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 bandlimited system followed by an absolute value squaring operation and a hardlimiter. This is the system that represents coherent image formation using a high-contrast printer. A complex-valued solution is found using an iterative method which employs alternating projections, first onto the set of bandlimited functions, then onto the set of functions whose magnitudes are above and below the desired theshold. Enlarging the class of allowed inputs to include complex-valued signals, instead of real-valued signals, widens the domain of possible solutions to include solutions for problems with smaller –bandwidth products.

© 1985 Optical Society of America

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References

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  1. B. E. A. Saleh, S. I. Sayegh, “Reduction of Errors of Microphotographic Reproductions by Optimal Corrections of Original Masks,” Opt. Eng. 20, 781 (1981).
    [Crossref]
  2. 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. 5, 441 (1983).
    [Crossref]
  3. 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 ( 33Apr.1985).
    [Crossref]
  4. K. M. Nashold, B. E. A. Saleh, “Image Construction Through Diffraction-Limited Systems: An Iterative Approach,” J. Opt. Soc. Am. A 2 (May1985).
    [Crossref]
  5. A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
    [Crossref]
  6. R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
    [Crossref]
  7. D. C. Youla, “Generalized Image Restoration by the Method of Alternative Orthogonal Projections,” IEEE Trans. Circuits Syst. CAS-25, 695 (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 (1982).
    [Crossref]
  9. J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758 (1982).
    [Crossref] [PubMed]
  10. M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving Resolution in Photolithography with a Phase-Shift Mask,” IEEE Trans. Electron Devices ED-29, 1828 (1982).
    [Crossref]
  11. D. S. Goodman, M. D. Levenson, H. Santini, V. Viswanathan, “Improved Photolithographic Resolution with a Phase-Shifting Mask,” in Proceedings, Microcircuit '82, the 1982 Grenoble Photolithography Conference.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 131.
  13. D. C. Youla, Department of Electrical Engineering, Polytechnic Institute of New York; private communications.
  14. Z. Opial, “Weak Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings,” Bull. Am. Math. Soc. 73, 591 (1967).
    [Crossref]

1985 (2)

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 ( 33Apr.1985).
[Crossref]

K. M. Nashold, B. E. A. Saleh, “Image Construction Through Diffraction-Limited Systems: An Iterative Approach,” J. Opt. Soc. Am. A 2 (May1985).
[Crossref]

1983 (1)

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. 5, 441 (1983).
[Crossref]

1982 (3)

D. C. Youla, H. Webb, “Image Restoration by the Method of Convex Projections: Part 1—Theory,” IEEE Trans. Med. Imaging MI-1, 81 (1982).
[Crossref]

J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758 (1982).
[Crossref] [PubMed]

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving Resolution in Photolithography with a Phase-Shift Mask,” IEEE Trans. Electron Devices ED-29, 1828 (1982).
[Crossref]

1981 (1)

B. E. A. Saleh, S. I. Sayegh, “Reduction of Errors of Microphotographic Reproductions by Optimal Corrections of Original Masks,” Opt. Eng. 20, 781 (1981).
[Crossref]

1978 (1)

D. C. Youla, “Generalized Image Restoration by the Method of Alternative Orthogonal Projections,” IEEE Trans. Circuits Syst. CAS-25, 695 (1978).

1975 (1)

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[Crossref]

1974 (1)

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[Crossref]

1967 (1)

Z. Opial, “Weak Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings,” Bull. Am. Math. Soc. 73, 591 (1967).
[Crossref]

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[Crossref]

Goodman, D. S.

D. S. Goodman, M. D. Levenson, H. Santini, V. Viswanathan, “Improved Photolithographic Resolution with a Phase-Shifting Mask,” in Proceedings, Microcircuit '82, the 1982 Grenoble Photolithography Conference.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 131.

Levenson, M. D.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving Resolution in Photolithography with a Phase-Shift Mask,” IEEE Trans. Electron Devices ED-29, 1828 (1982).
[Crossref]

D. S. Goodman, M. D. Levenson, H. Santini, V. Viswanathan, “Improved Photolithographic Resolution with a Phase-Shifting Mask,” in Proceedings, Microcircuit '82, the 1982 Grenoble Photolithography Conference.

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 ( 33Apr.1985).
[Crossref]

K. M. Nashold, B. E. A. Saleh, “Image Construction Through Diffraction-Limited Systems: An Iterative Approach,” J. Opt. Soc. Am. A 2 (May1985).
[Crossref]

Opial, Z.

Z. Opial, “Weak Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings,” Bull. Am. Math. Soc. 73, 591 (1967).
[Crossref]

Papoulis, A.

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[Crossref]

Saleh, B. E. A.

K. M. Nashold, B. E. A. Saleh, “Image Construction Through Diffraction-Limited Systems: An Iterative Approach,” J. Opt. Soc. Am. A 2 (May1985).
[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 ( 33Apr.1985).
[Crossref]

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. 5, 441 (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 (1981).
[Crossref]

Santini, H.

D. S. Goodman, M. D. Levenson, H. Santini, V. Viswanathan, “Improved Photolithographic Resolution with a Phase-Shifting Mask,” in Proceedings, Microcircuit '82, the 1982 Grenoble Photolithography Conference.

Sayegh, S. I.

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 ( 33Apr.1985).
[Crossref]

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. 5, 441 (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 (1981).
[Crossref]

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving Resolution in Photolithography with a Phase-Shift Mask,” IEEE Trans. Electron Devices ED-29, 1828 (1982).
[Crossref]

Viswanathan, N. S.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving Resolution in Photolithography with a Phase-Shift Mask,” IEEE Trans. Electron Devices ED-29, 1828 (1982).
[Crossref]

Viswanathan, V.

