Abstract

The insight into deconvolution and phase retrieval afforded by the concept of the zero sheet (of the spectrum of a compact image) is summarized. Difficulties associated with practical implementation of the zero-sheet concept are outlined. Possible means for overcoming these difficulties are suggested and are illustrated with selected examples for several types of deconvolution, i.e., standard, ensemble blind (i.e., blind deconvolution in Stockham’s sense), and pure blind (i.e., deconvolution of a single blurred image without prior knowledge of the point-spread function). A specialized procedure is described for blindly deconvolving a blurred image when the spectrum of the point-spread function is an unknown pure phase function. It is indicated how this procedure may facilitate phase retrieval for spectra of complex-valued images. Possible effects on Wiener filtering and on astronomical speckle imaging are discussed.

© 1990 Optical Society of America

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References

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  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  2. J. R. Fienup, C. C. Wackermen, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  3. G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef] [PubMed]
  4. B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
    [CrossRef]
  5. R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–184 (1987).
    [CrossRef]
  6. R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
    [CrossRef]
  7. R. G. Lane, R. H. T. Bates, “Relevance for blind deconvolution of recovering Fourier magnitude,” Opt. Commun. 63, 11–14 (1987).
    [CrossRef]
  8. 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]
  9. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  10. J. L. C. Sanz, T. S. Huang, “Polynomial system of equations and its applications to the study of the effect of noise on multidimensional Fourier transform phase retrieval from magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 997–1004 (1985).
    [CrossRef]
  11. I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2160 (1985).
    [CrossRef]
  12. 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]
  13. 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]
  14. Y. Q. Shi, N. K. Bose, “Some results in nonnegativity constrained spectral factorisation,” Opt. Commun. 68, 251–256 (1988).
    [CrossRef]
  15. 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]
  16. R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
    [CrossRef]
  17. J. R. Fienup, “Phase error correction by shear averaging,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 134–137.
  18. R. H. T. Bates, R. G. Lane, “Deblurring should now be automatic,” Scan. Microsc. Suppl. 2, 149–156 (1988).
  19. R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989) (corrected and updated reprinting).
  20. R. G. Lane, “Blind deconvolution and phase retrieval,” Ph.D. dissertation (Engineering Library, University of Canterbury, Christchurch, New Zealand, 1988).
  21. B. L. K. Davey, A. M. Sinton, R. H. T. Bates, “Zero-and-add,” Opt. Eng. 25, 765–771 (1986).
    [CrossRef]
  22. M. J. McDonnell, W. K. Kennedy, R. H. T. Bates, “Identifying and overcoming practical problems of digital image restoration,” N. Z. J. Sci. 19, 127–133 (1976).
  23. T. G. Stockham, T. M. Cannon, R. B. Ingebretson, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
    [CrossRef]
  24. R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep.90, 203–297 (1982).
    [CrossRef]
  25. A. M. Sinton, B. L. K. Davey, R. H. T. Bates, “Augmenting shift-and-add with zero-and-add,” J. Opt. Soc. Am. A 3, 1010–1017 (1986).
    [CrossRef]
  26. R. H. T. Bates, P. J. Napier, “Identification and removal of phase errors in interferometry,” Mon. Not. R. Astron. Soc. 158, 405–424 (1972).
  27. T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–11 (1990).
    [CrossRef]
  28. R. H. T. Bates, D. G. H. Tan, “Toward reconstructing phases of inverse-scattering signals,” J. Opt. Soc. Am. A 2, 2013–2018 (1985).
    [CrossRef]
  29. 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]
  30. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
    [CrossRef]
  31. R. H. T. Bates, B. L. K. Davey, “Towards making shift-and-add a versatile imaging technique,” in Digital Image Recovery and SynthesisP. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 87–94 (1987).
    [CrossRef]

1990 (1)

1989 (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

1988 (3)

G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
[CrossRef] [PubMed]

