Abstract

The use of support constraints for improving the quality of Fourier spectra and their associated images is discussed as well as the relationship between the two domains. Theoretical relationships are derived that predict the noise reduction in both the image domain and the Fourier domain achieved by single and repeated application of support constraints for the case of wide-sense stationary Fourier-domain noise. It is shown that the application of support constraints can increase noise inside the support constraint if the application is not done correctly. An iterative algorithm is proposed that enforces support constraints in such a way that noise is never increased inside the support constraint and the algorithm achieves the minimum possible noise in a finite number of steps.

© 1994 Optical Society of America

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References

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  1. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,”IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
    [CrossRef]
  2. M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
    [CrossRef]
  3. H. Stark, ed., Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987).
  4. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  5. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [CrossRef]
  6. M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—applications and numerical results,”IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
    [CrossRef]
  7. C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,”IEEE Trans. Signal Process. (to be published).
  8. H. J. Trussell, M. R. Civanlar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. 32, 201–212 (1984).
    [CrossRef]
  9. M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,”IEEE Trans. Acoust. Speech Signal Process. 34, 919–936 (1986).
    [CrossRef]

1990 (1)

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

1989 (1)

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,”IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

1986 (1)

M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,”IEEE Trans. Acoust. Speech Signal Process. 34, 919–936 (1986).
[CrossRef]

1984 (1)

H. J. Trussell, M. R. Civanlar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. 32, 201–212 (1984).
[CrossRef]

1982 (2)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—applications and numerical results,”IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

1978 (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Civanlar, M. R.

M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,”IEEE Trans. Acoust. Speech Signal Process. 34, 919–936 (1986).
[CrossRef]

H. J. Trussell, M. R. Civanlar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. 32, 201–212 (1984).
[CrossRef]

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,”IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

Matson, C. L.

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,”IEEE Trans. Signal Process. (to be published).

Sezan, M. I.

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—applications and numerical results,”IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Stark, H.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—applications and numerical results,”IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Tekalp, A. M.

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Trussell, H. J.

M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,”IEEE Trans. Acoust. Speech Signal Process. 34, 919–936 (1986).
[CrossRef]

H. J. Trussell, M. R. Civanlar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. 32, 201–212 (1984).
[CrossRef]

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (3)

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,”IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

H. J. Trussell, M. R. Civanlar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. 32, 201–212 (1984).
[CrossRef]

M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,”IEEE Trans. Acoust. Speech Signal Process. 34, 919–936 (1986).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

IEEE Trans. Med. Imaging (2)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—applications and numerical results,”IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Opt. Eng. (1)

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Other (2)

H. Stark, ed., Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987).

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,”IEEE Trans. Signal Process. (to be published).

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Figures (10)

Fig. 1
Fig. 1

Variance of individual elements of the re al part of the support-constrained Fourier spectrum as a function of the number of iterations for (a) σr2/σi2 = 1, (b) σr2/σi2 = 0.6, and (c) σr2/σi2 = 0.2. Curves, theoretical predictions; symbols, simulation data points.

Fig. 2
Fig. 2

Variance of individual elements of the imaginary part of the support-constrained Fourier spectrum as a function of the number of iterations for (a) σr2/σi2 = 1, (b) σr2/σi2 = 0.6, and (c) σr2/σi2 = 0.2. Curves, theoretical predictions; symbols, simulation data points.

Fig. 3
Fig. 3

Total variance in the asymmetric region of the support constraint as a function of the number of iterations for (a) σr2/σi2 = 1, (b) σr2/σi2 = 0.6, and (c) σr2/σi2 = 0.2. Curves, theoretical predictions; symbols, simulation data points.

Fig. 4
Fig. 4

Variances for σr2/σi2 = 1 as a function of the iteration number. Curves (a) and (b) are the total variances in the asymmetric region of the support constraint for the original algorithm and the modified algorithm, respectively. Curves (c) and (d) are the total variances in the symmetric region of the support constraint for the modified algorithm and the original algorithm, respectively.

