P. J. Bones, C. R. Parker, B. L. Satherly, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995); see also R. W. Watson, C. R. Parker, P. J. Jones, “Demonstration of two-dimensional consistent deconvolution,” Opt. Commun. 93, 359–365 (1992).

[CrossRef]

R. Barakat, “Zero crossing rate of differentiated speckle intensity,” J. Opt. Soc. Am. A 11, 671–673 (1994).

[CrossRef]

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]

P. T. Chen, M. A. Fiddy, “Image reconstruction from power spectral data with use of point zero locations,” J. Opt. Soc. Am. A 11, 2210–2214 (1994).

[CrossRef]

R. H. T. Bates, H. Jiang, B. L. K. Davey, “Multidimensional system identification through blind deconvolution,” Multidimen. Sys. Signal Process. 1, 127–142 (1990).

[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, 468–474 (1990).

[CrossRef]

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

[CrossRef]

H. M. Berenyi, H. V. Deighton, M. A. Fiddy, “The use of bivariate polynomial factorization algorithm in two dimensional phase problems,” Opt. Acta 32, 687–700 (1985).

[CrossRef]

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of miltidimensional band-limited functions,”J. Opt. Soc. Am. 73, 693–698 (1985).

[CrossRef]

M. Nieto-Vesperinas, “Inverse scattering problems: a study in terms of the zeros of entire functions,”J. Math. Phys. 25, 2109–2115 (1984).

[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–928 (1981).

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations,”J. Opt. Soc. Am. 73, 525–527 (1983).

[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–928 (1981).

R. H. T. Bates, H. Jiang, B. L. K. Davey, “Multidimensional system identification through blind deconvolution,” Multidimen. Sys. Signal Process. 1, 127–142 (1990).

[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, 468–474 (1990).

[CrossRef]

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

[CrossRef]

H. M. Berenyi, H. V. Deighton, M. A. Fiddy, “The use of bivariate polynomial factorization algorithm in two dimensional phase problems,” Opt. Acta 32, 687–700 (1985).

[CrossRef]

P. J. Bones, C. R. Parker, B. L. Satherly, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995); see also R. W. Watson, C. R. Parker, P. J. Jones, “Demonstration of two-dimensional consistent deconvolution,” Opt. Commun. 93, 359–365 (1992).

[CrossRef]

H. J. Bremermann, “Several complex variable,” Studies in Real and Complex Analysis, I. I. Hirschmann, ed., Vol. 3 of Studies in Mathematics (Mathematical Association of America, Washington, D.C., 1965).

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. SPIE2029, 16–24 (1993).

[CrossRef]

M. Nieto-Vesperinas, J. C. Dainty, “Phase recovery for two dimensional digital objects by polynomial factorization,” Opt. Commun. 58, 83–88 (1986).

[CrossRef]

R. H. T. Bates, H. Jiang, B. L. K. Davey, “Multidimensional system identification through blind deconvolution,” Multidimen. Sys. Signal Process. 1, 127–142 (1990).

[CrossRef]

H. M. Berenyi, H. V. Deighton, M. A. Fiddy, “The use of bivariate polynomial factorization algorithm in two dimensional phase problems,” Opt. Acta 32, 687–700 (1985).

[CrossRef]

P. T. Chen, M. A. Fiddy, “Image reconstruction from power spectral data with use of point zero locations,” J. Opt. Soc. Am. A 11, 2210–2214 (1994).

[CrossRef]

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of miltidimensional band-limited functions,”J. Opt. Soc. Am. 73, 693–698 (1985).

[CrossRef]

H. M. Berenyi, H. V. Deighton, M. A. Fiddy, “The use of bivariate polynomial factorization algorithm in two dimensional phase problems,” Opt. Acta 32, 687–700 (1985).

[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. SPIE2029, 16–24 (1993).

[CrossRef]

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 13, pp. 499–529.

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

[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. SPIE2029, 16–24 (1993).

[CrossRef]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).

[CrossRef]

R. H. T. Bates, H. Jiang, B. L. K. Davey, “Multidimensional system identification through blind deconvolution,” Multidimen. Sys. Signal Process. 1, 127–142 (1990).

[CrossRef]

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

[CrossRef]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).

[CrossRef]

M. Nieto-Vesperinas, J. C. Dainty, “Phase recovery for two dimensional digital objects by polynomial factorization,” Opt. Commun. 58, 83–88 (1986).