Y. Q. Shi, N. K. Bose, “Some results in nonnegativity constrained spectral factorisation,” Opt. Commun. 68, 251–256 (1988).
[CrossRef]

R. H. T. Bates, R. G. Lane, “Deblurring should now be automatic,” Scan. Microsc. Suppl. 2, 149–156 (1988).

1987 (6)

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]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–184 (1987).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Relevance for blind deconvolution of recovering Fourier magnitude,” Opt. Commun. 63, 11–14 (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]

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]

1986 (4)

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

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
[CrossRef]

J. R. Fienup, C. C. Wackermen, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

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

1985 (4)

J. L. C. Sanz, T. S. Huang, “Polynomial system of equations and its applications to the study of the effect of noise on multidimensional Fourier transform phase retrieval from magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 997–1004 (1985).
[CrossRef]

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2160 (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]

R. H. T. Bates, D. G. H. Tan, “Toward reconstructing phases of inverse-scattering signals,” J. Opt. Soc. Am. A 2, 2013–2018 (1985).
[CrossRef]

1983 (1)

1982 (1)

1979 (1)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1976 (1)

M. J. McDonnell, W. K. Kennedy, R. H. T. Bates, “Identifying and overcoming practical problems of digital image restoration,” N. Z. J. Sci. 19, 127–133 (1976).

1975 (1)

T. G. Stockham, T. M. Cannon, R. B. Ingebretson, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

1972 (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).

Ayers, G. R.

Bates, R. H. T.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

R. H. T. Bates, R. G. Lane, “Deblurring should now be automatic,” Scan. Microsc. Suppl. 2, 149–156 (1988).

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, “Relevance for blind deconvolution of recovering Fourier magnitude,” Opt. Commun. 63, 11–14 (1987).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–184 (1987).
[CrossRef]

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

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
[CrossRef]

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

R. H. T. Bates, D. G. H. Tan, “Toward reconstructing phases of inverse-scattering signals,” J. Opt. Soc. Am. A 2, 2013–2018 (1985).
[CrossRef]

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

M. J. McDonnell, W. K. Kennedy, R. H. T. Bates, “Identifying and overcoming practical problems of digital image restoration,” N. Z. J. Sci. 19, 127–133 (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, 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, B. L. K. Davey, “Towards making shift-and-add a versatile imaging technique,” in Digital Image Recovery and SynthesisP. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 87–94 (1987).
[CrossRef]

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

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989) (corrected and updated reprinting).

Bose, N. K.

Y. Q. Shi, N. K. Bose, “Some results in nonnegativity constrained spectral factorisation,” Opt. Commun. 68, 251–256 (1988).
[CrossRef]

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Cannon, T. M.

T. G. Stockham, T. M. Cannon, R. B. Ingebretson, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Crimmins, T. R.

Dainty, J. C.

Davey, B. L. K.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[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. A 3, 1010–1017 (1986).
[CrossRef]

R. H. T. Bates, B. L. K. Davey, “Towards making shift-and-add a versatile imaging technique,” in Digital Image Recovery and SynthesisP. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 87–94 (1987).
[CrossRef]

Deighton, H. V.

Fiddy, M. A.

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]

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

Huang, T. S.

J. L. C. Sanz, T. S. Huang, “Polynomial system of equations and its applications to the study of the effect of noise on multidimensional Fourier transform phase retrieval from magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 997–1004 (1985).
[CrossRef]

Ingebretson, R. B.

T. G. Stockham, T. M. Cannon, R. B. Ingebretson, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

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]

Kennedy, W. K.

M. J. McDonnell, W. K. Kennedy, R. H. T. Bates, “Identifying and overcoming practical problems of digital image restoration,” N. Z. J. Sci. 19, 127–133 (1976).

