Abstract

The iterative blind deconvolution algorithm proposed by Ayers and Dainty [ Opt. Lett. 13, 547 ( 1988)] and improved on by Davey et al. [ Opt. Commun. 69, 353 ( 1989)] is applied to the problem of phase retrieval, which is a special case of the blind deconvolution problem. A close relationship between this algorithm and the error-reduction version of the iterative Fourier-transform phase-retrieval algorithm is shown analytically. The performance of the blind deconvolution algorithm is compared with the error-reduction and hybrid input–output versions of the iterative Fourier-transform algorithm by reconstruction experiments on real-valued, nonnegative images with and without noise.

© 1990 Optical Society of America

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References

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  1. G. R. Ayers, J. C. Dainty, “An iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [Crossref]
  2. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref]
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref]
  4. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [Crossref]
  5. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
    [Crossref]
  6. B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).In Eq. (15) of that paper, by our logic, the term α/|Fi−1(u)|2in the denominator should be α/|Hi−1(u)|2.
    [Crossref]
  7. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [Crossref]
  8. T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 1–13 (1990).
    [Crossref]
  9. J. H. Seldin, J. R. Fienup, “Phase retrieval using Ayers/Dainty deconvolution,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 124–127.

1990 (1)

T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 1–13 (1990).
[Crossref]

1989 (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).In Eq. (15) of that paper, by our logic, the term α/|Fi−1(u)|2in the denominator should be α/|Hi−1(u)|2.
[Crossref]

1988 (1)

1987 (1)

1986 (1)

1982 (1)

1978 (1)

1967 (1)

Ayers, G. R.

Bates, R. H. T.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).In Eq. (15) of that paper, by our logic, the term α/|Fi−1(u)|2in the denominator should be α/|Hi−1(u)|2.
[Crossref]

Crimmins, T. R.

T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 1–13 (1990).
[Crossref]

Dainty, J. C.

Davey, B. L. K.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).In Eq. (15) of that paper, by our logic, the term α/|Fi−1(u)|2in the denominator should be α/|Hi−1(u)|2.
[Crossref]

Fienup, J. R.

T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 1–13 (1990).
[Crossref]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[Crossref]

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

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

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[Crossref]

J. H. Seldin, J. R. Fienup, “Phase retrieval using Ayers/Dainty deconvolution,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 124–127.

Helstrom, C. W.

Lane, R. G.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).In Eq. (15) of that paper, by our logic, the term α/|Fi−1(u)|2in the denominator should be α/|Hi−1(u)|2.
[Crossref]

Seldin, J. H.

J. H. Seldin, J. R. Fienup, “Phase retrieval using Ayers/Dainty deconvolution,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 124–127.

Thelen, B. J.

T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 1–13 (1990).
[Crossref]

Wackerman, C. C.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).In Eq. (15) of that paper, by our logic, the term α/|Fi−1(u)|2in the denominator should be α/|Hi−1(u)|2.
[Crossref]

Opt. Lett. (2)

Other (1)

J. H. Seldin, J. R. Fienup, “Phase retrieval using Ayers/Dainty deconvolution,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 124–127.

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Figures (10)

Fig. 1
Fig. 1

AD blind deconvolution algorithm.

Fig. 2
Fig. 2

AD blind deconvolution algorithm applied to phase retrieval.

Fig. 3
Fig. 3

Comparison of phase-retrieval using AD blind deconvolution with the HIO and ER iterative transform algorithms for a real-valued, nonnegative object with known support and no Fourier modulus error. Reconstructed images: (A) HIO/ER (indistinguishable from the original object); (B) ER; (C) AD with the Fourier constraint of Eq. (14); (D) AD with the Fourier constraint of Eq. (15).

Fig. 4
Fig. 4

ABSERR versus iteration number for the reconstructions of Fig. 3.

Fig. 5
Fig. 5

Comparison of the effect of the pre-Wiener filtering of noisy Fourier intensity data on reconstructions with the ER algorithm. Reconstructed images after 1000 iterations: (A) 5% FME, no pre-Wiener filtering; (B) 5% FME, pre-Wiener filtering; (C) 20% FME, no pre-Wiener filtering; (D) 20% FME, pre-Wiener filtering.

