Abstract

The mathematical generalization of image restoration by recursive methods furnished by D. C. Youla [IEEE Trans. Circuits Syst. <b>CAS 25</b>, 695–702 (1978)] is used to show that arbitrary <i>L</i><sub>2</sub> (i.e., square-integrable) images can be reconstructed from two projections without any <i>a priori</i> assumption regarding the mathematical properties of the object, such as space-limitedness or band-limitedness. Recursive algorithms are given to restore images from (1) extended segments and low-pass spectra and (2) short segments and high-pass spectra. Using the alternating projection theorem, we prove monotonic convergence (in the norm) to the original image.

© 1981 Optical Society of America

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  1. D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. CAS 25, 695–702 (1978).
  2. A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
  3. R. W. Gershberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
  4. J. von Neumann, The Geometry of Orthogonal Spaces, Vol. II (Princeton U. Press, Princeton, N.J., 1950).
  5. H. Stark et al., "Direct Fourier reconstruction in computer tomography," IEEE Trans. Acoust. Speech Signal Process. ASSP-29 (1981) (to be published).
  6. H. Stark, "Some film-noise measurements by diffraction of coherent light," Appl. Opt. 10, 333–337 (1971).
  7. A. Lent and H. Tuy, "An iterative method for the extrapolation of bandlimited functions," Tech. Rep. MITG 35 (State University of New York at Buffalo, Amherst, N.Y., 1979).
  8. M. Davison and F. A. Grunbaum, "Convolution algorithm for arbitary projection angles," IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).

1981

H. Stark et al., "Direct Fourier reconstruction in computer tomography," IEEE Trans. Acoust. Speech Signal Process. ASSP-29 (1981) (to be published).

1979

M. Davison and F. A. Grunbaum, "Convolution algorithm for arbitary projection angles," IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).

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).

1974

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

1971

Davison, M.

M. Davison and F. A. Grunbaum, "Convolution algorithm for arbitary projection angles," IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).

Gershberg, R. W.

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

Grunbaum, F. A.

M. Davison and F. A. Grunbaum, "Convolution algorithm for arbitary projection angles," IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).

Lent, A.

A. Lent and H. Tuy, "An iterative method for the extrapolation of bandlimited functions," Tech. Rep. MITG 35 (State University of New York at Buffalo, Amherst, N.Y., 1979).

Papoulis, A.

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

Stark, H.

H. Stark et al., "Direct Fourier reconstruction in computer tomography," IEEE Trans. Acoust. Speech Signal Process. ASSP-29 (1981) (to be published).

H. Stark, "Some film-noise measurements by diffraction of coherent light," Appl. Opt. 10, 333–337 (1971).

Tuy, H.

A. Lent and H. Tuy, "An iterative method for the extrapolation of bandlimited functions," Tech. Rep. MITG 35 (State University of New York at Buffalo, Amherst, N.Y., 1979).

von Neumann, J.

J. von Neumann, The Geometry of Orthogonal Spaces, Vol. II (Princeton U. Press, Princeton, N.J., 1950).

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).

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

H. Stark et al., "Direct Fourier reconstruction in computer tomography," IEEE Trans. Acoust. Speech Signal Process. ASSP-29 (1981) (to be published).

IEEE Trans. Circuits Syst.

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).

IEEE Trans. Nucl. Sci.

M. Davison and F. A. Grunbaum, "Convolution algorithm for arbitary projection angles," IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).

Opt. Acta

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

Other

J. von Neumann, The Geometry of Orthogonal Spaces, Vol. II (Princeton U. Press, Princeton, N.J., 1950).

A. Lent and H. Tuy, "An iterative method for the extrapolation of bandlimited functions," Tech. Rep. MITG 35 (State University of New York at Buffalo, Amherst, N.Y., 1979).

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