[CrossRef]

M. Nieto-Vesperinas, “Inverse scattering problems: a study in terms of the zeros of entire functions,”J. Math. Phys. 25, 2109–2115 (1984).

[CrossRef]

P. J. Bones, C. R. Parker, B. L. Satherly, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995); see also R. W. Watson, C. R. Parker, P. J. Jones, “Demonstration of two-dimensional consistent deconvolution,” Opt. Commun. 93, 359–365 (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, 468–474 (1990).

[CrossRef]

P. J. Bones, C. R. Parker, B. L. Satherly, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995); see also R. W. Watson, C. R. Parker, P. J. Jones, “Demonstration of two-dimensional consistent deconvolution,” Opt. Commun. 93, 359–365 (1992).

[CrossRef]

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of miltidimensional band-limited functions,”J. Opt. Soc. Am. 73, 693–698 (1985).

[CrossRef]

E. C. Titchmarsh, The Theory of Functions (Oxford U. Press, Oxford, 1939).

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]

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]

J. R. Feinup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).

[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. SPIE2029, 16–24 (1993).

[CrossRef]

P. J. Bones, C. R. Parker, B. L. Satherly, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995); see also R. W. Watson, C. R. Parker, P. J. Jones, “Demonstration of two-dimensional consistent deconvolution,” Opt. Commun. 93, 359–365 (1992).

[CrossRef]

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations,”J. Opt. Soc. Am. 73, 525–527 (1983).

[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–928 (1981).

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

[CrossRef]

M. Nieto-Vesperinas, “Inverse scattering problems: a study in terms of the zeros of entire functions,”J. Math. Phys. 25, 2109–2115 (1984).

[CrossRef]

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of miltidimensional band-limited functions,”J. Opt. Soc. Am. 73, 693–698 (1985).

[CrossRef]

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations,”J. Opt. Soc. Am. 73, 525–527 (1983).

[CrossRef]

R. Barakat, “Zero crossing rate of differentiated speckle intensity,” J. Opt. Soc. Am. A 11, 671–673 (1994).

[CrossRef]

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]

P. T. Chen, M. A. Fiddy, “Image reconstruction from power spectral data with use of point zero locations,” J. Opt. Soc. Am. A 11, 2210–2214 (1994).

[CrossRef]

J. R. Feinup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).

[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, 468–474 (1990).

[CrossRef]

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]

P. J. Bones, C. R. Parker, B. L. Satherly, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995); see also R. W. Watson, C. R. Parker, P. J. Jones, “Demonstration of two-dimensional consistent deconvolution,” Opt. Commun. 93, 359–365 (1992).

[CrossRef]

R. H. T. Bates, H. Jiang, B. L. K. Davey, “Multidimensional system identification through blind deconvolution,” Multidimen. Sys. Signal Process. 1, 127–142 (1990).

[CrossRef]

H. M. Berenyi, H. V. Deighton, M. A. Fiddy, “The use of bivariate polynomial factorization algorithm in two dimensional phase problems,” Opt. Acta 32, 687–700 (1985).

[CrossRef]

M. Nieto-Vesperinas, J. C. Dainty, “Phase recovery for two dimensional digital objects by polynomial factorization,” Opt. Commun. 58, 83–88 (1986).

[CrossRef]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).

[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–928 (1981).

E. C. Titchmarsh, The Theory of Functions (Oxford U. Press, Oxford, 1939).

H. J. Bremermann, “Several complex variable,” Studies in Real and Complex Analysis, I. I. Hirschmann, ed., Vol. 3 of Studies in Mathematics (Mathematical Association of America, Washington, D.C., 1965).

If one were to use zero locations on a line Θ passing through the x, yorigin in the real x–yplane, it follows that a coordinate system can be chosen such that the polynomial in Eq. (5) is factorizable into a simple 1D Hadamard product. The inverse Fourier transform of data on this single line corresponds to a projection of the image onto the line parallel to that in the spectral domain. Assuming that the image varies fairly smoothly, rotating this line Θ about the origin leads to additional Hadamard product representations for which the zero locations should migrate incrementally from one Θ to the next. When the projection-slice theorem is used, it follows that the sum of all these 1D Hadamard products on a set of lines Θ will recover the complete 2D image. Moreover, by fitting the evaluated product to the available data one can usually identify the order of the zeros and whether there are any missing. It also provides an opportunity to fine tune the zero locations, should they fall between sample values.

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 13, pp. 499–529.

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. SPIE2029, 16–24 (1993).

[CrossRef]