Fig. 5
Fig. 5

Variances for σr2/σi2 = 0.6 as a function of the iteration number. Curves (a) and (b) are the total variances in the asymmetric region of the support constraint for the original algorithm and the modified algorithm, respectively. Curves (c) and (d) are the total variances in the symmetric region of the support constraint for the modified algorithm and the original algorithm, respectively.

Fig. 6
Fig. 6

Variances for σr2/σi2 = 0.2 as a function of the iteration number. Curves (a) and (b) are the total variances in the asymmetric region of the support constraint for the original algorithm and the modified algorithm, respectively. Curves (c) and (d) are the total variances in the symmetric region of the support constraint for the modified algorithm and the original algorithm, respectively.

Fig. 7
Fig. 7

Uncorrupted satellite image.

Fig. 8
Fig. 8

Support region of satellite image shown in Fig. 7. The symmetric part of the support is shown in white, the asymmetric part in gray.

Fig. 9
Fig. 9

Noise-corrupted satellite image for σi2 = 107 times the dc value of the image and σr2 = 0.0.

Fig. 10
Fig. 10

Reconstructed satellite image after 20 iterations of the original algorithm.

Equations (35)

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o ( x ) = w ( x ) o ( x ) .
O ( u ) = - W ( ξ ) O ( u - ξ ) d ξ ,
O s ( u ) = - W ( ξ ) O e ( u - ξ ) d ξ = - W r ( ξ ) O e r ( u - ξ ) d ξ - - W i ( ξ ) O e i ( u - ξ ) d ξ + j [ - W r ( ξ ) O e i ( u - ξ ) d ξ + - W i ( ξ ) O e r ( u - ξ ) d ξ ] ,
Var { O s r ( u ) } [ O s r ( u ) ] 2 - O s r ( u ) 2 = - W r ( ξ ) O e r ( u - ξ ) d ξ - W r ( ζ ) O e r ( u - ζ ) d ζ - - W r ( ξ ) O e r ( u - ξ ) d ξ - W r ( ζ ) O e r ( u - ζ ) d ζ + - W i ( ξ ) O e i ( u - ξ ) d ξ - W i ( ζ ) O e i ( u - ζ ) d ζ - - W i ( ξ ) O e i ( u - ξ ) d ξ - W i ( ζ ) O e i ( u - ζ ) d ζ - 2 - W r ( ξ ) O e r ( u - ξ ) d ξ - W i ( ζ ) O e i ( u - ζ ) d ζ + 2 - W r ( ξ ) O e r ( u - ξ ) d ξ - W i ( ζ ) O e i ( u - ζ ) d ζ = - W r ( ξ ) W r ( ζ ) [ O e r ( u - ξ ) O e r ( u - ζ ) - O e r ( u - ξ ) O e r ( u - ζ ) ] d ξ d ζ + - W i ( ξ ) W i ( ζ ) [ O e i ( u - ξ ) O e i ( u - ζ ) - O e i ( u - ξ ) O e i ( u - ζ ) ] d ξ d ζ - 2 - W r ( ξ ) W i ( ζ ) [ O e r ( u - ξ ) O e i ( u - ζ ) - O e r ( u - ξ ) O e i ( u - ζ ) ] d ξ d ζ ,
