Abstract

Recent developments in the application of zeros (of the analytically continued spectrum of a compact two-dimensional image) in solving deconvolution and phase retrieval problems are reviewed. New algorithms for use in the presence of noise are described and demonstrated. These include algorithms for deconvolution where the point-spread function is approximately known, for ensemble blind deconvolution (such as is required for ensembles of astronomical speckle images), and for phase retrieval (itself a special case of blind deconvolution). Many of the ideas embodied in the algorithms were foreshadowed by Bates et al. [ J. Opt. Soc. Am. A 7, 468 ( 1990)]. Simulated images are employed in the examples shown, except for phase retrieval, where successful recovery of the phase error in the aperture of a radio telescope is demonstrated.

© 1995 Optical Society of America

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  1. R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
    [CrossRef]
  2. R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
    [CrossRef]
  3. H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
    [CrossRef] [PubMed]
  4. D. C. Ghiglia, L. A. Romero, G. A. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024–1036 (1993).
    [CrossRef]
  5. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
    [CrossRef]
  6. G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “A solution to the phase problem based on the theory of entire functions,” Optik (Stuttgart) 49, 71–80 (1977).
  7. C. R. Parker, “Zero-based phase retrieval,” Ph.D. dissertation (Department of Electrical and Electronic Engineering, University of Canterbury, Christchurch, New Zealand, 1994).
  8. P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 14–22 (1993).
    [CrossRef]
  9. P. Chen, M. A. Fiddy, “Two dimensional blind deconvolution from point zero locations,” in Inverse Optics III, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2241, 238–249 (1994).
    [CrossRef]
  10. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  11. C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros to overcome a phase retrieval stagnation,” J. Opt. Soc. Am. A 8, 1898–1904 (1991).
    [CrossRef]
  12. C. C. Wackerman, A. E. Yagle, “Phase retrieval and estimation with use of real-plane zeros,” J. Opt. Soc. Am. A 11, 2016–2026 (1994).
    [CrossRef]
  13. A. A. G. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
    [CrossRef]
  14. A. Zakhor, D. Izraelevitz, “A note on the sampling of zero crossings of two-dimensional signals,” Proc. IEEE 74, 1285–1287 (1986).
    [CrossRef]
  15. R. W. Watson, “Advances in zero-based consistent deconvolution and evaluation of human sensory-motor function,” Ph.D. dissertation (University of Canterbury, Christchurch, New Zealand, 1994).
  16. R. W. Watson, C. R. Parker, P. J. Bones, “Demonstration of two-dimensional consistent deconvolution using zeros,” Opt. Commun. 93, 359–365 (1992).
    [CrossRef]
  17. S. R. Curtis, A. V. Oppenheim, J. S. Lim, “Signal reconstruction from Fourier transform sign information,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643–657 (1985).
    [CrossRef]
  18. D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
    [CrossRef]
  19. B. L. Satherley, C. R. Parker, “Two-dimensional image reconstruction from zero sheets,” Opt. Lett. 18, 2053–2055 (1993).
    [CrossRef] [PubMed]
  20. C. R. Parker, B. L. Satherley, P. J. Bones, “Image reconstruction from zeros of the z-transform,” in IEEE International Conference on Acoustics, Speech and Signal Processing, Volume 5: Image and Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, New York, N.Y., 1994), pp. V-465–V-468.
  21. R. H. T. Bates, P. J. Napier, A. E. McKinnon, M. J. McDonnell, “Self-consistent deconvolution. I: Theory,” Optik (Stuttgart) 44, 183–201 (1976).
  22. Image generated with the matlab®Image Processing Tool-box, The MathWorks, Inc.
  23. M. A. Jenkins, J. F. Traub, “Algorithm 419: zeros of a complex polynomial [C2],” Commun. ACM 15, 97–99 (1972).
    [CrossRef]
  24. D. H. Withers, “Remark on algorithm 419,” Commun. ACM 17, 157 (1974).
  25. S. M. Kay, R. Sudhaker, “A zero crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
    [CrossRef]
  26. C. Saloma, P. Haeberli, “Optical spectrum analysis from zero crossings,” Opt. Lett. 16, 1535–1537 (1991).
    [CrossRef] [PubMed]
  27. R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
    [CrossRef]
  28. A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  29. K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
    [CrossRef]
  30. A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
    [CrossRef]
  31. R. H. T. Bates, “A stochastic image restoration procedure,” Opt. Commun. 19, 240–244 (1976).
    [CrossRef]
  32. B. L. K. Davey, A. M. Sinton, R. H. T. Bates, “Zero-and-add,” Opt. Eng. 25, 765–771 (1986).
    [CrossRef]
  33. A. M. Sinton, B. L. K. Davey, R. H. T. Bates, “Augmenting shift-and-add with zero-and-add,” J. Opt. Soc. Am. 3, 1010–1017 (1986).
    [CrossRef]
  34. R. H. T. Bates, R. G. Lane, “Automatic deconvolution and phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 158–164 (1987).
    [CrossRef]
  35. B. L. Satherley, P. J. Bones, “Zero tracks for blind deconvolution of blurred ensembles,” Appl. Opt. 33, 2197–2205 (1994).
    [CrossRef] [PubMed]
  36. B. L. Satherley, “Zero-based ensemble deconvolution and EEG spectral topography,” Ph.D. dissertation (University of Canterbury, Christchurch, New Zealand, 1994).
  37. C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
    [CrossRef]
  38. R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1986), Vol. 67, pp. 1–64.
    [CrossRef]
  39. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  40. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
    [CrossRef]
  41. B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
    [CrossRef]
  42. J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low resolution image,” J. Opt. Soc. Am. A 7, 450–458 (1990).
    [CrossRef]
  43. R. H. T. Bates, P. J. Napier, “Identification and removal of phase errors in interferometry,” Mon. Not. R. Astron. Soc. 158, 405–424 (1972).
  44. P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array: design and performance of a modern synthesis radio telescope,” Proc. IEEE 71, 1295–1320 (1983).
    [CrossRef]
  45. J. E. McCormack, G. Junkin, A. P. Anderson, “Microwave metrology of reflector antennas from a single amplitude,” IEE Proc. Part H 137, 276–284 (1990).

