Abstract

Recent work on the Fourier-phase problem is shown to be relevant to the reconstruction of the relative phases of fields scattered to a large number of separated inaccurately surveyed locations. A proposed approach to phase retrieval is illustrated with a computational example. It is assumed that the waveforms of the scattered fields can be recorded at each location.

© 1985 Optical Society of America

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References

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  1. L. Colin, ed., Mathematics of Profile Inversion, NASA Tech. Memo TM X-62 (NASA Ames Research Center, Moffet Field, Calif., August1972);K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1977);H. P. Baltes, ed., Inverse Source Problems in Optics (Springer-Verlag, New York, 1978);Inverse Scattering Problems in Optics (Springer-Verlag, New York, 1980);IEEE Trans. Antennas Propag. 29(2) (1981).
    [CrossRef]
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982);R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983);J. R. Fienup, in Indirect Imaging, J. A. Roberts, ed. (Cambridge U. Press, London, 1984);R. H. T. Bates, W. R. Fright, W. A. Norton, in Indirect Imaging, J. A. Roberts, ed. (Cambridge U. Press, London, 1984);R. H. T. Bates, W. R. Fright, “Reconstructing images from their Fourier intensities” in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1984), Vol. 1, pp. 227–264;M. C. Won, D. Mnyama, R. H. T. Bates, “Improving initial phase estimates for phase retrieval algorithms,” Opt. Acta (to be published).
    [CrossRef] [PubMed]
  3. G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).
  4. R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).
  5. R. H. T. Bates, F. L. Ng, “Polarization-source formulation of electromagnetism and dielectric-loaded waveguides,” Proc. Inst. Electr. Eng. 119, 1568–1574 (1972).
    [CrossRef]
  6. B. B. Baker, E. T. Copson, Mathematical Theory of Huygens’ Principle, 2nd. ed. (Oxford U. Press, New York, 1953).
  7. S. Silver, Microwave Antenna Theory and Design (Dover, New York, 1965), Chap. 5.
  8. R. M. Lewis, “Physical optics inverse diffraction,” IEEE Trans. Antennas Propag. AP-17, 308–314 (1969).
    [CrossRef]
  9. C. K. Chan, N. H. Farhat, “Frequency swept tomographic imaging of three dimensional perfectly conducting objects,” IEEE Trans. Antennas Propag. AP-29, 312–319 (1981).
    [CrossRef]
  10. B. D. Steinberg, Microwave Imaging with Large Antenna Arrays (Wiley, New York, 1983).
  11. R. H. T. Bates, B. S. Robinson, “Ultrasonic transmission speckle imaging,” Ultrasonic Imaging 3, 378–394 (1981);R. H. T. Bates, R. A. Minard, “Compensation for multiple reflection,” IEEE Trans. Sonics Ultrason. SU-31, 330–336 (1984);R. A. Minard, B. S. Robinson, R. H. T. Bates, “Full-wave computed tomography. Part 3: coherent shift-and-add imaging,” Proc. Inst. Electr. Eng. Part A 132, 50–58 (1985).
    [CrossRef] [PubMed]
  12. 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).
  13. R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections: IV: projection completion methods (computational examples),” Optik 50, 269–278 (1978);R. H. T. Bates, W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).
  14. E. C. Sutton, S. Subramanian, C. H. Townes, “Interferometric measurements of stellar positions in the infrared,” Astron. Astrophys. 110, 324–331 (1982);G. N. Gibson, J. Heyman, J. Jugten, W. Fitelson, C. H. Townes, “Optical path length fluctuations in the atmosphere,” preprint (Department of Physics, University of California, Calif. 94720, 1984).
  15. E. D. Brigham, Fast Fourier Transform (Prentice-Hall, Englewood-Cliffs, N.J., 1974).

1982

1981

C. K. Chan, N. H. Farhat, “Frequency swept tomographic imaging of three dimensional perfectly conducting objects,” IEEE Trans. Antennas Propag. AP-29, 312–319 (1981).
[CrossRef]

R. H. T. Bates, B. S. Robinson, “Ultrasonic transmission speckle imaging,” Ultrasonic Imaging 3, 378–394 (1981);R. H. T. Bates, R. A. Minard, “Compensation for multiple reflection,” IEEE Trans. Sonics Ultrason. SU-31, 330–336 (1984);R. A. Minard, B. S. Robinson, R. H. T. Bates, “Full-wave computed tomography. Part 3: coherent shift-and-add imaging,” Proc. Inst. Electr. Eng. Part A 132, 50–58 (1985).
[CrossRef] [PubMed]

1978

R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections: IV: projection completion methods (computational examples),” Optik 50, 269–278 (1978);R. H. T. Bates, W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

1976

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).

