Abstract

The analytic properties of multidimensional band-limited functions are described, with particular emphasis on the occurrence of zeros in the intensity distribution. It is shown that zeros at isolated points lead to ambiguities in the phase that have implications for phase retrieval, phase unwrapping, and phase-only reconstruction.

© 1985 Optical Society of America

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References

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  1. M. A. Fiddy, G. Ross, M. Nieto-Vesperinas, A. M. J. Huiser, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
    [CrossRef]
  2. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–46 (1963).
    [CrossRef]
  3. H. A. Ferwerda, “Fundamental aspects of the phase retrieval problem,”AIP Conf. Proc. 65, 402–411 (1981).
    [CrossRef]
  4. M. A. Fiddy, “The phase retrieval problem,” Proc. Soc. Photo-Opt. Instrum. Eng. 413, 176–181 (1983).
  5. P. Kiedron, “A phase retrieval technique in pupil synthesis,” in Digest of Topical Meeting on Signal Recovery (Optical Society of America, Washington, D.C., 1983), paper ThA13.
  6. A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
    [CrossRef]
  7. A. V. Oppenheim, M. H. Hayes, J. S. Lim, “Iterative procedures for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).
    [CrossRef]
  8. A. Levi, H. Stark, “Signal reconstruction from phase by projections onto convex sets,”J. Opt. Soc. Am. 73, 810–822 (1983).
    [CrossRef]
  9. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  10. A. V. Oppenheim, R. V. Scafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.
  11. A. J. Devaney, “Inverse scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
    [CrossRef] [PubMed]
  12. A. J. Devaney, “Geophysical diffraction tomography,”IEEE Trans. Geosci. Remote Sensing GE-22, 3–13 (1984).
    [CrossRef]
  13. R. P. Boas, Entire Functions (Academic, New York, 1954).
  14. E. C. Titchmarsh, “On the zeros of certain integral functions,” Proc. Lond. Math. Soc. Ser. A 25, 283–294 (1926).
    [CrossRef]
  15. A. A. G. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. IEEE 68, 308–328 (1980).
    [CrossRef]
  16. I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,”J. Math. Phys. 23, 2291–2298 (1982).
    [CrossRef]
  17. J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta. 31, 325–330 (1984).
    [CrossRef]
  18. J. L. C. Sanz, T. S. Huang, “Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude,”J. Opt. Soc. Am. 73, 1446–1450 (1983).
    [CrossRef]
  19. P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).
  20. M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one dimension,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  21. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
    [CrossRef]
  22. V. T. Tom, T. F. Quatieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative non-expansive signal reconstruction algorithms,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
    [CrossRef]
  23. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [CrossRef] [PubMed]
  24. A. M. J. Huiser, P. van Toorn, “Ambiguity of the phase reconstruction problem,” Opt. Lett. 5, 499–501 (1980).
    [CrossRef] [PubMed]
  25. E. C. Titchbmarsh, The Theory of Functions, 2nd ed. (Oxford U. Press, Oxford, 1939).
  26. N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wavefront surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).
  27. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations,”J. Opt. Soc. Am. 73, 525–528 (1983).
    [CrossRef]
  28. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  29. R. C. Gunning, H. Rossi, Analytic Functions of Several Complex Variables (Prentice-Hall, Englewood Cliffs, N.J., 1965).
  30. M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of band-limited functions and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
    [CrossRef]
  31. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik 61, 247–262 (1982).
  32. J. R. Fienup, Environmental Research Institute of Michigan, P.O. Box 8186, Ann Arbor, Michigan 48107 (personal communication).

1984

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

A. J. Devaney, “Geophysical diffraction tomography,”IEEE Trans. Geosci. Remote Sensing GE-22, 3–13 (1984).
[CrossRef]

J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta. 31, 325–330 (1984).
[CrossRef]

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of band-limited functions and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

1983

1982

M. A. Fiddy, G. Ross, M. Nieto-Vesperinas, A. M. J. Huiser, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

A. V. Oppenheim, M. H. Hayes, J. S. Lim, “Iterative procedures for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).
[CrossRef]

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

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,”J. Math. Phys. 23, 2291–2298 (1982).
[CrossRef]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik 61, 247–262 (1982).

1981

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wavefront surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

V. T. Tom, T. F. Quatieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative non-expansive signal reconstruction algorithms,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

H. A. Ferwerda, “Fundamental aspects of the phase retrieval problem,”AIP Conf. Proc. 65, 402–411 (1981).
[CrossRef]

A. J. Devaney, “Inverse scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
[CrossRef] [PubMed]

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1980

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

A. M. J. Huiser, P. van Toorn, “Ambiguity of the phase reconstruction problem,” Opt. Lett. 5, 499–501 (1980).
[CrossRef] [PubMed]

1974

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).

