Abstract

This paper deals with the long-standing problem of reconstructing an optical object of known finite spatial extent from an image that is a noisy low-pass-filtered version of the object. The development of a number of iterative reconstruction algorithms in recent years has created a resurgence of interest in this topic. We consider several reconstruction algorithms from a point of view that illuminates their performance in the presence of noise. We also introduce a new iterative reconstruction algorithm and compare its performance with that of an earlier algorithm proposed by Gerchberg.

© 1980 Optical Society of America

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  1. B. R. Frieden, "Image Enhancement and Restoration," in Topics in Applied Physics, Picture Processing and Digital Filters, edited by T. S. Huang (Springer-Verlag, New York, 1975), Vol. 6.
  2. R. W. Gerchberg, "Super-Resolution through Error-Energy Reduction," Opt. Acta 21, 709–720 (1974).
  3. D. C. Youla, "Generalized Image Restoration by the Method of Alternating Orthogonal Projections," IEEE Trans. Circ. Syst. CAS-25, 694–702 (1978).
  4. C. K. Rushforth and R. W. Harris, "Restoration, Resolution, and Noise," J. Opt. Soc. Am. 58, 539–545 (1968).
  5. D. Slepian and H. O. Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty—I," Bell Syst. Tech. J. 40, 43–63 (1961).
  6. A. Papoulis, "A New Algorithm in Spectral Analysis and Bandlimited Extrapolation," IEEE Trans. Circ. Syst. CAS-22, 735–742 (1975).
  7. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  8. M.S. Sabri and W. Steenaart, "An Approach to Bandlimited Signal Extrapolation—The Extrapolation Matrix," IEEE Trans. Circ. Sys. CAS-25, 74–78 (1978).
  9. G. Cesini, G. Guattani, G. Lucarini, and C. Palma, "An Iterative Method for Restoring Noisy Images," Opt. Acta 25, 501–508 (1978).

1978

D. C. Youla, "Generalized Image Restoration by the Method of Alternating Orthogonal Projections," IEEE Trans. Circ. Syst. CAS-25, 694–702 (1978).

M.S. Sabri and W. Steenaart, "An Approach to Bandlimited Signal Extrapolation—The Extrapolation Matrix," IEEE Trans. Circ. Sys. CAS-25, 74–78 (1978).

G. Cesini, G. Guattani, G. Lucarini, and C. Palma, "An Iterative Method for Restoring Noisy Images," Opt. Acta 25, 501–508 (1978).

1975

A. Papoulis, "A New Algorithm in Spectral Analysis and Bandlimited Extrapolation," IEEE Trans. Circ. Syst. CAS-22, 735–742 (1975).

B. R. Frieden, "Image Enhancement and Restoration," in Topics in Applied Physics, Picture Processing and Digital Filters, edited by T. S. Huang (Springer-Verlag, New York, 1975), Vol. 6.

1974

R. W. Gerchberg, "Super-Resolution through Error-Energy Reduction," Opt. Acta 21, 709–720 (1974).

1968

1961

D. Slepian and H. O. Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty—I," Bell Syst. Tech. J. 40, 43–63 (1961).

Cesini, G.

G. Cesini, G. Guattani, G. Lucarini, and C. Palma, "An Iterative Method for Restoring Noisy Images," Opt. Acta 25, 501–508 (1978).

Frieden, B. R.

B. R. Frieden, "Image Enhancement and Restoration," in Topics in Applied Physics, Picture Processing and Digital Filters, edited by T. S. Huang (Springer-Verlag, New York, 1975), Vol. 6.

Gerchberg, R. W.

R. W. Gerchberg, "Super-Resolution through Error-Energy Reduction," Opt. Acta 21, 709–720 (1974).

Guattani, G.

G. Cesini, G. Guattani, G. Lucarini, and C. Palma, "An Iterative Method for Restoring Noisy Images," Opt. Acta 25, 501–508 (1978).

Harris, R. W.

Lucarini, G.

G. Cesini, G. Guattani, G. Lucarini, and C. Palma, "An Iterative Method for Restoring Noisy Images," Opt. Acta 25, 501–508 (1978).

Palma, C.

G. Cesini, G. Guattani, G. Lucarini, and C. Palma, "An Iterative Method for Restoring Noisy Images," Opt. Acta 25, 501–508 (1978).

Papoulis, A.

A. Papoulis, "A New Algorithm in Spectral Analysis and Bandlimited Extrapolation," IEEE Trans. Circ. Syst. CAS-22, 735–742 (1975).

Pollak, H. O.

D. Slepian and H. O. Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty—I," Bell Syst. Tech. J. 40, 43–63 (1961).

Rushforth, C. K.

Sabri, M.S.

M.S. Sabri and W. Steenaart, "An Approach to Bandlimited Signal Extrapolation—The Extrapolation Matrix," IEEE Trans. Circ. Sys. CAS-25, 74–78 (1978).

Saxton, W. O.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Slepian, D.

D. Slepian and H. O. Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty—I," Bell Syst. Tech. J. 40, 43–63 (1961).

Steenaart, W.

M.S. Sabri and W. Steenaart, "An Approach to Bandlimited Signal Extrapolation—The Extrapolation Matrix," IEEE Trans. Circ. Sys. CAS-25, 74–78 (1978).

Youla, D. C.

D. C. Youla, "Generalized Image Restoration by the Method of Alternating Orthogonal Projections," IEEE Trans. Circ. Syst. CAS-25, 694–702 (1978).

Bell Syst. Tech. J.

D. Slepian and H. O. Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty—I," Bell Syst. Tech. J. 40, 43–63 (1961).

IEEE Trans. Circ. Sys.

M.S. Sabri and W. Steenaart, "An Approach to Bandlimited Signal Extrapolation—The Extrapolation Matrix," IEEE Trans. Circ. Sys. CAS-25, 74–78 (1978).

IEEE Trans. Circ. Syst.

A. Papoulis, "A New Algorithm in Spectral Analysis and Bandlimited Extrapolation," IEEE Trans. Circ. Syst. CAS-22, 735–742 (1975).

D. C. Youla, "Generalized Image Restoration by the Method of Alternating Orthogonal Projections," IEEE Trans. Circ. Syst. CAS-25, 694–702 (1978).

J. Opt. Soc. Am.

Opt. Acta

R. W. Gerchberg, "Super-Resolution through Error-Energy Reduction," Opt. Acta 21, 709–720 (1974).

G. Cesini, G. Guattani, G. Lucarini, and C. Palma, "An Iterative Method for Restoring Noisy Images," Opt. Acta 25, 501–508 (1978).

Other

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

B. R. Frieden, "Image Enhancement and Restoration," in Topics in Applied Physics, Picture Processing and Digital Filters, edited by T. S. Huang (Springer-Verlag, New York, 1975), Vol. 6.

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