1972

R. H. T. Bates, F. L. Ng, “Polarization-source formulation of electromagnetism and dielectric-loaded waveguides,” Proc. Inst. Electr. Eng. 119, 1568–1574 (1972).
[CrossRef]

1969

R. M. Lewis, “Physical optics inverse diffraction,” IEEE Trans. Antennas Propag. AP-17, 308–314 (1969).
[CrossRef]

Baker, B. B.

B. B. Baker, E. T. Copson, Mathematical Theory of Huygens’ Principle, 2nd. ed. (Oxford U. Press, New York, 1953).

Bates, R. H. T.

R. H. T. Bates, B. S. Robinson, “Ultrasonic transmission speckle imaging,” Ultrasonic Imaging 3, 378–394 (1981);R. H. T. Bates, R. A. Minard, “Compensation for multiple reflection,” IEEE Trans. Sonics Ultrason. SU-31, 330–336 (1984);R. A. Minard, B. S. Robinson, R. H. T. Bates, “Full-wave computed tomography. Part 3: coherent shift-and-add imaging,” Proc. Inst. Electr. Eng. Part A 132, 50–58 (1985).
[CrossRef] [PubMed]

R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections: IV: projection completion methods (computational examples),” Optik 50, 269–278 (1978);R. H. T. Bates, W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

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, F. L. Ng, “Polarization-source formulation of electromagnetism and dielectric-loaded waveguides,” Proc. Inst. Electr. Eng. 119, 1568–1574 (1972).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

Brigham, E. D.

E. D. Brigham, Fast Fourier Transform (Prentice-Hall, Englewood-Cliffs, N.J., 1974).

Chan, C. K.

C. K. Chan, N. H. Farhat, “Frequency swept tomographic imaging of three dimensional perfectly conducting objects,” IEEE Trans. Antennas Propag. AP-29, 312–319 (1981).
[CrossRef]

Copson, E. T.

B. B. Baker, E. T. Copson, Mathematical Theory of Huygens’ Principle, 2nd. ed. (Oxford U. Press, New York, 1953).

Farhat, N. H.

C. K. Chan, N. H. Farhat, “Frequency swept tomographic imaging of three dimensional perfectly conducting objects,” IEEE Trans. Antennas Propag. AP-29, 312–319 (1981).
[CrossRef]

Fienup, J. R.

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).

Lewis, R. M.

R. M. Lewis, “Physical optics inverse diffraction,” IEEE Trans. Antennas Propag. AP-17, 308–314 (1969).
[CrossRef]

Lewitt, R. M.

R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections: IV: projection completion methods (computational examples),” Optik 50, 269–278 (1978);R. H. T. Bates, W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

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).

Ng, F. L.

R. H. T. Bates, F. L. Ng, “Polarization-source formulation of electromagnetism and dielectric-loaded waveguides,” Proc. Inst. Electr. Eng. 119, 1568–1574 (1972).
[CrossRef]

Ramachandran, G. N.

G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).

Robinson, B. S.

R. H. T. Bates, B. S. Robinson, “Ultrasonic transmission speckle imaging,” Ultrasonic Imaging 3, 378–394 (1981);R. H. T. Bates, R. A. Minard, “Compensation for multiple reflection,” IEEE Trans. Sonics Ultrason. SU-31, 330–336 (1984);R. A. Minard, B. S. Robinson, R. H. T. Bates, “Full-wave computed tomography. Part 3: coherent shift-and-add imaging,” Proc. Inst. Electr. Eng. Part A 132, 50–58 (1985).
[CrossRef] [PubMed]

Silver, S.

S. Silver, Microwave Antenna Theory and Design (Dover, New York, 1965), Chap. 5.

Srinivasan, R.

G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).

Steinberg, B. D.

B. D. Steinberg, Microwave Imaging with Large Antenna Arrays (Wiley, New York, 1983).

Subramanian, S.

E. C. Sutton, S. Subramanian, C. H. Townes, “Interferometric measurements of stellar positions in the infrared,” Astron. Astrophys. 110, 324–331 (1982);G. N. Gibson, J. Heyman, J. Jugten, W. Fitelson, C. H. Townes, “Optical path length fluctuations in the atmosphere,” preprint (Department of Physics, University of California, Calif. 94720, 1984).

Sutton, E. C.

E. C. Sutton, S. Subramanian, C. H. Townes, “Interferometric measurements of stellar positions in the infrared,” Astron. Astrophys. 110, 324–331 (1982);G. N. Gibson, J. Heyman, J. Jugten, W. Fitelson, C. H. Townes, “Optical path length fluctuations in the atmosphere,” preprint (Department of Physics, University of California, Calif. 94720, 1984).

Townes, C. H.

E. C. Sutton, S. Subramanian, C. H. Townes, “Interferometric measurements of stellar positions in the infrared,” Astron. Astrophys. 110, 324–331 (1982);G. N. Gibson, J. Heyman, J. Jugten, W. Fitelson, C. H. Townes, “Optical path length fluctuations in the atmosphere,” preprint (Department of Physics, University of California, Calif. 94720, 1984).