Lane, R. G.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

R. H. T. Bates, R. G. Lane, “Deblurring should now be automatic,” Scan. Microsc. Suppl. 2, 149–156 (1988).

R. G. Lane, R. H. T. Bates, “Relevance for blind deconvolution of recovering Fourier magnitude,” Opt. Commun. 63, 11–14 (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, 180–184 (1987).
[CrossRef]

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

R. G. Lane, “Blind deconvolution and phase retrieval,” Ph.D. dissertation (Engineering Library, University of Canterbury, Christchurch, New Zealand, 1988).

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]

McDonnell, M. J.

M. J. McDonnell, W. K. Kennedy, R. H. T. Bates, “Identifying and overcoming practical problems of digital image restoration,” N. Z. J. Sci. 19, 127–133 (1976).

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989) (corrected and updated reprinting).

Mnyama, D.

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
[CrossRef]

Napier, P. J.

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

Sanz, J. L. C.

J. L. C. Sanz, T. S. Huang, “Polynomial system of equations and its applications to the study of the effect of noise on multidimensional Fourier transform phase retrieval from magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 997–1004 (1985).
[CrossRef]

Scivier, M. S.

Shi, Y. Q.

Y. Q. Shi, N. K. Bose, “Some results in nonnegativity constrained spectral factorisation,” Opt. Commun. 68, 251–256 (1988).
[CrossRef]

Sinton, A. M.

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Stefanescu, I. S.

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

Stockham, T. G.

T. G. Stockham, T. M. Cannon, R. B. Ingebretson, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Tan, D. G. H.

Thelen, B. J.

Wackermen, C. C.

Adv. Electron. Electron Phys. (1)

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” Adv. Electron. Electron Phys. 67, 1–64 (1986).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (3)

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]

J. L. C. Sanz, T. S. Huang, “Polynomial system of equations and its applications to the study of the effect of noise on multidimensional Fourier transform phase retrieval from magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 997–1004 (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. Math. Phys. (1)

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

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).

N. Z. J. Sci. (1)

M. J. McDonnell, W. K. Kennedy, R. H. T. Bates, “Identifying and overcoming practical problems of digital image restoration,” N. Z. J. Sci. 19, 127–133 (1976).

Opt. Commun. (5)

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

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Relevance for blind deconvolution of recovering Fourier magnitude,” Opt. Commun. 63, 11–14 (1987).
[CrossRef]

Y. Q. Shi, N. K. Bose, “Some results in nonnegativity constrained spectral factorisation,” Opt. Commun. 68, 251–256 (1988).
[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. (2)

Proc. IEEE (1)

T. G. Stockham, T. M. Cannon, R. B. Ingebretson, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Scan. Microsc. Suppl. (1)

R. H. T. Bates, R. G. Lane, “Deblurring should now be automatic,” Scan. Microsc. Suppl. 2, 149–156 (1988).

Other (6)

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989) (corrected and updated reprinting).

R. G. Lane, “Blind deconvolution and phase retrieval,” Ph.D. dissertation (Engineering Library, University of Canterbury, Christchurch, New Zealand, 1988).

J. R. Fienup, “Phase error correction by shear averaging,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 134–137.

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

R. H. T. Bates, B. L. K. Davey, “Towards making shift-and-add a versatile imaging technique,” in Digital Image Recovery and SynthesisP. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 87–94 (1987).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Zero maps for the first one-dimensional example. Zeros belonging to Z(G, 1) and Z(H, 1) are identified by ○ and +, respectively: (a) cl = 0.003, (b) cl = 0.03.

Fig. 2
Fig. 2

Two-dimensional example: f(x) and h(x) are at left and right, respectively; (a) magnitudes (white to black represents levels 0 to 1.0 in 5 bits), (b) phases (white to black represents 0 to 6.28 in 0.2-rad steps). Note that both f(x) and h(x) are complex valued.

Fig. 3
Fig. 3

Zero tracks for the two-dimensional (with cl = 0.03) example. Tracks belonging to Z(G, 2) and Z(H, 2) are identified by ○ — and ×—, respectively.