1963

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–46 (1963).
[CrossRef]

1926

E. C. Titchmarsh, “On the zeros of certain integral functions,” Proc. Lond. Math. Soc. Ser. A 25, 283–294 (1926).
[CrossRef]

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, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wavefront surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik 61, 247–262 (1982).

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Boas, R. P.

R. P. Boas, Entire Functions (Academic, New York, 1954).

Brames, B. J.

Dainty, J. C.

J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta. 31, 325–330 (1984).
[CrossRef]

M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
[CrossRef] [PubMed]

Devaney, A. J.

A. J. Devaney, “Geophysical diffraction tomography,”IEEE Trans. Geosci. Remote Sensing GE-22, 3–13 (1984).
[CrossRef]

A. J. Devaney, “Inverse scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
[CrossRef] [PubMed]

Ferwerda, H. A.

H. A. Ferwerda, “Fundamental aspects of the phase retrieval problem,”AIP Conf. Proc. 65, 402–411 (1981).
[CrossRef]

Fiddy, M. A.

J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta. 31, 325–330 (1984).
[CrossRef]

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of band-limited functions and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
[CrossRef] [PubMed]

M. A. Fiddy, “The phase retrieval problem,” Proc. Soc. Photo-Opt. Instrum. Eng. 413, 176–181 (1983).

M. A. Fiddy, G. Ross, M. Nieto-Vesperinas, A. M. J. Huiser, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

Fienup, J. R.

J. R. Fienup, Environmental Research Institute of Michigan, P.O. Box 8186, Ann Arbor, Michigan 48107 (personal communication).

Gianino, P. D.

Gunning, R. C.

R. C. Gunning, H. Rossi, Analytic Functions of Several Complex Variables (Prentice-Hall, Englewood Cliffs, N.J., 1965).

Hall, T. J.

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of band-limited functions and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

Hayes, M. H.

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

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

A. V. Oppenheim, M. H. Hayes, J. S. Lim, “Iterative procedures for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).
[CrossRef]

V. T. Tom, T. F. Quatieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative non-expansive signal reconstruction algorithms,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

Horner, J. L.

Huang, T. S.

Huiser, A. M. J.

M. A. Fiddy, G. Ross, M. Nieto-Vesperinas, A. M. J. Huiser, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

A. M. J. Huiser, P. van Toorn, “Ambiguity of the phase reconstruction problem,” Opt. Lett. 5, 499–501 (1980).
[CrossRef] [PubMed]

Kiedron, P.

P. Kiedron, “A phase retrieval technique in pupil synthesis,” in Digest of Topical Meeting on Signal Recovery (Optical Society of America, Washington, D.C., 1983), paper ThA13.

Levi, A.

Lim, J. S.

A. V. Oppenheim, M. H. Hayes, J. S. Lim, “Iterative procedures for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).
[CrossRef]

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Mamaev, A. V.

Manolitsakis, I.

I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,”J. Math. Phys. 23, 2291–2298 (1982).
[CrossRef]

McClellan, J. H.

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

V. T. Tom, T. F. Quatieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative non-expansive signal reconstruction algorithms,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

Napier, P. J.

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).

Nieto-Vesperinas, M.

M. A. Fiddy, G. Ross, M. Nieto-Vesperinas, A. M. J. Huiser, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, M. H. Hayes, J. S. Lim, “Iterative procedures for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).
[CrossRef]

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

A. V. Oppenheim, R. V. Scafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

Pilipetsky, N. F.

Quatieri, T. F.

V. T. Tom, T. F. Quatieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative non-expansive signal reconstruction algorithms,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

Requicha, A. A. G.

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

Ross, G.

M. A. Fiddy, G. Ross, M. Nieto-Vesperinas, A. M. J. Huiser, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

Rossi, H.

R. C. Gunning, H. Rossi, Analytic Functions of Several Complex Variables (Prentice-Hall, Englewood Cliffs, N.J., 1965).

Sanz, J. L. C.

Scafer, R. V.

A. V. Oppenheim, R. V. Scafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

Scivier, M. S.

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of band-limited functions and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

Shkunov, V. V.

Stark, H.

Titchbmarsh, E. C.

E. C. Titchbmarsh, The Theory of Functions, 2nd ed. (Oxford U. Press, Oxford, 1939).

Titchmarsh, E. C.

