Abstract

The points at which two-dimensional spectra of band-limited functions are zero can be used to generate a polynomial approximation of the complex spectrum. These point-zero locations are common to both the spectrum and the associated power spectrum. The phase of the complex-valued polynomial along with the measured intensity data is used to generate an initial guess for the function that an error-reduction algorithm can improve. This approach has practical utility for Fourier phase retrieval as well as for blind deconvolution. The limitations of the model are discussed, and examples are given.

© 1996 Optical Society of America

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References

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  1. 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.
  2. M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  3. 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]
  4. M. Nieto-Vesperinas, J. C. Dainty, “Phase recovery for two dimensional digital objects by polynomial factorization,” Opt. Commun. 58, 83–88 (1986).
    [CrossRef]
  5. 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]
  6. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. 35, 520–526 (1987).
    [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  12. J. R. Feinup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  13. 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]
  14. 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]
  15. 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]
  16. 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]
  17. 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).
  18. R. Barakat, “Zero crossing rate of differentiated speckle intensity,” J. Opt. Soc. Am. A 11, 671–673 (1994).
    [CrossRef]
  19. E. C. Titchmarsh, The Theory of Functions (Oxford U. Press, Oxford, 1939).
  20. 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).
  21. 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.
  22. M. Nieto-Vesperinas, “Inverse scattering problems: a study in terms of the zeros of entire functions,”J. Math. Phys. 25, 2109–2115 (1984).
    [CrossRef]

1995

1994

1991

1990

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]

1987

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

1986

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

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

1985

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]

1984

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

1983

1982

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

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

1981

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).

Barakat, R.

Baranova, N. B.

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).

Bates, R. H. T.

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]

Berenyi, H. M.

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]

Bones, P. J.

Bremermann, H. J.

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).

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. SPIE2029, 16–24 (1993).
[CrossRef]

Chen, P. T.

Dainty, J. C.

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

Davey, B. L. K.

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]

Deighton, H. V.

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]

Feinup, J. R.

Fiddy, M. A.

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.

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. 35, 520–526 (1987).
[CrossRef]

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. SPIE2029, 16–24 (1993).
[CrossRef]

Hayes, M. H.

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

Jiang, H.

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]

Lane, R. G.

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

Mamaev, A. V.

McClellan, J. H.

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

Nieto-Vesperinas, M.

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]

Parker, C. R.

Pilipetsky, N. F.

Quek, B. K.

Satherly, B. L.

Scivier, M. S.

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]

Shkunov, V. V.

Titchmarsh, E. C.

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

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. SPIE2029, 16–24 (1993).
[CrossRef]

Watson, R. W.

Yagle, A. E.

Zel’dovich, B. Ya.

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).

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process

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

J. Math. Phys.

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

J. Opt. Soc. Am.

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]

J. Opt. Soc. Am. A

Multidimen. Sys. Signal Process.

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]

Opt. Acta

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]

Opt. Commun.

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

Proc. IEEE

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

Sov. Phys. JETP

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).

Other

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]

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Figures (5)

Fig. 1
Fig. 1

(a) Magnitude of the function x3 + y3 + xyc = 0, (b) factorizable zero set thresholded to a value close to zero. In (b) a real-zero ellipse and a real-zero line have been chosen to match closely the real-zero crossing of the irreducible function.

Fig. 2
Fig. 2

(a) The inverse z transform of the function in Fig. 1 gives the associated image, which is just described by the coefficients of the polynomial. The object consists of 3 pixels of unit height at coordinates (1, 3), (3, 1), and (1, 1) and a pixel of height c at (0, 0). (b) On multiplying out the factors representing the zeros in Fig. 1(b) and inverse-z transforming, one finds this distribution of pixels, which is very similar to (a).

Fig. 3
Fig. 3

(a) The object originally located in a 64 × 64 array of zeros; only the central portion of the array is shown here. (b) The power spectrum of this object, (c) its (wrapped) phase. (d) Distribution of real point zeros, (e) phase function recovered from this factorizable model. (f) Image obtained by use of this phase and the known Fourier magnitude data. (g) The phase after 30 iterations of the error-reduction algorithm and (h) the reconstruction. Reconstructions are also shown after 30 iterations by use of (i) the random phase and (j) a zero phase as initial phase estimates. These objects are reconstructed into 64 × 64 arrays, but only the central portion of the array is shown here. The error-reduction algorithm assumed an object support that was a square 1 pixel wider than the widest dimension of the actual object support.

Fig. 4
Fig. 4

(a) Central portion of the array of an object originally located in a 64 × 64 array of zeros, (b) its power spectrum, and (c) its true phase function. (d) Distribution of real point zeros, (e) phase function recovered from this factorizable model, and (f) reconstruction using this phase with known Fourier magnitude data. (g) The phase after 600 iterations of the error-reduction algorithm. Image estimates are obtained from the Fourier magnitude data and (h) this iterated phase, as well as by use of (i) a random phase and (j) a zero phase as initial phase estimates. These objects are reconstructed into 64 × 64 arrays, but only the central portion of the array is shown here. The error-reduction algorithm assumed an object support that was a square 1 pixel wider than the widest dimension of the actual object support.

Fig. 5
Fig. 5

(a), (b) Two object plane functions, an image and an unknown point-spread function. (c) Their convolution, (d) the corresponding power spectrum. (e) The real point zeros. Using a second point-spread function, we find (f) a different power spectrum from which (g) a second set of zero locations is found. (h) Those zeros common to both distributions, which are the same as those shown in (d), from which the reconstruction of the object is obtained by iteration.

Equations (9)

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2 i δ E ( r , z ) / δ z + Δ E ( r , z ) = 0 ,
Δ = δ 2 / δ x 2 + δ 2 / δ y 2 .
E ( r , z ) = exp ( i α ) ( A x x + i A y y ) .
F ( x , y ) = j ( A j + i B j ) ,
F ( x , y ) = g ( x , y ) [ y m + p 1 ( x ) y m - 1 + + p m ( x ) ] ,
1 / 2 π i 1 / F δ F / δ y d y ,             y = r .
F ( x , y ) ( y - ζ 1 ) ( y - ζ 2 ) ( y - ζ 3 ) ( y - ζ m ) ,
F ( x , y ) ( y - ζ 1 ) ( y - ζ 2 ) ( y - ζ 3 ) ( y - ζ m ) ,
x 3 + y 3 + x y - c = 0.

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