Abstract

We investigate the problem of representing an arbitrary class of real functions ƒ(·) in terms of their sampled values along the radius <i>r</i> and at equal angular increments of the azimuthal angle θ. Two different bandwidth constraints on ƒ(<i>r</i>,θ) are considered: Fourier and Hankel. The end result is two theorems which enable images to be reconstructed from their samples. The theorems have potential application in image storage, image encoding, and computer-aided tomography.

© 1979 Optical Society of America

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  1. G. H. Watson, Theory of Bessel Functions (Cambridge University, New York, 1944), pp. 576–596.
  2. Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 254.
  3. H. Gamo, Matrix Treatment of Partial Coherence, Progress in Optics, Vol. III, edited by E. Wolf (North-Holland, Amsterdam, 1964), pp. 302–304.
  4. H. Gamo, Jpn J. Appl. Phys. 26, 102–114 (1957).
  5. A. J. Jerri, Proc. IEEE 65, 1565–1592 (1977).
  6. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 163.
  7. M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions (Dover, New York, 1965), p. 371.
  8. H. Stark and C. S. Sarna, Appl. Opt. 18, 2086–2088 (1979).
  9. L. R. Rabiner and B. Guld, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, New Jersey, 1975), p. 393.
  10. C. Braccini and A. V. Oppenheim, IEEE Trans. Acoust. Speech Signal Process. ASSP-22, 236–244 (1974).
  11. H. Stark, I. N. Paul, and C. S. Sarna, "Fourier-Transform Reconstruction in CAT by Exact Interpolation," to be presented at IEEE Conference on Engineering in Medicine and Biology, Oct. 6–7, 1979, Denver, Colorado (unpublished).
  12. F. W. J. Olver (editor), Royal Society Mathematical Tables, 7, (Cambridge University, Cambridge, 1960).
  13. British Association Mathematical Tables VI, Part I, (Cambridge University, Cambridge, 1958); additional information on Hankel and other self-reciprocal transforms may be found in E. C. Titmarsh, Introduction to the Theory of Fourier Integrals, (Oxford University, Oxford, 1937).

Braccini, C.

C. Braccini and A. V. Oppenheim, IEEE Trans. Acoust. Speech Signal Process. ASSP-22, 236–244 (1974).

Gamo, H.

H. Gamo, Matrix Treatment of Partial Coherence, Progress in Optics, Vol. III, edited by E. Wolf (North-Holland, Amsterdam, 1964), pp. 302–304.

H. Gamo, Jpn J. Appl. Phys. 26, 102–114 (1957).

Guld, B.

L. R. Rabiner and B. Guld, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, New Jersey, 1975), p. 393.

Jerri, A. J.

A. J. Jerri, Proc. IEEE 65, 1565–1592 (1977).

Luke, Y. L.

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 254.

Oppenheim, A. V.

C. Braccini and A. V. Oppenheim, IEEE Trans. Acoust. Speech Signal Process. ASSP-22, 236–244 (1974).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 163.

Paul, I. N.

H. Stark, I. N. Paul, and C. S. Sarna, "Fourier-Transform Reconstruction in CAT by Exact Interpolation," to be presented at IEEE Conference on Engineering in Medicine and Biology, Oct. 6–7, 1979, Denver, Colorado (unpublished).

Rabiner, L. R.

L. R. Rabiner and B. Guld, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, New Jersey, 1975), p. 393.

Sarna, C. S.

H. Stark and C. S. Sarna, Appl. Opt. 18, 2086–2088 (1979).

H. Stark, I. N. Paul, and C. S. Sarna, "Fourier-Transform Reconstruction in CAT by Exact Interpolation," to be presented at IEEE Conference on Engineering in Medicine and Biology, Oct. 6–7, 1979, Denver, Colorado (unpublished).

Stark, H.

H. Stark, I. N. Paul, and C. S. Sarna, "Fourier-Transform Reconstruction in CAT by Exact Interpolation," to be presented at IEEE Conference on Engineering in Medicine and Biology, Oct. 6–7, 1979, Denver, Colorado (unpublished).

H. Stark and C. S. Sarna, Appl. Opt. 18, 2086–2088 (1979).

Watson, G. H.

G. H. Watson, Theory of Bessel Functions (Cambridge University, New York, 1944), pp. 576–596.

Other (13)

G. H. Watson, Theory of Bessel Functions (Cambridge University, New York, 1944), pp. 576–596.

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 254.

H. Gamo, Matrix Treatment of Partial Coherence, Progress in Optics, Vol. III, edited by E. Wolf (North-Holland, Amsterdam, 1964), pp. 302–304.

H. Gamo, Jpn J. Appl. Phys. 26, 102–114 (1957).

A. J. Jerri, Proc. IEEE 65, 1565–1592 (1977).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 163.

M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions (Dover, New York, 1965), p. 371.

H. Stark and C. S. Sarna, Appl. Opt. 18, 2086–2088 (1979).

L. R. Rabiner and B. Guld, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, New Jersey, 1975), p. 393.

C. Braccini and A. V. Oppenheim, IEEE Trans. Acoust. Speech Signal Process. ASSP-22, 236–244 (1974).

H. Stark, I. N. Paul, and C. S. Sarna, "Fourier-Transform Reconstruction in CAT by Exact Interpolation," to be presented at IEEE Conference on Engineering in Medicine and Biology, Oct. 6–7, 1979, Denver, Colorado (unpublished).

F. W. J. Olver (editor), Royal Society Mathematical Tables, 7, (Cambridge University, Cambridge, 1960).

British Association Mathematical Tables VI, Part I, (Cambridge University, Cambridge, 1958); additional information on Hankel and other self-reciprocal transforms may be found in E. C. Titmarsh, Introduction to the Theory of Fourier Integrals, (Oxford University, Oxford, 1937).

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