Abstract

A new method for reconstructing a compact pixellated image from the zero sheet of its z transform is presented. Existing methods either are computationally expensive, restrictive regarding the set of points from the zero sheet used in the reconstruction, or require a good estimate of the image for success. The new method avoids these shortcomings and is illustrated with an example. The potential of the method to improve algorithms that use contaminated zero sheets is briefly discussed.

© 1993 Optical Society of America

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  1. R. H. T. Bates, B. K. Quek, C. R. Parker, J. Opt. Soc. Am. A 7, 468 (1990).
    [CrossRef]
  2. R. H. T. Bates, R. G. Lane, Scanning Microsc. Suppl. 2, 149 (1988).
  3. R. G. Lane, R. H. T. Bates, J. Opt. Soc. Am. A 4, 180 (1987).
    [CrossRef]
  4. R. G. Lane, W. R. Fright, R. H. T. Bates, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520 (1987).
    [CrossRef]
  5. C. R. Parker, P. J. Bones, Opt. Commun. 92, 209 (1992).
    [CrossRef]
  6. R. W. Watson, C. R. Parker, P. J. Bones, Opt. Commun. 93, 359 (1992).
    [CrossRef]
  7. P. J. Bones, B. L. Satherley, R. W. Watson, IEE Conf. Publ. 354, 353 (1992).
  8. S. R. Curtis, A. V. Oppenheim, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643 (1985).
    [CrossRef]
  9. D. Izraelevitz, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511 (1987).
    [CrossRef]
  10. S. I. Sayegh, Y. L. Kok, J. H. Hong, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 107 (1987).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1993 (1)

1992 (3)

C. R. Parker, P. J. Bones, Opt. Commun. 92, 209 (1992).
[CrossRef]

R. W. Watson, C. R. Parker, P. J. Bones, Opt. Commun. 93, 359 (1992).
[CrossRef]

P. J. Bones, B. L. Satherley, R. W. Watson, IEE Conf. Publ. 354, 353 (1992).

1990 (1)

1988 (1)

R. H. T. Bates, R. G. Lane, Scanning Microsc. Suppl. 2, 149 (1988).

1987 (4)

R. G. Lane, R. H. T. Bates, J. Opt. Soc. Am. A 4, 180 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520 (1987).
[CrossRef]

D. Izraelevitz, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511 (1987).
[CrossRef]

S. I. Sayegh, Y. L. Kok, J. H. Hong, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 107 (1987).
[CrossRef]

1985 (2)

M. S. Scivier, M. A. Fiddy, J. Opt. Soc. Am. A 2, 693 (1985).
[CrossRef]

S. R. Curtis, A. V. Oppenheim, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643 (1985).
[CrossRef]

1974 (1)

D. H. Withers, Commun. ACM 17, 157 (1974).
[CrossRef]

1972 (1)

M. A. Jenkins, J. F. Traub, Commun. ACM 15, 97 (1972).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, B. K. Quek, C. R. Parker, J. Opt. Soc. Am. A 7, 468 (1990).
[CrossRef]

R. H. T. Bates, R. G. Lane, Scanning Microsc. Suppl. 2, 149 (1988).

R. G. Lane, R. H. T. Bates, J. Opt. Soc. Am. A 4, 180 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520 (1987).
[CrossRef]

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989), Chap. 2, p. 34.

Bones, P. J.

C. R. Parker, P. J. Bones, Opt. Commun. 92, 209 (1992).
[CrossRef]

R. W. Watson, C. R. Parker, P. J. Bones, Opt. Commun. 93, 359 (1992).
[CrossRef]

P. J. Bones, B. L. Satherley, R. W. Watson, IEE Conf. Publ. 354, 353 (1992).

C. R. Parker, P. J. Bones, in Proceedings of the Seventh New Zealand Image Processing Workshop, P. J. Bones, ed. (University of Canterbury, Christchurch–Lincoln, New Zealand, 1992), p. 5.

Curtis, S. R.

S. R. Curtis, A. V. Oppenheim, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643 (1985).
[CrossRef]

Fiddy, M. A.

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520 (1987).
[CrossRef]

Ghiglia, D. C.

Hong, J. H.

S. I. Sayegh, Y. L. Kok, J. H. Hong, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 107 (1987).
[CrossRef]

Izraelevitz, D.

D. Izraelevitz, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511 (1987).
[CrossRef]

Jenkins, M. A.

M. A. Jenkins, J. F. Traub, Commun. ACM 15, 97 (1972).
[CrossRef]

Kok, Y. L.

S. I. Sayegh, Y. L. Kok, J. H. Hong, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 107 (1987).
[CrossRef]

Lane, R. G.

