Abstract

It is desired to preprocess an input image so that, when it is distorted by an imaging system, a prescribed output image is produced. The system of interest is a linear, shift-invariant, band-limited system followed by a hard limiter. Such a system is found, for example, in microphotography, when a camera that is band limited by diffraction effects is used to print on very-high-contrast film. In this paper, an iterative solution that employs alternating projections is presented. Two variations of the procedure, which is similar to the Gerchberg–Papoulis algorithm, are applied to several examples having different space–bandwidth products. Also, a method of overrelaxing the projections to improve convergence is studied.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Eastman Kodak Company, Techniques of Microphotography, Pub. P-52 (Eastman Kodak Company, Rochester, N.Y., 1967).
  2. B. E. A. Saleh, S. I. Sayegh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
    [CrossRef]
  3. S. I. Sayegh, B. E. A. Saleh, “Image design: generation of a prescribed image at the output of a band-limited system,”IEEE Trans. Pattern Recognition Mach. Intell. PAMI-5, 441–445 (1983).
    [CrossRef]
  4. S. I. Sayegh, B. E. A. Saleh, K. M. Nashold, “Image design: generation of a prescribed image through a diffraction-limited system with high-contrast recording,” IEEE Trans. Acoust. Speech Signal Process. (to be published).
  5. A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  6. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  7. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 695–702 (1978).
  8. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—Theory,”IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [CrossRef]
  9. Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Am. Math. Soc. 73, 591–597 (1967).
    [CrossRef]
  10. S. I. Sayegh, Department of Electrical Engineering and Systems Science, Michigan State University, East Lansing, Michigan 48824 (personal communication).

1983

S. I. Sayegh, B. E. A. Saleh, “Image design: generation of a prescribed image at the output of a band-limited system,”IEEE Trans. Pattern Recognition Mach. Intell. PAMI-5, 441–445 (1983).
[CrossRef]

1982

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

1981

B. E. A. Saleh, S. I. Sayegh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
[CrossRef]

1978

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 695–702 (1978).

1975

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1967

Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Am. Math. Soc. 73, 591–597 (1967).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Nashold, K. M.

S. I. Sayegh, B. E. A. Saleh, K. M. Nashold, “Image design: generation of a prescribed image through a diffraction-limited system with high-contrast recording,” IEEE Trans. Acoust. Speech Signal Process. (to be published).

Opial, Z.

Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Am. Math. Soc. 73, 591–597 (1967).
[CrossRef]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Saleh, B. E. A.

S. I. Sayegh, B. E. A. Saleh, “Image design: generation of a prescribed image at the output of a band-limited system,”IEEE Trans. Pattern Recognition Mach. Intell. PAMI-5, 441–445 (1983).
[CrossRef]

B. E. A. Saleh, S. I. Sayegh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
[CrossRef]

S. I. Sayegh, B. E. A. Saleh, K. M. Nashold, “Image design: generation of a prescribed image through a diffraction-limited system with high-contrast recording,” IEEE Trans. Acoust. Speech Signal Process. (to be published).

Sayegh, S. I.

S. I. Sayegh, B. E. A. Saleh, “Image design: generation of a prescribed image at the output of a band-limited system,”IEEE Trans. Pattern Recognition Mach. Intell. PAMI-5, 441–445 (1983).
[CrossRef]

B. E. A. Saleh, S. I. Sayegh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
[CrossRef]

S. I. Sayegh, B. E. A. Saleh, K. M. Nashold, “Image design: generation of a prescribed image through a diffraction-limited system with high-contrast recording,” IEEE Trans. Acoust. Speech Signal Process. (to be published).

S. I. Sayegh, Department of Electrical Engineering and Systems Science, Michigan State University, East Lansing, Michigan 48824 (personal communication).

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—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 1—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, 695–702 (1978).

Bull. Am. Math. Soc.

Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Am. Math. Soc. 73, 591–597 (1967).
[CrossRef]

IEEE Trans. Circuits Syst.

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 695–702 (1978).

IEEE Trans. Med. Imaging

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

IEEE Trans. Pattern Recognition Mach. Intell.

S. I. Sayegh, B. E. A. Saleh, “Image design: generation of a prescribed image at the output of a band-limited system,”IEEE Trans. Pattern Recognition Mach. Intell. PAMI-5, 441–445 (1983).
[CrossRef]

Opt. Acta

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Eng.

B. E. A. Saleh, S. I. Sayegh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
[CrossRef]

Other

Eastman Kodak Company, Techniques of Microphotography, Pub. P-52 (Eastman Kodak Company, Rochester, N.Y., 1967).

S. I. Sayegh, B. E. A. Saleh, K. M. Nashold, “Image design: generation of a prescribed image through a diffraction-limited system with high-contrast recording,” IEEE Trans. Acoust. Speech Signal Process. (to be published).