D. S. Goodman, M. D. Levenson, H. Santini, V. Viswanathan, “Improved Photolithographic Resolution with a Phase-Shifting Mask,” in Proceedings, Microcircuit '82, the 1982 Grenoble Photolithography Conference.

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 (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 (1982).
[Crossref]

D. C. Youla, “Generalized Image Restoration by the Method of Alternative Orthogonal Projections,” IEEE Trans. Circuits Syst. CAS-25, 695 (1978).

D. C. Youla, Department of Electrical Engineering, Polytechnic Institute of New York; private communications.

Appl. Opt. (1)

Bull. Am. Math. Soc. (1)

Z. Opial, “Weak Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings,” Bull. Am. Math. Soc. 73, 591 (1967).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process (1)

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 ( 33Apr.1985).
[Crossref]

IEEE Trans. Circuits Syst. (2)

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[Crossref]

D. C. Youla, “Generalized Image Restoration by the Method of Alternative Orthogonal Projections,” IEEE Trans. Circuits Syst. CAS-25, 695 (1978).

IEEE Trans. Electron Devices (1)

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving Resolution in Photolithography with a Phase-Shift Mask,” IEEE Trans. Electron Devices ED-29, 1828 (1982).
[Crossref]

IEEE Trans. Med. Imaging (1)

D. C. Youla, H. Webb, “Image Restoration by the Method of Convex Projections: Part 1—Theory,” IEEE Trans. Med. Imaging MI-1, 81 (1982).
[Crossref]

IEEE Trans. Pattern Recognition Mach. Intell. (1)

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. 5, 441 (1983).
[Crossref]

J. Opt. Soc. Am. A (1)

K. M. Nashold, B. E. A. Saleh, “Image Construction Through Diffraction-Limited Systems: An Iterative Approach,” J. Opt. Soc. Am. A 2 (May1985).
[Crossref]

Opt. Acta (1)

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[Crossref]

Opt. Eng. (1)

B. E. A. Saleh, S. I. Sayegh, “Reduction of Errors of Microphotographic Reproductions by Optimal Corrections of Original Masks,” Opt. Eng. 20, 781 (1981).
[Crossref]

Other (3)

D. S. Goodman, M. D. Levenson, H. Santini, V. Viswanathan, “Improved Photolithographic Resolution with a Phase-Shifting Mask,” in Proceedings, Microcircuit '82, the 1982 Grenoble Photolithography Conference.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 131.

D. C. Youla, Department of Electrical Engineering, Polytechnic Institute of New York; private communications.

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Figures (11)

Fig. 1
Fig. 1

System.

Fig. 2
Fig. 2

Algorithm.

Fig. 3
Fig. 3

Desired output.

Fig. 4
Fig. 4

(a) Magnitude of output of bandlimiter when desired output is used as input. (b) Resulting output from hardlimiter when input to hardlimiter is as in (a).

Fig. 5
Fig. 5

(a) Magnitude of output of bandlimiter when input of Fig. 6 is used; (b) resulting output from hardlimiter.

Fig. 6
Fig. 6

Magnitude and phase of input as a function of distance x.

Fig. 7
Fig. 7

Constructed input found after 100 iterations when the initial input used in the algorithm was the real valued function in Fig. 3.

Fig. 8
Fig. 8

Output obtained using the constructed input of Fig. 7.

Fig. 9
Fig. 9

Magnitude and phase of initial input which resulted in the best constructed input.

Fig. 10
Fig. 10

(a) Magnitude of constructed input found after 100 iterations when initial input had a 90° phase difference between pulses; (b) phase of constructed input.

Fig. 11
Fig. 11

Output obtained using the constructed input of Fig. 10.

Equations (19)

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g = S f ,
g = S f , S = L B ,
g b ( x ) = B f ( x ) = F 1 [ H ( ω ) F ( ω ) ] ,
H ( ω ) = 1 , for ω Ω = 0 , elsewhere ,
g ( x ) = 1 , for | g b ( x ) | 2 T , = 0 , otherwise ,
g ( x ) = 1 , x Δ + = 0 , x Δ .
f ( x ) = A 0 δ ( x d 2 ) + A 0 δ ( x + d 2 ) .
f b ( x ) = A 0 2 ν 0 sinc [ ( x d 2 ) 2 ν 0 ] + A 0 2 ν 0 sinc [ ( x + d 2 ) 2 ν 0 ] ,
f ( x ) = A 0 δ ( x d 2 ) + A 0 exp ( j θ 0 ) δ ( x + d 2 )
| f b ( x ) | 2 = | A 0 2 ν | 2 { sinc 2 [ ( x d 2 ) 2 ν 0 ] + sinc 2 [ ( x + d 2 ) 2 ν 0 ] + 2 sinc [ ( x d 2 ) 2 ν 0 ] sinc [ ( x + d 2 ) 2 ν 0 ] cos ( θ 0 ) } .
| f ( x ) | T for x Δ + ,
| f ( x ) | T for x Δ .
f C 0 = i = 1 m C i .
P 1 f = F 1 [ H ( ω ) F ( ω ) ]
P 2 f = f ( x ) x Δ , = f ( x ) otherwise ,
P 3 f = f + ( x ) x Δ + = f ( x ) otherwise ,
f ( x ) = f ( x ) if | f ( x ) | T , = T exp [ j θ ( x ) ] if | f ( x ) | > T ,
f + ( x ) = f ( x ) if | f ( x ) | T , = T exp [ j θ ( x ) ] if | f ( x ) | < T .
α n = k = ν 0 ν 0 F n ( k Ω ) / F n ( k Ω )

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