Abstract

In an attempt to improve the convergence rates of band-limited image restoration, we derive and apply two algorithms employing nonlinear projections onto closed convex sets. The performances of these algorithms are compared with the well-known Gerchberg—Papoulis (GP) procedures for several cases and are shown to have superior initial convergence rates although eventually they behave like the GP procedures. Both algorithms are shown to converge weakly (i.e., inner product convergence) to the unknown image.

© 1982 Optical Society of America

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  1. R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
  2. A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
  3. T. Sato et al., "Tomographic image reconstruction in limited projections using iterative revisions in image and transform spaces," Appl. Opt. 20, 395–399 (1980).
  4. D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. CAS-25, 695–702 (1978).
  5. D. C. Youla, "Image restoration by the method of projections onto convex sets—part I," Rep. POLY-MRI- (Polytechnich Institute of New York, New York, 1981), pp. 1420–1481.
  6. V. T. Tom, T. F. Quatieri, M. H. Hayes, and J. H. McClellan, "Convergence of iterative nonexpansive signal reconstruction algorithms," IEEE Trans. Acoust. Speech Signal Proc. ASSP-29, 1052–1058 (1981).
  7. A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).

1981 (2)

V. T. Tom, T. F. Quatieri, M. H. Hayes, and J. H. McClellan, "Convergence of iterative nonexpansive signal reconstruction algorithms," IEEE Trans. Acoust. Speech Signal Proc. ASSP-29, 1052–1058 (1981).

A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).

1980 (1)

1978 (1)

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

1975 (1)

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

1974 (1)

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).

Gerchberg, R. W.

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).

Hayes, M. H.

V. T. Tom, T. F. Quatieri, M. H. Hayes, and J. H. McClellan, "Convergence of iterative nonexpansive signal reconstruction algorithms," IEEE Trans. Acoust. Speech Signal Proc. ASSP-29, 1052–1058 (1981).

Lent, A.

A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).

McClellan, J. H.

V. T. Tom, T. F. Quatieri, M. H. Hayes, and J. H. McClellan, "Convergence of iterative nonexpansive signal reconstruction algorithms," IEEE Trans. Acoust. Speech Signal Proc. ASSP-29, 1052–1058 (1981).

Papoulis, A.

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

Quatieri, T. F.

V. T. Tom, T. F. Quatieri, M. H. Hayes, and J. H. McClellan, "Convergence of iterative nonexpansive signal reconstruction algorithms," IEEE Trans. Acoust. Speech Signal Proc. ASSP-29, 1052–1058 (1981).

Sato, T.

Tom, V. T.

V. T. Tom, T. F. Quatieri, M. H. Hayes, and J. H. McClellan, "Convergence of iterative nonexpansive signal reconstruction algorithms," IEEE Trans. Acoust. Speech Signal Proc. ASSP-29, 1052–1058 (1981).

Tuy, H.

A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).

Youla, D. C.

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

D. C. Youla, "Image restoration by the method of projections onto convex sets—part I," Rep. POLY-MRI- (Polytechnich Institute of New York, New York, 1981), pp. 1420–1481.

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Proc. (1)

V. T. Tom, T. F. Quatieri, M. H. Hayes, and J. H. McClellan, "Convergence of iterative nonexpansive signal reconstruction algorithms," IEEE Trans. Acoust. Speech Signal Proc. ASSP-29, 1052–1058 (1981).

IEEE Trans. Circuits Syst. (2)

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

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

J. Math. Anal. Appl. (1)

A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).

Opt. Acta (1)

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).

Other (1)

D. C. Youla, "Image restoration by the method of projections onto convex sets—part I," Rep. POLY-MRI- (Polytechnich Institute of New York, New York, 1981), pp. 1420–1481.

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