Abstract

Locations at which the Fourier transform F(u, υ) of an image equals zero have been called real-plane zeros, since they are the intersections of the zero curves of the analytic extension of F(u, υ) with the real–real (u, υ) plane. It has been shown that real-plane zero locations have a significant effect on the Fourier phase in that they are the end points of phase branch cuts, and it has been shown that real-plane zero locations can be estimated from Fourier magnitude data. Thus real-plane zeros can be utilized in phase retrieval algorithms to help constrain the possible Fourier phases. First we show a simplified procedure for estimating real-plane zeros from the Fourier magnitude. Then we present a new phase retrieval algorithm that uses real-plane zero locations to generate a simple parameterization of the Fourier phase and uses knowledge about the image to estimate the Fourier phase parameters. We show by example that this algorithm generates improved phase retrieval results when it is used as an initial guess into existing iterative algorithms. We assume that the image is real valued.

© 1994 Optical Society of America

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References

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  1. C. Y. C. Lui, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
    [CrossRef]
  2. J. Karle, “Recovering phase information from intensity data,” Science 232, 837–843 (1986).
    [CrossRef] [PubMed]
  3. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
    [CrossRef]
  4. E. J. Akutowicz, “On the determination of the phase of a Fourier integral,” Trans. Am. Math. Soc. 83, 179–192 (1956).
  5. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  6. H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase retrieval problem,” Opt. Let. 10, 250–251 (1985).
    [CrossRef]
  7. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  8. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  9. D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
    [CrossRef]
  10. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
    [CrossRef]
  11. N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).
    [CrossRef]
  12. H. Stark, ed., Image Recovery: Theory and Applications (Academic, San Diego, Calif., 1987).
  13. I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,” J. Math. Phys. 23, 2291–2298 (1982).
    [CrossRef]
  14. I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2159 (1985).
    [CrossRef]
  15. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [CrossRef]
  16. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. 3, 1897–1907 (1986).
    [CrossRef]
  17. 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]
  18. S. Weissbach, F. Wyrowski, O. Byngdahl, “Errordiffusion algorithm in phase synthesis and retrieval techniques,” Opt. Lett. 17, 235–237 (1992).
    [CrossRef] [PubMed]
  19. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]

1992

1991

1988

1987

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

1986

J. Karle, “Recovering phase information from intensity data,” Science 232, 837–843 (1986).
[CrossRef] [PubMed]

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

1985

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
[CrossRef]

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2159 (1985).
[CrossRef]

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase retrieval problem,” Opt. Let. 10, 250–251 (1985).
[CrossRef]

1982

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

1979

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1973

C. Y. C. Lui, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

1972

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

1956

E. J. Akutowicz, “On the determination of the phase of a Fourier integral,” Trans. Am. Math. Soc. 83, 179–192 (1956).

Akutowicz, E. J.

E. J. Akutowicz, “On the determination of the phase of a Fourier integral,” Trans. Am. Math. Soc. 83, 179–192 (1956).

Bates, R. H. T.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Bryngdahl, O.

Byngdahl, O.

Deighton, H. V.

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase retrieval problem,” Opt. Let. 10, 250–251 (1985).
[CrossRef]

Fiddy, M. A.

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase retrieval problem,” Opt. Let. 10, 250–251 (1985).
[CrossRef]

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
[CrossRef]

Fienup, J. R.

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

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

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

Gerchberg, R. W.

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

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).
[CrossRef]

Izraelevitz, D.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Karle, J.

J. Karle, “Recovering phase information from intensity data,” Science 232, 837–843 (1986).
[CrossRef] [PubMed]

Lane, R. G.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

Lim, J. S.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Lohmann, A. W.

C. Y. C. Lui, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Lui, C. Y. C.

C. Y. C. Lui, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

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]

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Saxton, W. O.

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

Scivier, M. S.

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
[CrossRef]

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase retrieval problem,” Opt. Let. 10, 250–251 (1985).
[CrossRef]

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Stefanescu, I. S.

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2159 (1985).
[CrossRef]

Wackerman, C. C.

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]

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

Weissbach, S.

Wyrowski, F.

Yagle, A. E.

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[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]

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2159 (1985).
[CrossRef]

J. Opt. Soc. Am.

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

J. Opt. Soc. Am. A

J. Phys. D

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Opt. Commun.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

C. Y. C. Lui, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Opt. Let.

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase retrieval problem,” Opt. Let. 10, 250–251 (1985).
[CrossRef]

Opt. Lett.