1994 (2)

1993 (2)

1992 (2)

R. W. Watson, C. R. Parker, P. J. Bones, “Demonstration of two-dimensional consistent deconvolution using zeros,” Opt. Commun. 93, 359–365 (1992).
[CrossRef]

C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
[CrossRef]

1991 (2)

1990 (3)

1989 (1)

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

1987 (5)

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

1986 (4)

A. Zakhor, D. Izraelevitz, “A note on the sampling of zero crossings of two-dimensional signals,” Proc. IEEE 74, 1285–1287 (1986).
[CrossRef]

S. M. Kay, R. Sudhaker, “A zero crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
[CrossRef]

B. L. K. Davey, A. M. Sinton, R. H. T. Bates, “Zero-and-add,” Opt. Eng. 25, 765–771 (1986).
[CrossRef]

A. M. Sinton, B. L. K. Davey, R. H. T. Bates, “Augmenting shift-and-add with zero-and-add,” J. Opt. Soc. Am. 3, 1010–1017 (1986).
[CrossRef]

1985 (2)

S. R. Curtis, A. V. Oppenheim, J. S. Lim, “Signal reconstruction from Fourier transform sign information,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
[CrossRef] [PubMed]

1984 (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

1983 (1)

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array: design and performance of a modern synthesis radio telescope,” Proc. IEEE 71, 1295–1320 (1983).
[CrossRef]

1982 (2)

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1980 (1)

A. A. G. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
[CrossRef]

1977 (1)

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “A solution to the phase problem based on the theory of entire functions,” Optik (Stuttgart) 49, 71–80 (1977).

