) cos [ 2 π ( m - n ) j / N ] } / N             m = 0 , 1 , , N - 1 , n = 0 , 1 , , L - 1.

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