Abstract

The phase-retrieval problem for discrete multidimensional fields is investigated. In particular, a recursive procedure is developed for reconstructing a signal from the modulus of its Fourier transform. The information necessary to begin the recursion is the boundary values of the signal. Although it is not always possible to determine these boundary values from Fourier modulus data only, if the sequence has a region of support with a certain geometry then these boundary values can be determined. These geometries represent a generalization of the conditions for off-axis holography.

© 1983 Optical Society of America

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References

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  1. H. A. Ferwerda, "The phase reconstruction problem for wave amplitudes and coherence functions," in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
    [CrossRef]
  2. L. S. Taylor, "The phase retrieval problem," IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
    [CrossRef]
  3. 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]
  4. M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
    [CrossRef]
  5. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).
  6. J. R. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. 18, 529–534 (1979).
    [CrossRef]
  7. M. H. Hayes, "The representation of signals in terms of spectral amplitude," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 1446–1449.
  8. P. L. Van Hove, M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from signed Fourier transform magnitude," IEEE Trans. Acoust. Speech Signal Process. (to be published).
  9. M. H. Hayes and T. F. Quatieri, "The importance of boundary conditions in the phase retrieval problem," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.
  10. A. Mostowski and M. Stark, Introduction to Higher Algebra (Pergamon, New York, 1964).
  11. E. M. Hofstetter, "Construction of time-limited functions with specified autocorrelation functions," IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
    [CrossRef]
  12. M. H. Hayes and J. H. McClellan, "Reducible polynomials in more than one variable," Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  13. M. A. Fiddy, B. J. Brames, and J. C. Dainty, "Enforcing irreducibility for phase retrieval in two dimensions," Opt. Lett. 8, 96–98 (1983).
    [CrossRef] [PubMed]

1983 (1)

1982 (1)

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]

1981 (1)

L. S. Taylor, "The phase retrieval problem," IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

1980 (1)

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

1979 (1)

J. R. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. 18, 529–534 (1979).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).

1964 (1)

E. M. Hofstetter, "Construction of time-limited functions with specified autocorrelation functions," IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

Brames, B. J.

Dainty, J. C.

Ferwerda, H. A.

H. A. Ferwerda, "The phase reconstruction problem for wave amplitudes and coherence functions," in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
[CrossRef]

Fiddy, M. A.

Fienup, J. R.

J. R. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. 18, 529–534 (1979).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).

Hayes, M. H.

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]

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

M. H. Hayes, "The representation of signals in terms of spectral amplitude," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 1446–1449.

P. L. Van Hove, M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from signed Fourier transform magnitude," IEEE Trans. Acoust. Speech Signal Process. (to be published).

M. H. Hayes and T. F. Quatieri, "The importance of boundary conditions in the phase retrieval problem," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

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

Hofstetter, E. M.

E. M. Hofstetter, "Construction of time-limited functions with specified autocorrelation functions," IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

Lim, J. S.

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

P. L. Van Hove, M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from signed Fourier transform magnitude," IEEE Trans. Acoust. Speech Signal Process. (to be published).

McClellan, J. H.

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

Mostowski, A.

A. Mostowski and M. Stark, Introduction to Higher Algebra (Pergamon, New York, 1964).

Oppenheim, A. V.

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

P. L. Van Hove, M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from signed Fourier transform magnitude," IEEE Trans. Acoust. Speech Signal Process. (to be published).

Quatieri, T. F.

M. H. Hayes and T. F. Quatieri, "The importance of boundary conditions in the phase retrieval problem," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).

Stark, M.

A. Mostowski and M. Stark, Introduction to Higher Algebra (Pergamon, New York, 1964).

Taylor, L. S.

L. S. Taylor, "The phase retrieval problem," IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

Van Hove, P. L.

P. L. Van Hove, M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from signed Fourier transform magnitude," IEEE Trans. Acoust. Speech Signal Process. (to be published).

IEEE Trans. Acoust. Speech Signal Process. (2)

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]

M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

L. S. Taylor, "The phase retrieval problem," IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

IEEE Trans. Inf. Theory (1)

E. M. Hofstetter, "Construction of time-limited functions with specified autocorrelation functions," IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. 18, 529–534 (1979).
[CrossRef]

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).

Other (6)

H. A. Ferwerda, "The phase reconstruction problem for wave amplitudes and coherence functions," in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
[CrossRef]

M. H. Hayes, "The representation of signals in terms of spectral amplitude," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 1446–1449.

P. L. Van Hove, M. H. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from signed Fourier transform magnitude," IEEE Trans. Acoust. Speech Signal Process. (to be published).

M. H. Hayes and T. F. Quatieri, "The importance of boundary conditions in the phase retrieval problem," in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

A. Mostowski and M. Stark, Introduction to Higher Algebra (Pergamon, New York, 1964).

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

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Figures (6)

Fig. 1
Fig. 1

(a) A region of support, R(M, N), for a two-dimensional sequence. (b) The region of support of its autocorrelation.

Fig. 2
Fig. 2

System interpretation of the linear equations that define the recursive phase-retrieval algorithm.