1976 (2)

R. H. T. Bates, P. J. Napier, A. E. McKinnon, M. J. McDonnell, “Self-consistent deconvolution. I: Theory,” Optik (Stuttgart) 44, 183–201 (1976).

R. H. T. Bates, “A stochastic image restoration procedure,” Opt. Commun. 19, 240–244 (1976).
[CrossRef]

1974 (2)

K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

D. H. Withers, “Remark on algorithm 419,” Commun. ACM 17, 157 (1974).

1972 (2)

M. A. Jenkins, J. F. Traub, “Algorithm 419: zeros of a complex polynomial [C2],” Commun. ACM 15, 97–99 (1972).
[CrossRef]

R. H. T. Bates, P. J. Napier, “Identification and removal of phase errors in interferometry,” Mon. Not. R. Astron. Soc. 158, 405–424 (1972).

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Anderson, A. P.

J. E. McCormack, G. Junkin, A. P. Anderson, “Microwave metrology of reflector antennas from a single amplitude,” IEE Proc. Part H 137, 276–284 (1990).

Bates, R. H. T.

R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
[CrossRef]

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

B. L. K. Davey, A. M. Sinton, R. H. T. Bates, “Zero-and-add,” Opt. Eng. 25, 765–771 (1986).
[CrossRef]

A. M. Sinton, B. L. K. Davey, R. H. T. Bates, “Augmenting shift-and-add with zero-and-add,” J. Opt. Soc. Am. 3, 1010–1017 (1986).
[CrossRef]

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

R. H. T. Bates, “A stochastic image restoration procedure,” Opt. Commun. 19, 240–244 (1976).
[CrossRef]

R. H. T. Bates, P. J. Napier, A. E. McKinnon, M. J. McDonnell, “Self-consistent deconvolution. I: Theory,” Optik (Stuttgart) 44, 183–201 (1976).

R. H. T. Bates, P. J. Napier, “Identification and removal of phase errors in interferometry,” Mon. Not. R. Astron. Soc. 158, 405–424 (1972).

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1986), Vol. 67, pp. 1–64.
[CrossRef]

R. H. T. Bates, R. G. Lane, “Automatic deconvolution and phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 158–164 (1987).
[CrossRef]

Bones, P. J.

B. L. Satherley, P. J. Bones, “Zero tracks for blind deconvolution of blurred ensembles,” Appl. Opt. 33, 2197–2205 (1994).
[CrossRef] [PubMed]

C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
[CrossRef]

R. W. Watson, C. R. Parker, P. J. Bones, “Demonstration of two-dimensional consistent deconvolution using zeros,” Opt. Commun. 93, 359–365 (1992).
[CrossRef]

C. R. Parker, B. L. Satherley, P. J. Bones, “Image reconstruction from zeros of the z-transform,” in IEEE International Conference on Acoustics, Speech and Signal Processing, Volume 5: Image and Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, New York, N.Y., 1994), pp. V-465–V-468.

Chen, P.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 14–22 (1993).
[CrossRef]

P. Chen, M. A. Fiddy, “Two dimensional blind deconvolution from point zero locations,” in Inverse Optics III, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2241, 238–249 (1994).
[CrossRef]

Curtis, S. R.

S. R. Curtis, A. V. Oppenheim, J. S. Lim, “Signal reconstruction from Fourier transform sign information,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

Davey, B. L. K.

A. M. Sinton, B. L. K. Davey, R. H. T. Bates, “Augmenting shift-and-add with zero-and-add,” J. Opt. Soc. Am. 3, 1010–1017 (1986).
[CrossRef]

B. L. K. Davey, A. M. Sinton, R. H. T. Bates, “Zero-and-add,” Opt. Eng. 25, 765–771 (1986).
[CrossRef]

Deighton, H. V.

Ekers, R. D.