E. C. Titchmarsh, “On the zeros of certain integral functions,” Proc. Lond. Math. Soc. Ser. A 25, 283–294 (1926).
[CrossRef]

Tom, V. T.

V. T. Tom, T. F. Quatieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative non-expansive signal reconstruction algorithms,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

van Toorn, P.

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–46 (1963).
[CrossRef]

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, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wavefront surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

AIP Conf. Proc.

H. A. Ferwerda, “Fundamental aspects of the phase retrieval problem,”AIP Conf. Proc. 65, 402–411 (1981).
[CrossRef]

Appl. Opt.

Astron. Astrophys. Suppl.

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).

IEEE Trans. Acoust. Speech Signal Process.

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

V. T. Tom, T. F. Quatieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative non-expansive signal reconstruction algorithms,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing

A. J. Devaney, “Geophysical diffraction tomography,”IEEE Trans. Geosci. Remote Sensing GE-22, 3–13 (1984).
[CrossRef]

J. Math. Phys.

I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,”J. Math. Phys. 23, 2291–2298 (1982).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of band-limited functions and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

M. A. Fiddy, G. Ross, M. Nieto-Vesperinas, A. M. J. Huiser, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–46 (1963).
[CrossRef]

Opt. Acta.

J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta. 31, 325–330 (1984).
[CrossRef]

Opt. Eng.

A. V. Oppenheim, M. H. Hayes, J. S. Lim, “Iterative procedures for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).
[CrossRef]

Opt. Lett.

Optik

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik 61, 247–262 (1982).

Proc. IEEE

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

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

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

Proc. Lond. Math. Soc. Ser. A

E. C. Titchmarsh, “On the zeros of certain integral functions,” Proc. Lond. Math. Soc. Ser. A 25, 283–294 (1926).
[CrossRef]

Proc. R. Soc. London Ser. A

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

M. A. Fiddy, “The phase retrieval problem,” Proc. Soc. Photo-Opt. Instrum. Eng. 413, 176–181 (1983).

Sov. Phys. JETP

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wavefront surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Other

R. C. Gunning, H. Rossi, Analytic Functions of Several Complex Variables (Prentice-Hall, Englewood Cliffs, N.J., 1965).

J. R. Fienup, Environmental Research Institute of Michigan, P.O. Box 8186, Ann Arbor, Michigan 48107 (personal communication).

E. C. Titchbmarsh, The Theory of Functions, 2nd ed. (Oxford U. Press, Oxford, 1939).

P. Kiedron, “A phase retrieval technique in pupil synthesis,” in Digest of Topical Meeting on Signal Recovery (Optical Society of America, Washington, D.C., 1983), paper ThA13.

R. P. Boas, Entire Functions (Academic, New York, 1954).

A. V. Oppenheim, R. V. Scafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

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Figures (6)

Fig. 1
Fig. 1

The zero lines in mod(F) on the x1x2 plane for a featureless square object.

Fig. 2
Fig. 2

To illustrate Eisenstein’s criterion.

Fig. 3
Fig. 3

The zeros of Re[F(x1, x2)], where f0M is nonzero but small with respect to the rest of the object, which is the featureless square of Fig. 1.

Fig. 4
Fig. 4

The zeros of Im[F(x1, x2)], where f0M is the same as for Fig. 3.

Fig. 5
Fig. 5

The zeros of mod[F(x1, x2)], with f0M chosen as for Figs. 3 and 4.

Fig. 6
Fig. 6

To show the branch cut in phase in the neighborhood of a point zero.

Equations (10)

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

F ( x ) = a b f ( t ) exp ( i x t ) d t ,
F ( z ) = a b f ( t ) exp ( i z t ) d t ,
F ( z ) = F ( 0 ) j ( 1 - z / z j ) ,
z j = ( 2 j π - i ln [ f ( b ) / f ( a ) ] / ( b - a ) .
F ( z ) = j N { F j ( z ) exp [ γ j ( z ) ] } L j ,             N
F ( z ) = j f j exp ( i j z ) ,
z 1 + i ( 1 - r 2 / a 2 ) ,
F ( x 1 , x 2 ) = ( f 00 + f 10 x 1 + + f M 0 x 1 M ) + ( f 01 + f 11 x 1 + + f M 1 x 1 M ) x 2 + + ( f 0 M + f 1 M x 1 + + f M M x 1 M ) x 2 M .
F ( x 1 , x 2 ) = exp ( i a ) ( A 1 x 1 + i A 2 x 2 ) .
arg ( F ) = a + arctan ( A 2 x 2 / A 1 x 1 )

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