Var { O s i ( u ) } [ O s i ( u ) ] 2 - O s i ( u ) 2 = - W r ( ξ ) W r ( ζ ) [ O e i ( u - ξ ) O e i ( u - ζ ) - O e i ( u - ξ ) O e i ( u - ζ ) ] d ξ d ζ + - W i ( ξ ) W i ( ζ ) [ O e r ( u - ξ ) O e r ( u - ζ ) - O e r ( u - ξ O e r ( u - ζ ) ] d ξ d ζ + 2 - W r ( ξ ) W i ( ζ ) [ O e i ( u - ξ ) O e i ( u - ζ ) - O e i ( u - ξ ) O e r ( u - ζ ) ] d ξ d ζ .
N r ( u ) = O e r ( u ) - O e r ( u ) , N i ( u ) = O e i ( u ) - O e i ( u ) ,
N r ( u 1 ) N r ( u 2 ) = R r ( u 1 - u 2 ) + R r ( u 1 + u 2 ) , N i ( u 1 ) N i ( u 2 ) = R i ( u 1 - u 2 ) - R i ( u 1 + u 2 ) , N i ( u 1 ) N r ( u 2 ) = N r ( u 1 ) N i ( u 2 ) = 0.
Var { O s r ( u ) } = - [ W r ( ξ ) W r ( ζ ) R r ( ξ - ζ ) + W i ( ξ ) W i ( ζ ) R i ( ξ - ζ ) ] d ξ d ζ ,
Var { O s i ( u ) } = - [ W r ( ξ ) W r ( ζ ) R i ( ξ - ζ ) + W i ( ξ ) W i ( ζ ) R r ( ξ - ζ ) ] d ξ d ζ .
Var { O s r ( u ) } = - w e 2 ( x ) r r ( x ) d x + - w 0 2 ( x ) r i ( x ) d x ,
Var { O s i ( u ) } = - w e 2 ( x ) r i ( x ) d x + - w o 2 ( x ) r r ( x ) d x ,
w e ( x ) = w s ( x ) + ½ [ w a ( x ) + w - a ( x ) ] , w o ( x ) = ½ [ w a ( x ) - w - a ( x ) ] ,
w e n ( x ) = w s ( x ) + 2 - n [ w a ( x ) + w - a ( x ) ] , w o n ( x ) = 2 - n w a ( x ) + ( - 2 ) - n w - a ( x ) ,
Var { O s r ( u ) } = w s ( x ) r r ( x ) d x + 1 2 w a ( x ) [ r i ( x ) + r r ( x ) ] d x ,
Var { O s i ( u ) } = w s ( x ) r i ( x ) d x + 1 2 w a ( x ) [ r i ( x ) + r r ( x ) ] d x .
Var { O s r ( u ) } = - r r ( x ) d x = R r ( 0 ) ,
Var { O s i ( u ) } = - r i ( x ) d x = R i ( 0 ) .
Var { O s r ( u ) } r r ( 0 ) w s ( x ) d x + 1 2 [ r i ( 0 ) + r r ( 0 ) ] w a ( x ) d x ,
Var { O s i ( u ) } r i ( 0 ) w s ( x ) d x + 1 2 [ r i ( 0 ) + r r ( 0 ) ] w a ( x ) d x ,
Var { O s r ( u ) } = w s ( x ) r r ( x ) d x ,
Var { O s i ( u ) } = w s ( x ) r i ( x ) d x .
Var { O s r ( u ) } = w ( x ) r r ( x ) d x = Var { O s i ( u ) } .
O s i ( m ) ( u ) = F { w e m ( x ) F - 1 [ O e i ( u ) ] } - j k = 0 m - 1 F { w e k ( x ) w o ( x ) F - 1 [ O e r ( u ) ] } ,
Var { O s i ( m ) ( u ) } = - { w s ( x ) + 2 - 2 m [ w a ( x ) + w - a ( x ) ] } r i ( x ) d x + - 1 4 [ w a ( x ) + w - a ( x ) ] × k = 0 m - 1 { w s ( x ) + k + 1 2 k [ w a ( x ) + w - a ( x ) ] } r r ( x ) d x + - 1 4 [ w a ( x ) + w - a ( x ) ] × k = m 2 m - 2 { w s ( x ) + 2 m - 1 - k 2 k [ w a ( x ) + w - a ( x ) ] } r r ( x ) d x = w s ( x ) r i ( x ) d x + w a ( x ) 2 ( 1 - 2 m ) r i ( x ) d x + 1 2 ( k = 0 m - 1 k + 1 2 k + k = m 2 m - 2 2 m - 1 - k 2 k ) w a ( x ) r r ( x ) d x ,