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array: design and performance of a modern synthesis radio telescope,” Proc. IEEE 71, 1295–1320 (1983).
[CrossRef]

Fiddy, M. A.

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
[CrossRef] [PubMed]

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “A solution to the phase problem based on the theory of entire functions,” Optik (Stuttgart) 49, 71–80 (1977).

P. Chen, M. A. Fiddy, “Two dimensional blind deconvolution from point zero locations,” in Inverse Optics III, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2241, 238–249 (1994).
[CrossRef]

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 14–22 (1993).
[CrossRef]

Fienup, J. R.

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

Ghiglia, D. C.

Greenaway, A. H.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 14–22 (1993).
[CrossRef]

Haeberli, P.

Izraelevitz, D.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

A. Zakhor, D. Izraelevitz, “A note on the sampling of zero crossings of two-dimensional signals,” Proc. IEEE 74, 1285–1287 (1986).
[CrossRef]

Jenkins, M. A.

M. A. Jenkins, J. F. Traub, “Algorithm 419: zeros of a complex polynomial [C2],” Commun. ACM 15, 97–99 (1972).
[CrossRef]

Junkin, G.

J. E. McCormack, G. Junkin, A. P. Anderson, “Microwave metrology of reflector antennas from a single amplitude,” IEE Proc. Part H 137, 276–284 (1990).

Kay, S. M.

S. M. Kay, R. Sudhaker, “A zero crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
[CrossRef]

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Kowalczyk, A. M.

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lane, R. G.

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

R. H. T. Bates, R. G. Lane, “Automatic deconvolution and phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 158–164 (1987).
[CrossRef]

Lim, J. S.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

S. R. Curtis, A. V. Oppenheim, J. S. Lim, “Signal reconstruction from Fourier transform sign information,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Mastin, G. A.

McCallum, B. C.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

McCormack, J. E.

J. E. McCormack, G. Junkin, A. P. Anderson, “Microwave metrology of reflector antennas from a single amplitude,” IEE Proc. Part H 137, 276–284 (1990).

McDonnell, M. J.

R. H. T. Bates, P. J. Napier, A. E. McKinnon, M. J. McDonnell, “Self-consistent deconvolution. I: Theory,” Optik (Stuttgart) 44, 183–201 (1976).

McKinnon, A. E.

R. H. T. Bates, P. J. Napier, A. E. McKinnon, M. J. McDonnell, “Self-consistent deconvolution. I: Theory,” Optik (Stuttgart) 44, 183–201 (1976).

Mnyama, D.

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1986), Vol. 67, pp. 1–64.
[CrossRef]

Napier, P. J.

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array: design and performance of a modern synthesis radio telescope,” Proc. IEEE 71, 1295–1320 (1983).
[CrossRef]

R. H. T. Bates, P. J. Napier, A. E. McKinnon, M. J. McDonnell, “Self-consistent deconvolution. I: Theory,” Optik (Stuttgart) 44, 183–201 (1976).

R. H. T. Bates, P. J. Napier, “Identification and removal of phase errors in interferometry,” Mon. Not. R. Astron. Soc. 158, 405–424 (1972).

Nieto-Vesperinas, M.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “A solution to the phase problem based on the theory of entire functions,” Optik (Stuttgart) 49, 71–80 (1977).

Oppenheim, A. V.

S. R. Curtis, A. V. Oppenheim, J. S. Lim, “Signal reconstruction from Fourier transform sign information,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

Parker, C. R.

B. L. Satherley, C. R. Parker, “Two-dimensional image reconstruction from zero sheets,” Opt. Lett. 18, 2053–2055 (1993).
[CrossRef] [PubMed]

R. W. Watson, C. R. Parker, P. J. Bones, “Demonstration of two-dimensional consistent deconvolution using zeros,” Opt. Commun. 93, 359–365 (1992).
[CrossRef]

C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
[CrossRef]

R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
[CrossRef]

C. R. Parker, “Zero-based phase retrieval,” Ph.D. dissertation (Department of Electrical and Electronic Engineering, University of Canterbury, Christchurch, New Zealand, 1994).