Fig. 3
Fig. 3

Phase retrieval using known boundary conditions. (a) Original image. (b) Reconstructed image.

Fig. 4
Fig. 4

(a) A triangular region of support for a two-dimensional sequence. (b) The region of support of its autocorrelation.

Fig. 5
Fig. 5

The division of the two-dimensional plane into eight regions and the rectangular region R(M, N).

Fig. 6
Fig. 6

Point sources sufficient for the determination of the boundary values of a two-dimensional sequence, x(m, n). (a) A single point source in region I at (M, N). (b) Two point sources, one in region II at (m0, N) and one in region VIII at (M, n0).

Equations (29)

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X ( z 1 , z 2 ) = m = - n = - x ( m , n ) z 1 - m z 2 - n ,
X ( e j ω 1 , e j ω 2 ) = m = - n = - x ( m , n ) e - j ω 1 m e - j ω 2 n ,
X ( e j ω 1 , e j ω 2 ) = X ( e j ω 1 , e j ω 2 ) e j ϕ x ( ω 1 , ω 2 ) .
R ( M , N ) = [ 0 , M - 1 ] × [ 0 , N - 1 ] ,
X ( z 1 , z 2 ) = α z 1 - n 1 z 2 - n 2 k = 1 p X k ( z 1 , z 2 ) ,
y ( m , n ) = ± x ( ± m + k , ± n + l )
x ( m , n ) ~ y ( m , n ) .
X ( e j ω 1 , e j ω 2 ) = Y ( e j ω 1 , e j ω 2 ) for ω 1 = a 1 , a 2 , , a M ω 2 = b 1 , b 2 , , b N ,
r ( m , n ) = x ( m , n ) * * x ( - m , - n ) = k = 0 M - 1 l = 0 N - 1 x ( k , l ) × ( m + k , n + l ) ,
r ( m , N - 1 ) = k = 0 M - 1 x ( k , 0 ) × ( m + k , N - 1 ) = x ( m , 0 ) * x ( - m , N - 1 )
r ( M - 1 , n ) = l = 0 N - 1 x ( 0 , l ) × ( M - 1 , n + l ) = x ( 0 , n ) * x ( M - 1 , n ) ,
x k ( m ) = x ( m , k )             for m = 0 , 1 , , M - 1
α ( m ) = x 0 ( m ) ,             β ( m ) = x N - 1 ( m ) ,
k = 0 M - 1 x N - 2 ( k ) α ( m + k ) + k = 1 M - 1 β ( k ) x 1 ( m + k ) = r N - 2 ( m )
x N - 2 ( m ) * α ( - m ) + β ( m ) * x 1 ( - m ) = r N - 2 ( m ) .
x n = [ x n ( 0 ) , x n ( 1 ) , , x n ( M - 1 ) ] ,
r n = [ r n ( 1 - M ) , r n ( 2 - M ) , , r n ( M - 1 ) ] .
[ A B ] [ x N - 2 x 1 ] = [ r N - 2 ] ,
[ 0 0 0 α 0 β 3 0 0 0 0 0 α 0 α 1 β 2 β 3 0 0 0 α 0 α 1 α 2 β 1 β 2 β 3 0 α 0 α 1 α 2 α 3 β 0 β 1 β 2 β 3 α 1 α 2 α 3 0 0 β 0 β 1 β 2 α 2 α 3 0 0 0 0 β 0 β 1 α 3 0 0 0 0 0 0 β 0 ] [ x 2 ( 0 ) x 2 ( 1 ) x 2 ( 2 ) x 2 ( 3 ) x 1 ( 0 ) x 1 ( 1 ) x 1 ( 2 ) x 1 ( 3 ) ] = [ r 2 ( - 3 ) r 2 ( - 2 ) r 2 ( - 1 ) r 2 ( 0 ) r 2 ( 1 ) r 2 ( 2 ) r 2 ( 3 ) ] .
r N - k ( m ) = x N - k ( m ) * α ( - m ) + β ( m ) * x k - 1 ( - m ) + l = 1 k - 2 x N - k + l ( m ) * x l ( - m ) ,
x N - k ( m ) * α ( - m ) + β ( m ) * x k - 1 ( - m ) = r ˜ N - k ( m ) ,
r ˜ N - k ( m ) = r N - k ( m ) - l = 1 k - 2 x N - k + l ( m ) * x l ( - m )
[ A B ] [ x N - k x k - 1 ] = [ r ˜ N - k ] ,
r ( M , 0 = x ( M , 0 ) x ( 0 , 0 ) , r ( 0 , N ) = x ( 0 , N ) x ( 0 , 0 ) , r ( M , - N ) = x ( 0 , N ) x ( M , 0 ) .
r ( M , N - n ) = x ( 0 , n )             for n = 0 , 1 , , N - 1
r ( M - m , N ) = x ( m , 0 )             for m = 0 , 1 , , M - 1.
r ( m 0 - m , N ) = x ( m , 0 )             for m = 0 , 1 , , M - 1.
r ( m 0 - m , N ) = x ( m , 0 )             for m = 0 , 1 , , M - 1
r ( M , n 0 - n ) = x ( 0 , n )             for n = 0 , 1 , , N - 1.