Abstract

Is it possible to determine a function with a finite support from the modulus of its Fourier transform? This problem, the so-called phase problem, is studied theoretically and numerically. It is shown theoretically that, at least for a wide class of functions, such determination is not possible. The theory developed in this Letter is essentially two dimensional. Examples are given and studied numerically.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. E. Burge et al., “The phase problem,” Proc. R. Soc. London A350, 1911976).
  2. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27 (1979).
    [CrossRef]
  3. Y. M. Burck, L. G. Sodin, “On the ambiguity of the phase reconstruction problem,” Opt. Commun. 30, 304 (1979).
    [CrossRef]
  4. W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973), p. 181.
  5. A. H. Greenaway, “Proposal for phase recovery from a single intensity distribution,” Opt. Lett. 1, 10 (1977). In this Letter the case is considered in which S consists of two disconnected domains.
    [CrossRef] [PubMed]
  6. R. E. A. C. Paley, N. Wiener, “Fourier transforms in the complex domain,” Am. Math. Soc. Colloquium Proc. 29 (1934), p. 12.

1979 (2)

Y. M. Burck, L. G. Sodin, “On the ambiguity of the phase reconstruction problem,” Opt. Commun. 30, 304 (1979).
[CrossRef]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27 (1979).
[CrossRef]

1977 (1)

1976 (1)

R. E. Burge et al., “The phase problem,” Proc. R. Soc. London A350, 1911976).

1934 (1)

R. E. A. C. Paley, N. Wiener, “Fourier transforms in the complex domain,” Am. Math. Soc. Colloquium Proc. 29 (1934), p. 12.

Burck, Y. M.

Y. M. Burck, L. G. Sodin, “On the ambiguity of the phase reconstruction problem,” Opt. Commun. 30, 304 (1979).
[CrossRef]

Burge, R. E.

R. E. Burge et al., “The phase problem,” Proc. R. Soc. London A350, 1911976).

Fienup, J. R.

Greenaway, A. H.

Paley, R. E. A. C.

R. E. A. C. Paley, N. Wiener, “Fourier transforms in the complex domain,” Am. Math. Soc. Colloquium Proc. 29 (1934), p. 12.

Rudin, W.

W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973), p. 181.

Sodin, L. G.

Y. M. Burck, L. G. Sodin, “On the ambiguity of the phase reconstruction problem,” Opt. Commun. 30, 304 (1979).
[CrossRef]

Wiener, N.

R. E. A. C. Paley, N. Wiener, “Fourier transforms in the complex domain,” Am. Math. Soc. Colloquium Proc. 29 (1934), p. 12.

Am. Math. Soc. Colloquium Proc. (1)

R. E. A. C. Paley, N. Wiener, “Fourier transforms in the complex domain,” Am. Math. Soc. Colloquium Proc. 29 (1934), p. 12.

Opt. Commun. (1)

Y. M. Burck, L. G. Sodin, “On the ambiguity of the phase reconstruction problem,” Opt. Commun. 30, 304 (1979).
[CrossRef]

Opt. Lett. (2)

Proc. R. Soc. London (1)

R. E. Burge et al., “The phase problem,” Proc. R. Soc. London A350, 1911976).

Other (1)

W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973), p. 181.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) The original real and positive object: exp(−ξ). (b) The object obtained by flipping zero ν1 according to Eq. (15). The resulting object is real.

Fig. 2
Fig. 2

(a) The object obtained by flipping zero ν5. The result is positive. (b) As in (a), where νn is ν6.

Fig. 3
Fig. 3

(a) Both zero ν5 and ν6 have been flipped. (b) The modulus of the Fourier transforms of the objects displayed in Figs. 1(a)3(a).

Fig. 4
Fig. 4

(a) The result obtained by replacing a zero (ν6) by a different polynomial of degree 2. (b) The modulus of the Fourier transform of (a).

Equations (17)

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

h ( x , y ) = S d ξ d η g ( ξ , η ) exp ( i x ξ + i y η ) ,
h ( x , y ) = p ( x , y ) f ( x , y ) ,
h ˜ ( x , y ) = q ( x , y ) f ( x , y ) ,
h ( x , y ) = S d ξ d η g ( ξ η ) exp ( i x ξ + i y η ) .
β χ = max ( ξ ) for ( ξ , η ) S
α χ = min ( ξ ) for ( ξ , η ) S .
g y ( ξ ) = d η g ( ξ , η ) exp ( i y η )
h y ( x ) = α β d ξ g y ( ξ ) exp ( i x ξ ) .
h ˜ y ( x ) = h ˜ y ( x ) = h ˜ ( x , y ) .
h ˜ y ( x ) = α β d ξ g ˜ y ( ξ ) exp ( i x ξ ) .
g ˜ ( ξ , η ) d x d y h ˜ y ( x ) ( x ) exp ( i x ξ i y η )
g ˜ ( ξ , η ) = d y g ˜ y ( ξ ) exp ( i y η )
= 0 for ξ < α and ξ > β .
g ˜ ( ξ , η ) = d x d y h ˜ ( x , y ) exp ( i x ξ i y η ) ,
h ( x , y ) = n = 1 ( 1 R 2 / ν n 2 ) ,
h ˜ n ( x , y ) = { [ ν n 2 ( x i ) 2 y 2 ] / [ ν n 2 R ] } × h ( x , y )
h ˜ ( x , y ) = ν 5 2 [ ( x i ) 2 + y 2 ] ν 5 2 [ ( x + i ) 2 + y 2 ] × ν 6 2 [ ( x i ) 2 + y 2 ] ν 6 2 [ ( x + i ) 2 + y 2 ] h ( x , y ) .

Metrics