C. R. Parker, B. L. Satherley, P. J. Bones, “Image reconstruction from zeros of the z-transform,” in IEEE International Conference on Acoustics, Speech and Signal Processing, Volume 5: Image and Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, New York, N.Y., 1994), pp. V-465–V-468.

Quek, B. K.

Requicha, A. A. G.

A. A. G. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
[CrossRef]

Romero, L. A.

Ross, G.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “A solution to the phase problem based on the theory of entire functions,” Optik (Stuttgart) 49, 71–80 (1977).

Saloma, C.

Satherley, B. L.

B. L. Satherley, P. J. Bones, “Zero tracks for blind deconvolution of blurred ensembles,” Appl. Opt. 33, 2197–2205 (1994).
[CrossRef] [PubMed]

B. L. Satherley, C. R. Parker, “Two-dimensional image reconstruction from zero sheets,” Opt. Lett. 18, 2053–2055 (1993).
[CrossRef] [PubMed]

C. R. Parker, B. L. Satherley, P. J. Bones, “Image reconstruction from zeros of the z-transform,” in IEEE International Conference on Acoustics, Speech and Signal Processing, Volume 5: Image and Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, New York, N.Y., 1994), pp. V-465–V-468.

B. L. Satherley, “Zero-based ensemble deconvolution and EEG spectral topography,” Ph.D. dissertation (University of Canterbury, Christchurch, New Zealand, 1994).

Scivier, M. S.

Sinton, A. M.

B. L. K. Davey, A. M. Sinton, R. H. T. Bates, “Zero-and-add,” Opt. Eng. 25, 765–771 (1986).
[CrossRef]

A. M. Sinton, B. L. K. Davey, R. H. T. Bates, “Augmenting shift-and-add with zero-and-add,” J. Opt. Soc. Am. 3, 1010–1017 (1986).
[CrossRef]

Sudhaker, R.

S. M. Kay, R. Sudhaker, “A zero crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
[CrossRef]

Thompson, A. R.

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array: design and performance of a modern synthesis radio telescope,” Proc. IEEE 71, 1295–1320 (1983).
[CrossRef]

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Traub, J. F.

M. A. Jenkins, J. F. Traub, “Algorithm 419: zeros of a complex polynomial [C2],” Commun. ACM 15, 97–99 (1972).
[CrossRef]

Wackerman, C. C.

Wang, Y.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 14–22 (1993).
[CrossRef]

Watson, R. W.

R. W. Watson, C. R. Parker, P. J. Bones, “Demonstration of two-dimensional consistent deconvolution using zeros,” Opt. Commun. 93, 359–365 (1992).
[CrossRef]

R. W. Watson, “Advances in zero-based consistent deconvolution and evaluation of human sensory-motor function,” Ph.D. dissertation (University of Canterbury, Christchurch, New Zealand, 1994).

Wheeler, M. W. L.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “A solution to the phase problem based on the theory of entire functions,” Optik (Stuttgart) 49, 71–80 (1977).

Wirnitzer, B.

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Withers, D. H.

D. H. Withers, “Remark on algorithm 419,” Commun. ACM 17, 157 (1974).

Yagle, A. E.

Zakhor, A.

A. Zakhor, D. Izraelevitz, “A note on the sampling of zero crossings of two-dimensional signals,” Proc. IEEE 74, 1285–1287 (1986).
[CrossRef]

Appl. Opt. (2)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Astrophys. J. Lett. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Commun. ACM (2)

M. A. Jenkins, J. F. Traub, “Algorithm 419: zeros of a complex polynomial [C2],” Commun. ACM 15, 97–99 (1972).
[CrossRef]

D. H. Withers, “Remark on algorithm 419,” Commun. ACM 17, 157 (1974).

IEE Proc. Part H (1)

J. E. McCormack, G. Junkin, A. P. Anderson, “Microwave metrology of reflector antennas from a single amplitude,” IEE Proc. Part H 137, 276–284 (1990).