Fig. 6
Fig. 6

Comparison of phase retrieval using AD, HIO, and ER for a real-valued, nonnegative object with known support and 5% FME. Reconstructed images: (A) HIO/ER, (B) ER, (C) AD with the Fourier constraint of Eq. (14), (D) AD with the Fourier constraint of Eq. (15).

Fig. 7
Fig. 7

Comparison of phase retrieval using AD, HIO, and ER for a real-valued, nonnegative object with known support and 20% FME. Reconstructed images: (A) HIO/ER, (B) ER, (C) AD with the Fourier constraint of Eq. (14), (D) AD with the Fourier constraint of Eq. (15).

Fig. 8
Fig. 8

ABSERR versus iteration number for the reconstructions of Fig. 7.

Fig. 9
Fig. 9

Comparison of phase retrieval using AD, HIO, and ER for a real-valued, nonnegative object with unknown support and no FME. (A) Object. Reconstructed images: (B) HIO/ER, (C) ER, (D) AD with the Fourier constraint of Eq. (14), (E) AD with the Fourier constraint of Eq. (15).

Fig. 10
Fig. 10

ABSERR versus iteration number for the reconstructions of Fig. 9.

Equations (23)

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c ( x ¯ ) = f ( x ¯ ) g ( x ¯ x ) d x + n ( x ¯ ) = f ( x ¯ ) * g ( x ¯ ) + n ( x ¯ ) ,
C ( ū ) = F ( ū ) G ( ū ) + N ( ū ) ,
F ( ū ) = | F ( ū ) | exp [ i ψ ( ū ) ] = [ f ( x ¯ ) ] = f ( x ¯ ) exp [ i 2 π ( ū · x ) ] d x .
r ( x ¯ ) = f ( x ¯ ) f * ( x ¯ x ) d x = 1 [ F ( ū ) F * ( ū ) ] = 1 [ | F ( ū ) | 2 ] .
F k ( ū ) = F k ( ū ) ;
F k ( ū ) = ( 1 β ) F k ( ū ) + β C ( ū ) G k ( ū ) ;
1 F k ( ū ) = 1 β F k ( ū ) + β G k ( ū ) C ( ū ) .
c ( x ¯ ) = s ( x ¯ ) * f ( x ¯ ) + n ( x ¯ ) ,
C ( ū ) = S ( ū ) F ( ū ) + N ( ū ) ,
F ̂ ( ū ) = W ( ū ) C ( ū ) ,
W ( ū ) = S * ( ū ) | S ( ū ) | 2 + | N ( ū ) | 2 / | F ( ū ) | 2 ,
F k ( ū ) = G k * ( ū ) | G k ( ū ) | 2 + σ 2 / | F k ( ū ) | 2 C ( ū ) ,
G k ( ū ) = F k * ( u ) | F k ( ū ) | 2 + σ 2 / | G k 1 ( ū ) | 2 C ( ū ) .
F k ( ū ) = G k * ( ū ) | G k ( ū ) | 2 + α C ( ū ) .
G k ( ū ) = F k * ( ū ) | F k ( ū ) | 2 + α C ( ū ) .
F k ( ū ) = G k * ( ū ) = F k ( ū ) | F k ( ū ) | 2 + α | F ( ū ) | 2 .
F k ( ū ) = F k ( ū ) | F k ( ū ) | 2 + σ 2 / | F k ( ū ) | 2 | F ( ū ) | 2 .
F k ( ū ) = | F ( ū ) | exp [ i Φ k ( ū ) ] = F k ( ū ) | F ( ū ) | | F k ( ū ) | .
F k ( ū ) = F k ( ū ) | F ( ū ) | 2 | F k ( ū ) | 2 .
ABSERR [ x ¯ | α f k ( x ¯ x ¯ 0 ) f ( x ¯ ) | 2 x ¯ | f ( x ¯ ) | 2 ] 1 / 2 ,
α = x ¯ f ( x ¯ ) f k * ( x ¯ x ¯ 0 ) x ¯ | f k ( x ) | 2
FME { ū [ | F ( ū ) | n | F ( ū ) | ] 2 ū | F ( ū ) | 2 } 1 / 2 .
| F ̂ ( ū ) | = [ 1 1 + σ 2 / | F ( ū ) | n 4 | F ( ū ) | n 2 ] 1 / 2 ,

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