Fig. 4
Fig. 4

Further zero tracks for the two-dimensional example. Tracks for cl = 0, 0.03, 0.1 are identified by ×—, ○ —, and □ —, respectively.

Fig. 5
Fig. 5

Zero maps of spectra of contaminated blurred versions of the true image f(x) whose real pixel values are listed in Table 2. The psf and contamination are also real for each blurred image. The first two zero maps are superpositions of the maps of the spectra of four differently blurred versions of f(x), whereas the third zero map is that of the spectrum of the uncontaminated f(x), which is why the positions of the four zeros in (c) are represented by single pixels. Because of the contamination, each zero in (a) and (b) is represented by a blob (the size of each blob is indicated by those blobs that are separated from the other blobs): (a) cl = 0.003, (b) cl = 0.03, (c) Z(F, 1).

Fig. 6
Fig. 6

Zero tracks of spectra of uncontaminated blurred versions of a two-dimensional true image f(x): (a) zero tracks, identified by *—, of Z(F, 2); (b) superposed zero tracks of four Z(Gm, 2) with cl = 0.03 for each gm(x), tracks belonging to Z(G1, 2), Z(G2, 2), Z(G3, 2), Z(G4, 2) identified by ○—, +—, □—, and ×—, respectively, with closed dashed curves encircling tracks corresponding to zero tracks of Z(F, 2).

Fig. 7
Fig. 7

Example of iterative pure blind deconvolution: (a) f(x), (b) h(x), (c) f ̂ ( x , 0.03 ), (d) ĥ(x, 0.03). Note that f(x) and h(x) are real (and also nonnegative).

Fig. 8
Fig. 8

Zero tracks relating to example of iterative pure blind deconvolution: ○—, Z(G, 2); +—, Z(F, 2); □—, Z ( F ̂ , 2 ); × —,Z(Ĥ,2).

Fig. 9
Fig. 9

Third one-dimensional example: (a) pixel magnitudes; (b) pixel phases: —, f(x);—, g(x).

Fig. 10
Fig. 10

Zero maps under Condition A for cl = 0: (a) ω = 1.0, (b) ω = 0.5. Labeling of the zeros belonging to the three zero maps: ●, Z(F,1);+,Z(G,1);○,Z(|F|2,1).

Fig. 11
Fig. 11

Zero maps for ω =0.3 and cl = 0.1: (a) Condition A, (b) Condition B. Labeling of the zeros as for Fig. 10.

Tables (2)

Tables Icon

Table 1 First One-Dimensional Examplea

Tables Icon

Table 2 Second One-Dimensional Examplea

Equations (18)

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γ = α + i β ,
a ( x ) = a 1 ( x ) a 2 ( x ) ,
a ( x ) = a 1 ( x ) a 2 ( x ) , a N ( x ) ,
Z ( A , K ) = n = 1 N Z ( A n , K ) ,
Z ( | A | 2 , K ) = Z ( A * , K ) Z ( A , K ) ,
g ( x ) = f ( x ) h ( x ) + c ( x ) ,
a ( x ) = m 1 , m 2 , , m K = 0 T m 1 , m 2 , , m K k = 1 K m k b ( x ) / x k m k ,
g m ( x ) = f ( x ) h m ( x ) + c m ( x ) ,
H ( λ ) = exp [ i ϕ ( λ ) ] ,
cl = ( | c ( x ) | 2 d x / | g ( x ) | 2 d x ) 1 / 2 ,
Z ( | G | 2 , K ) = Z ( F , K ) Z ( F * , K ) .
Z ( G , K ) = Z ( F , K ) .
F ̂ ( λ ) = F ( λ ) exp [ i ψ ( λ ) ] ,
G ( λ ) = F ( λ ) exp [ i θ ( λ ) ] ,
E g = [ | g ( x ) | 2 d x / s | g ( x ) | 2 d x ] 1 / 2 ,
f ̂ ( x , 0.003 )
f ̂ ( x , 0.03 )
f ̂ ( x , 0.003 )

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