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

S. M. Kay, R. Sudhaker, “A zero crossing-based spectrum analyzer,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 96–104 (1986).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

S. R. Curtis, A. V. Oppenheim, J. S. Lim, “Signal reconstruction from Fourier transform sign information,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

J. Mod. Opt. (1)

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

A. M. Sinton, B. L. K. Davey, R. H. T. Bates, “Augmenting shift-and-add with zero-and-add,” J. Opt. Soc. Am. 3, 1010–1017 (1986).
[CrossRef]

J. Opt. Soc. Am. A (7)

Mon. Not. R. Astron. Soc. (1)

R. H. T. Bates, P. J. Napier, “Identification and removal of phase errors in interferometry,” Mon. Not. R. Astron. Soc. 158, 405–424 (1972).

Opt. Commun. (4)

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

R. W. Watson, C. R. Parker, P. J. Bones, “Demonstration of two-dimensional consistent deconvolution using zeros,” Opt. Commun. 93, 359–365 (1992).
[CrossRef]

R. H. T. Bates, “A stochastic image restoration procedure,” Opt. Commun. 19, 240–244 (1976).
[CrossRef]

C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
[CrossRef]

Opt. Eng. (1)

B. L. K. Davey, A. M. Sinton, R. H. T. Bates, “Zero-and-add,” Opt. Eng. 25, 765–771 (1986).
[CrossRef]

Opt. Lett. (3)

Optik (Stuttgart) (2)

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “A solution to the phase problem based on the theory of entire functions,” Optik (Stuttgart) 49, 71–80 (1977).

R. H. T. Bates, P. J. Napier, A. E. McKinnon, M. J. McDonnell, “Self-consistent deconvolution. I: Theory,” Optik (Stuttgart) 44, 183–201 (1976).

Phys. Rep. (1)

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

Proc. IEEE (4)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

A. A. G. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
[CrossRef]

A. Zakhor, D. Izraelevitz, “A note on the sampling of zero crossings of two-dimensional signals,” Proc. IEEE 74, 1285–1287 (1986).
[CrossRef]

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array: design and performance of a modern synthesis radio telescope,” Proc. IEEE 71, 1295–1320 (1983).
[CrossRef]

Other (9)

R. W. Watson, “Advances in zero-based consistent deconvolution and evaluation of human sensory-motor function,” Ph.D. dissertation (University of Canterbury, Christchurch, New Zealand, 1994).

C. R. Parker, “Zero-based phase retrieval,” Ph.D. dissertation (Department of Electrical and Electronic Engineering, University of Canterbury, Christchurch, New Zealand, 1994).

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2029, 14–22 (1993).
[CrossRef]

P. Chen, M. A. Fiddy, “Two dimensional blind deconvolution from point zero locations,” in Inverse Optics III, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2241, 238–249 (1994).
[CrossRef]

C. R. Parker, B. L. Satherley, P. J. Bones, “Image reconstruction from zeros of the z-transform,” in IEEE International Conference on Acoustics, Speech and Signal Processing, Volume 5: Image and Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, New York, N.Y., 1994), pp. V-465–V-468.

Image generated with the matlab®Image Processing Tool-box, The MathWorks, Inc.

R. H. T. Bates, R. G. Lane, “Automatic deconvolution and phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 158–164 (1987).
[CrossRef]

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1986), Vol. 67, pp. 1–64.
[CrossRef]

B. L. Satherley, “Zero-based ensemble deconvolution and EEG spectral topography,” Ph.D. dissertation (University of Canterbury, Christchurch, New Zealand, 1994).