Optik

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

Science

J. Karle, “Recovering phase information from intensity data,” Science 232, 837–843 (1986).
[CrossRef] [PubMed]

Trans. Am. Math. Soc.

E. J. Akutowicz, “On the determination of the phase of a Fourier integral,” Trans. Am. Math. Soc. 83, 179–192 (1956).

Other

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Netherlands, 1989).
[CrossRef]

H. Stark, ed., Image Recovery: Theory and Applications (Academic, San Diego, Calif., 1987).

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Figures (7)

Fig. 1
Fig. 1

Zero curves of the Re[F(u, υ)] (dark) and the Im[F(u, υ)] (bright) superimposed upon locations that the Fourier magnitude indicates are real-plane zeros (white squares).

Fig. 2
Fig. 2

Examples of the best-possible reconstruction made with use of the linear zero phase (LZP) parameterization of the Fourier phase versus the original object.

Fig. 3
Fig. 3

Reconstructions made with use of various algorithms of the satellite object: comparison of traditional phase retrieval versus use of real-plane zeros.

Fig. 4
Fig. 4

Reconstructions made with use of various algorithms of the turbine object: comparison of traditional phase retrieval versus use of real-plane zeros.

Fig. 5
Fig. 5

Reconstructions made with use of various algorithms of the mandril object: comparison of traditional phase retrieval versus use of real-plane zeros.

Fig. 6
Fig. 6

Effect of noise in the Fourier modulus on the Fourier phase parameterization with use of real-plane zeros.

Fig. 7
Fig. 7

Effect of noise in the Fourier modulus on the traditional phase retrieval algorithm, for comparison with Fig. 6.

Tables (1)

Tables Icon

Table 1 Image Constraint Error Metric E0

Equations (16)

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F ( u , υ ) = | F ( u , υ ) | exp [ i p ( u , υ ) ] = f ( x , y ) exp [ i 2 π ( u x + υ y ) ] d x d y ,
| F ( u 1 , υ 1 ) | = max 1 δ u , δ υ 1 | F ( u 0 + δ u , υ 0 + δ υ ) | ,
| F ( u 0 , υ 0 ) | ½ ( | F ( u 1 , υ 1 ) | | F ( u 0 , υ 0 ) | ) .
T j ( u , υ ) = I j ( u , υ ) | F ( u , υ ) | exp [ i tan 1 ( υ υ j u u j ) ] ,
F j ( u , υ ) = exp ( i a j ) [ δ j T j ( u , υ ) ] + ( 1 δ j ) T j * ( u , υ ) ] ,
MSE = 1 N u υ | I j ( u , υ ) F ( u , υ ) F j ( u , υ ) | 2 ,
MSE = 2 u υ | F ( u , υ ) | 2 2 δ j Re [ exp ( i a j ) u υ F * ( u , υ ) T j ( u , υ ) ] 2 ( 1 δ j ) Re [ exp ( i a j ) u υ F * ( u , υ ) T j * ( u , υ ) ] ,
D 1 = u υ F ( u , υ ) T j * ( u , υ ) , D 2 = u υ F ( u , υ ) T j ( u , υ ) ,
NRMSE = { x y [ f ( x , y ) f ( x , y ) ] 2 x y f 2 ( x , y ) } 1 / 2 ,
F ( u , υ ) = j = 1 N z F j ( u , υ ) ,
f ( x , y ) = j = 1 N z { exp ( i a j ) [ δ j t j ( x , y ) + ( 1 δ j ) t j * ( x , y ) ] } .
E t = x y [ f ( x , y ) ] 2 , ( x , y ) S ,
E t = j j exp ( i a j ) exp ( i a j ) [ δ j δ j K 1 + δ j ( 1 δ j ) K 2 + ( 1 δ j ) δ j K 3 + ( 1 δ j ) ( 1 δ j ) K 4 ] ,
K 1 = x y t j ( x , y ) t j * ( x , y ) , K 2 = x y t j ( x , y ) t j ( x , y ) , K 3 = x y t j * ( x , y ) t j * ( x , y ) , K 4 = x y t j * ( x , y ) t j ( x , y )
E p 1 = x y [ f + ( x , y ) ] 2 , ( x , y ) S , E p 2 = x y [ f + ( x , y ) ] 2 x y [ f ( x , y ) ] 2 , ( x , y ) S ,
NRMSE = { u υ [ | F ( u , υ ) | 2 | F ( u , υ ) | 2 ] 2 u υ | F ( u , υ ) | 4 } 1 / 2 ,

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