Appl. Opt.

Astron. Astrophys.

E. C. Sutton, S. Subramanian, C. H. Townes, “Interferometric measurements of stellar positions in the infrared,” Astron. Astrophys. 110, 324–331 (1982);G. N. Gibson, J. Heyman, J. Jugten, W. Fitelson, C. H. Townes, “Optical path length fluctuations in the atmosphere,” preprint (Department of Physics, University of California, Calif. 94720, 1984).

IEEE Trans. Antennas Propag.

R. M. Lewis, “Physical optics inverse diffraction,” IEEE Trans. Antennas Propag. AP-17, 308–314 (1969).
[CrossRef]

C. K. Chan, N. H. Farhat, “Frequency swept tomographic imaging of three dimensional perfectly conducting objects,” IEEE Trans. Antennas Propag. AP-29, 312–319 (1981).
[CrossRef]

N. Z. J. Sci.

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).

Optik

R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections: IV: projection completion methods (computational examples),” Optik 50, 269–278 (1978);R. H. T. Bates, W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

Proc. Inst. Electr. Eng.

R. H. T. Bates, F. L. Ng, “Polarization-source formulation of electromagnetism and dielectric-loaded waveguides,” Proc. Inst. Electr. Eng. 119, 1568–1574 (1972).
[CrossRef]

Ultrasonic Imaging

R. H. T. Bates, B. S. Robinson, “Ultrasonic transmission speckle imaging,” Ultrasonic Imaging 3, 378–394 (1981);R. H. T. Bates, R. A. Minard, “Compensation for multiple reflection,” IEEE Trans. Sonics Ultrason. SU-31, 330–336 (1984);R. A. Minard, B. S. Robinson, R. H. T. Bates, “Full-wave computed tomography. Part 3: coherent shift-and-add imaging,” Proc. Inst. Electr. Eng. Part A 132, 50–58 (1985).
[CrossRef] [PubMed]

Other

L. Colin, ed., Mathematics of Profile Inversion, NASA Tech. Memo TM X-62 (NASA Ames Research Center, Moffet Field, Calif., August1972);K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1977);H. P. Baltes, ed., Inverse Source Problems in Optics (Springer-Verlag, New York, 1978);Inverse Scattering Problems in Optics (Springer-Verlag, New York, 1980);IEEE Trans. Antennas Propag. 29(2) (1981).
[CrossRef]

B. D. Steinberg, Microwave Imaging with Large Antenna Arrays (Wiley, New York, 1983).

B. B. Baker, E. T. Copson, Mathematical Theory of Huygens’ Principle, 2nd. ed. (Oxford U. Press, New York, 1953).

S. Silver, Microwave Antenna Theory and Design (Dover, New York, 1965), Chap. 5.

G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).

R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

E. D. Brigham, Fast Fourier Transform (Prentice-Hall, Englewood-Cliffs, N.J., 1974).

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Figures (3)

Fig. 1
Fig. 1

Body and coordinates.

Fig. 2
Fig. 2

Illustration of the performance of the modified Fienup algorithm: (a) |f(x)|, (b) first version of | f ( x ) |, and (c) fourth version of | f ( x ) |.

Fig. 3
Fig. 3

Phase error E (degrees) versus number N of iterations: ————, modified Fienup algorithm; — — — — —, standard Fienup algorithm.

Equations (14)

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

[ I ( u ) ] 1 / 2 exp [ i Φ ( u ) ] = F ( u ) L f ( x ) ,
| f ( x ) | < for all x ϒ
I ( u ) L f f ( x ) ,
| f f ( x ) | < η for all x ϒ a ,
a l = 2 b l for all l { 1 , 2 , 3 , , L } .
F ( u ) = ( L ) f ( x ) exp ( i 2 π u · x ) d ϒ with u · x = l = 1 L u l x l ,
s ( ϕ , θ , k ) 1 s ( ϕ , θ , t ) .
F ( u ) = W { S ( u ) } ,
Φ ̂ ( u , k ) = Φ ( u , k ) + α ( u ) + k β ( u ) .
b 1 = b 2 = 1 so that a 1 = a 2 = 2 .
Φ ̂ ( u , k ) = Φ ( u , k ) + α ( u ) .
F ̂ ( u , k ) = | F ( u , k ) | exp ( i Φ ̂ ( u , k ) ) 2 f ̂ ( x , t ) = f ̂ ( x ) ,
f ( x ) = f ( x ) for x Λ = 0 for x Λ ¯ a ,
Δ ( u ) = [ 1 / N ( u ) ] m [ Φ ̂ ( u , m / 2 ) Φ ( u , m / 2 ) ] ,

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