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Figures (11)

Fig. 1
Fig. 1

Magnitude of the 38 × 38 pixel convolution of a complex 32 × 32 pixel image of a panda with a 7 × 7 pixel complex pseudo-random psf. Recovery of the image by zero-map matching is illustrated in Fig. 2.

Fig. 2
Fig. 2

Effect of psf error on the zero-map matching algorithm. (a), (b), and (c) are the magnitudes of the reconstructions from the convolution shown in Fig. 1 achieved with psf errors of 30, 20, and 10 dB, respectively. The values of Q for these reconstructions are 104, 18, and 2 dB, respectively. The convolution was uncontaminated in the cases shown.

Fig. 3
Fig. 3

(a) 256 × 256 pixel convolution of a 250 × 250 pixel image with a 7-pixel-diameter disk, (b) real part of the reconstruction by zero-map matching (virtually indistinguishable from the true image; Q = 31 dB).

Fig. 4
Fig. 4

Demonstration of the zero-track matching algorithm applied to standard deconvolution. We convolved a 16 × 16 pixel bilevel image of the letter G with a 5 × 5 pixel bilevel psf (square pattern) to produce (a). Also shown are reconstructions from (b) 40-dB convolution SNR and (c) 30-dB convolution SNR.

Fig. 5
Fig. 5

Ensembles of zero-track sections representing an ensemble of 20 images and calculated for ζ = 1.0 exp(), with ϕ varied from 0 to π/2 rad: (a) uncontaminated case, (b) 60-dB SNR, (c) 50-dB SNR, (d) 40-dB SNR.

Fig. 6
Fig. 6

Demonstration of the performance of the zero-track-based ZAA algorithm. (a) f[m, n]; reconstructions obtained from ensembles with contamination levels of (b) 60 dB, (c) 50 dB, and (d) 40 dB. The reconstruction accuracies of the estimates are 42.20, 30.08, and 20.11 dB, respectively.

Fig. 7
Fig. 7

Example of the use of composite GDFT reconstructions and the zero-track-based ZAA algorithm. (a) Reconstruction derived from the 40-dB SNR ensemble with ρ, r = 0.7; Q = 11.07 dB. (b) Reconstruction obtained from the rotated ensemble; Q = 13.55 dB. (c) Composite reconstruction; Q = 18.15 dB.

Fig. 8
Fig. 8

Zero-map matching applied after the iteration at which Ei first falls below 10−5 during phase retrieval of a 16 × 16 pixel complex pseudorandom original image from uncontaminated magnitude data. The upper and lower solid curves depict Et and Ei, respectively, for HIO only. The dashed and dotted curves depict the effect upon Et and Ei, respectively, of applying zero-map matching and continuing HIO until 500 iterations in total.

Fig. 9
Fig. 9

Magnitude of the complex holographic far-field measurement for a 25-m-diameter telescope in the Very Large Array used for the phase retrieval experiment.

Fig. 10
Fig. 10

Convergence of iterative phase retrieval from telescope data by HIO alone (solid curves) and with assistance from zero-map matching followed by 20 more iterations of HIO, then ER (dashed and dotted curves); Eh is above, Ei below.

Fig. 11
Fig. 11

Magnitude and phase for the antenna aperture, obtained directly from the holographic data (left) and recovered from magnitude data through zero-map matching (right).

Tables (2)

Tables Icon

Table 1 Results from the Zero-Map Matching Deconvolution Algorithm

Tables Icon

Table 2 Comparison of the Results for the Zero-Track Matching Deconvolution Algorithm without Splicinga and with Splicing

Equations (28)