S. I. Sayegh, Department of Electrical Engineering and Systems Science, Michigan State University, East Lansing, Michigan 48824 (personal communication).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

The imaging system.

Fig. 2
Fig. 2

An illustration of the steps of the algorithm, starting with the zeroth iteration f0(x), computing the first iteration f1, and preceding to compute the next iteration. (a) Method 1; (b) method 2.

Fig. 3
Fig. 3

(a) The desired signal f(x); (b) its band-limited version g1(x) when the bandwidth ν0 is such that the space–bandwidth product 0 = 0.8.

Fig. 4
Fig. 4

As in Fig. 3, but 0 = 0.6.

Fig. 5
Fig. 5

As in Figs. 3 and 4, but 0 = 0.4.

Fig. 6
Fig. 6

The constructed signal (before hard-limiting) g1(x) after α had converged to 1.0 for 0 = 0.8.

Fig. 7
Fig. 7

As in Fig. 6, but 0 = 0.6.

Fig. 8
Fig. 8

As in Figs. 6 and 7, but 0 = 0.4.

Fig. 9
Fig. 9

A two-dimensional signal f(x) to be constructed.

Fig. 10
Fig. 10

The output of the imaging system if the desired output of Fig. 9 is used as the input, 0 = 0.8.

Fig. 11
Fig. 11

The constructed signal for 0 = 0.8.

Fig. 12
Fig. 12

The accuracy parameter αm versus the number of iterations m. For different values of the overrelaxation parameter λ = 1.0 (squares); 1.2 (circles); 1.4 (triangles); 1.6 (diamonds); 1.8 (crosses). (a) Method 2, 0 = 0.8; (b) method 2, 0 = 0.6; (c) method 2, 0 = 0.4; (d) method 1, 0 = 0.8; (e) method 1, 0 = 0.6; (f) method 1, 0 = 0.4.

Tables (4)

Tables Icon

Table 1 Number of Iterations m Required to Reach αm = 0.9999

Tables Icon

Table 2 Accuracy after the Zeroth and First Iteration

Tables Icon

Table 3 Accuracy of the First Iteration α1 and Number of Iterations m Needed to Reach αm = 09999 and αm = 1.0000 as a Function of Overrelaxation Constant λa

Tables Icon

Table 4 Accuracy of the Second Iteration α2 and Number of Iterations m Needed to Reach αm = 09999 and αm = 1.0000 as a Function of Overrelaxation Constant λa

Equations (22)

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

f C 0 = i = 1 m C i .
T = P m P m - 1 , , P 1 ,
T i = 1 + λ i ( P i - 1 ) ,             i = 1 , 2 , , m .
T = T m T m - 1 T 1 .
f ( x ) θ , x Δ , f ( x ) θ , x Δ c ,
P 1 f ( x ) = F - 1 [ H ( ω ) F ( ω ) ] ,
H ( ω ) = 1 , ω ω 0 , = 0 , elsewhere ,
if f ( x ) > θ , P 2 f ( x ) = f ( x ) , otherwise , P 2 f ( x ) = θ .
if f ( x ) < θ , P 2 f ( x ) = f ( x ) , otherwise , P 2 f ( x ) = θ .
f m ( x ) = T m f 0 ( x ) .
α m = - ω 0 ω 0 F m ( ω ) 2 d ω / - F m ( ω ) 2 d ω ,
if f ( x ) > θ , P 2 f ( x ) = f ( x ) , otherwise , P 2 f ( x ) = 2 θ - f ( x ) ,
if f ( x ) < θ , P f 2 ( x ) = f ( x ) , otherwise , P 2 f ( x ) = 2 θ - f ( x ) .
P 1 f ( n ) = F - 1 [ H ( k ) F ( k ) ] ,             n = 0 , 1 , , N - 1 , k = - N 2 , , 0 , , N 2 - 1.
H ( k ) = 1 , k k 0 , = 0 , elsewhere ,
α m = k = - k 0 k 0 F m ( k ) 2 / k = - F m ( k ) 2 ,
P f = f ( x ) s ( x ) ,             for f ( x ) > > 0 , = s ( x ) ,             elsewhere ,
T f - T g f - g             for all f , g C ,
P 2 f ( x ) = f ( x ) s ( x ) ,
f 1 - f 2 2 = - f 1 ( x ) - f 2 ( x ) 2 d x - [ f 1 ( x ) - f 2 ( x ) ] 2 d x = - [ f 1 ( x ) s ( x ) - f 2 ( x ) s ( x ) ] 2 d x = - P 2 f 1 - P 2 f 2 2 d x = P 2 f 1 - P 2 f 2 2 .
f 1 - f 2 P 2 f 1 - P 2 f 2 .             Q . E . D .
P 2 n f = f ( x ) × s ( x ) s ( x ) × s ( x ) = f ( x ) × s ( x ) = P 2 n + 1 f .

Metrics