R. H. T. Bates, R. G. Lane, Scanning Microsc. Suppl. 2, 149 (1988).

R. G. Lane, R. H. T. Bates, J. Opt. Soc. Am. A 4, 180 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520 (1987).
[CrossRef]

Lim, J. S.

D. Izraelevitz, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511 (1987).
[CrossRef]

S. R. Curtis, A. V. Oppenheim, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643 (1985).
[CrossRef]

Mastin, G. A.

McDonnell, M. J.

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989), Chap. 2, p. 34.

Oppenheim, A. V.

S. R. Curtis, A. V. Oppenheim, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643 (1985).
[CrossRef]

Parker, C. R.

R. W. Watson, C. R. Parker, P. J. Bones, Opt. Commun. 93, 359 (1992).
[CrossRef]

C. R. Parker, P. J. Bones, Opt. Commun. 92, 209 (1992).
[CrossRef]

R. H. T. Bates, B. K. Quek, C. R. Parker, J. Opt. Soc. Am. A 7, 468 (1990).
[CrossRef]

C. R. Parker, P. J. Bones, in Proceedings of the Seventh New Zealand Image Processing Workshop, P. J. Bones, ed. (University of Canterbury, Christchurch–Lincoln, New Zealand, 1992), p. 5.

Quek, B. K.

Romero, L. A.

Satherley, B. L.

P. J. Bones, B. L. Satherley, R. W. Watson, IEE Conf. Publ. 354, 353 (1992).

Sayegh, S. I.

S. I. Sayegh, Y. L. Kok, J. H. Hong, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 107 (1987).
[CrossRef]

Scivier, M. S.

Traub, J. F.

M. A. Jenkins, J. F. Traub, Commun. ACM 15, 97 (1972).
[CrossRef]

Watson, R. W.

P. J. Bones, B. L. Satherley, R. W. Watson, IEE Conf. Publ. 354, 353 (1992).

R. W. Watson, C. R. Parker, P. J. Bones, Opt. Commun. 93, 359 (1992).
[CrossRef]

Withers, D. H.

D. H. Withers, Commun. ACM 17, 157 (1974).
[CrossRef]

Commun. ACM (2)

M. A. Jenkins, J. F. Traub, Commun. ACM 15, 97 (1972).
[CrossRef]

D. H. Withers, Commun. ACM 17, 157 (1974).
[CrossRef]

IEE Conf. Publ. (1)

P. J. Bones, B. L. Satherley, R. W. Watson, IEE Conf. Publ. 354, 353 (1992).

IEEE Trans. Acoust. Speech Signal Process. (4)

S. R. Curtis, A. V. Oppenheim, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643 (1985).
[CrossRef]

D. Izraelevitz, J. S. Lim, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511 (1987).
[CrossRef]

S. I. Sayegh, Y. L. Kok, J. H. Hong, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 107 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520 (1987).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

C. R. Parker, P. J. Bones, Opt. Commun. 92, 209 (1992).
[CrossRef]

R. W. Watson, C. R. Parker, P. J. Bones, Opt. Commun. 93, 359 (1992).
[CrossRef]

Scanning Microsc. Suppl. (1)

R. H. T. Bates, R. G. Lane, Scanning Microsc. Suppl. 2, 149 (1988).

Other (2)

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989), Chap. 2, p. 34.

C. R. Parker, P. J. Bones, in Proceedings of the Seventh New Zealand Image Processing Workshop, P. J. Bones, ed. (University of Canterbury, Christchurch–Lincoln, New Zealand, 1992), p. 5.

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

F ( u , υ ) = m , n = 0 N 1 f [ m , n ] exp [ i 2 π Δ ( u m + υ n ) ] ,
( ζ , γ ) = m , n = 0 N 1 f [ m , n ] ζ m γ n .
( ζ , γ ) = k C ( ζ ) l = 1 N 1 [ γ γ ˆ l ( ζ ) ] ,
( ζ , γ c ) = k ( γ c ) l = 1 N 1 [ ζ ζ ˆ l ( γ c ) ] ,
C ( ζ p ) = l = 1 N 1 [ ζ p ζ ˆ l ( γ c ) ] / l = 1 N 1 [ γ c γ ˆ l ( ζ p ) ] , p = 0 , 1 , , N 1 .
C ( ζ p ) l = 1 N 1 [ γ q γ ˆ l ( ζ p ) ] = k 1 ( ζ p , γ q ) , p , q = 0 , 1 , , N 1 , = k 1 F ( p + i a , q + i b ) , = DFT { k 1 exp [ 2 π ( a m + b n ) / N ] f [ m , n ] } .
r [ m , n ] = k 1 f [ m , n ] = IDFT [ k 1 F ( p + i a , q + i b ) ] exp [ 2 π ( a m + b n ) / N ] .

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