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F ( u , v ) = m = 0 N - 1 n = 0 N - 1 f [ m , n ] exp [ i 2 π Δ ( u m + v n ) ] ,
F [ p , q ] = m = 0 N - 1 n = 0 N - 1 f [ m , n ] exp [ i 2 π ( p m + q n ) / N ] , p , q = 0 , 1 , , N - 1 ,
f [ m , n ] = p = 0 N - 1 q = 0 N - 1 F [ p , q ] exp [ - i 2 π ( p m + q n ) / N ] , m , n = 0 , 1 , , N - 1 ,
ζ = exp ( i 2 π Δ u ) ,             γ = exp ( i 2 π Δ v )
F ( ζ , γ ) = m = 0 N - 1 n = 0 N - 1 f [ m , n ] ζ m γ n ,
g ( x , y ) = f ( x , y ) h ( x , y ) .
G ( ζ , γ ) = F ( ζ , γ ) H ( ζ , γ ) .
Z [ G ( ζ , γ ) ] = Z [ F ( ζ , γ ) ] Z [ H ( ζ , γ ) ] .
ζ = ξ + i η = ρ exp ( i ϕ ) ,
γ = α + i β = r exp ( i θ ) .
F ( ζ , γ ) = k C ( ζ ) l = 1 N - 1 [ γ - γ ¯ l ( ζ ) ] ,
SNR = 10 log 10 ( m = 0 N - 1 n = 0 N - 1 f [ m , n ] 2 m = 0 N - 1 n = 0 N - 1 c [ m , n ] 2 )             ( dB ) .
Q = 10 log 10 ( m = 0 N - 1 n = 0 N - 1 f [ m , n ] 2 m = 0 N - 1 n = 0 N - 1 f [ m , n ] - f ^ [ m , n ] 2 )             ( dB ) .
F ( ζ , γ ) k l = 1 L [ ( ζ - ζ l ) + i ( γ - γ l ) ] ,
F ( ζ p , γ ) = n = 0 N - 1 ( m = 0 N - 1 f [ m , n ] ζ p m ) γ n ,
F ( ζ , γ q ) = m = 0 N - 1 ( n = 0 N - 1 f [ m , n ] γ q n ) ζ m = k ( γ q ) l = 1 N - 1 [ ζ - ζ ¯ l ( γ q ) ] .
F ( ζ , γ c ) = k l = 1 N - 1 [ ζ - ζ ¯ l ( γ c ) ] .
C ( ζ p ) = l = 1 N - 1 [ ζ p - ζ ¯ l ( γ c ) ] l = 1 N - 1 [ γ c - γ ¯ l ( ζ p ) ] ,             p = 0 , 1 , , N - 1.
r [ m , n ] = k - 1 f [ m , n ] = IDFT { C ( ζ p ) l = 1 N - 1 [ γ q - γ ¯ l ( ζ p ) ] } exp [ - 2 π ( a m + b n ) / N ] .
a [ m , n ] = g [ m , n ] + c [ m , n ] ,
d = min ( v 1 - v 2 , v 1 - v 2 + 1 / Δ , v 2 - v 1 + 1 / Δ )
d p q = j = 1 J Z p [ H ^ ( ζ ϕ j , γ ) ] - Z q [ A ( ζ ϕ j , γ ) ] ,
Tang ( H ^ , p , j ) - Tang ( A , q , j ) < π 4 ,
g k [ m , n ] = f [ m , n ] h k [ m , n ] ,             k = 1 , 2 , , K ,
Z [ F ( ζ , γ ) ] = Z [ G 1 ( ζ , γ ) ] Z [ G 2 ( ζ , γ ) ] Z [ G K ( ζ , γ ) ] ,
E i = m , n Γ g k [ m , n ] 2 m = 0 N - 1 n = 0 N - 1 g k [ m , n ] 2 ,
E t = m = 0 N - 1 n = 0 N - 1 g ˜ k [ m , n ] - f [ m , n ] 2 m = 0 N - 1 n = 0 N - 1 g k [ m , n ] 2 ,
E h = m , n Γ g ˜ k [ m , n ] - f h [ m , n ] 2 + m , n Γ g k [ m , n ] 2 m = 0 N - 1 n = 0 N - 1 g k [ m , n ] 2 ,

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