Var { O s i ( ) ( u ) } = w s ( x ) r i ( x ) d x + 2 w a ( x ) r r ( x ) d x .
O s r ( m ) ( u ) = F { w e ( x ) F - 1 [ O e r ( u ) ] } + j F { w o ( x ) w e m - 1 ( x ) F - 1 [ O e i ( u ) ] } + k = 0 m - 2 F { w e k ( x ) w o 2 ( x ) F - 1 [ O e r ( u ) ] } ,
Var { O s r ( m ) ( u ) } = w s ( x ) r r ( x ) d x + w a ( x ) 2 ( 1 - 2 m ) r i ( x ) d x + [ 1 2 + k = 1 m - 1 1 2 k + 1 8 k = 0 m - 2 k + 1 2 k + 1 8 k = m - 1 2 m - 4 2 m - 3 - k 2 k ] w a ( x ) r r ( x ) d x .
Var { O s r ( ) ( u ) } = w s ( x ) r r ( x ) d x + 2 w a ( x ) r r ( x ) d x .
n m ( x 1 ) n m ( x 2 ) = - [ N r ( m ) ( u 1 ) + j N i ( m ) ( u 1 ) ] exp ( j 2 π x 1 u 1 ) d u 1 × - [ N r ( m ) ( u 2 ) - j N i ( m ) ( u 2 ) ] exp ( - j 2 π x 2 u 2 ) d u 2 = - - [ N r ( m ) ( u 1 ) N r ( m ) ( u 2 ) + N i ( m ) ( u 1 ) N i ( m ) ( u 2 ) - j N r ( m ) ( u 1 ) N i ( m ) ( u 2 ) + j N i ( m ) ( u 1 ) N r ( m ) ( u 2 ) ] × exp ( j 2 π x 1 u 1 ) exp ( - j 2 π x 2 u 2 ) d u 2 d u 1 .
n m ( x 1 ) n m ( x 2 ) = - - [ R r ( m ) ( Δ u ) + R i ( m ) ( Δ u ) + j R i r ( m ) ( Δ u ) - j R r i ( m ) ( Δ u ) ] × exp ( j 2 π x 1 Δ u ) exp [ j 2 π ( x 1 - x 2 ) u 2 ] d ( Δ u ) d u 2 = F - 1 { R r ( m ) ( u ) + R i ( m ) ( u ) + j R i r ( m ) ( u ) - j R r i ( m ) ( u ) } × δ ( x 1 - x 2 ) ,
σ m 2 ( x ) F - 1 { R r ( m ) ( u ) + R i ( m ) ( u ) + j R i r ( m ) ( u ) - j R r i ( m ) ( u ) } = [ r r ( x ) + r i ( x ) ] w s ( x ) + 2 ( 2 - 2 m ) r i ( x ) w a ( x ) + 1 2 ( k = 0 m - 1 k + 1 2 k + k = m 2 m - 2 2 m - 1 - k 2 k ) w a ( x ) r r ( x ) + ( 1 2 + k = 1 m - 1 1 2 k + 1 8 k = 0 m - 2 k + 1 2 k + 1 8 k = m - 1 2 m - 4 2 m - 3 - k 2 k ) w a ( x ) r r ( x ) ,
σ 2 ( x ) = [ r r ( x ) + r i ( x ) ] w s ( x ) + 4 r r ( x ) w a ( x ) .
σ m 2 ( x ) - σ m - 1 2 ( x ) = 1 2 2 m { [ ( 5 + k = 0 m - 4 1 2 k + k = 0 m - 5 1 2 k + 1 ) 2 m + 20 ] r r ( x ) - 12 r i ( x ) } × w a ( x ) , m 4 , = 1 16 [ 13 r r ( x ) - 3 r i ( x ) ] w a ( x ) , m = 3 , = 1 4 [ 5 r r ( x ) - 3 r i ( x ) ] w a ( x ) , m = 2.
r r ( x ) r i ( x ) = 3 5 .
w a ( x ) r r ( x ) d x w a ( x ) r i ( x ) d x = 3 5 ,

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