Abstract

A method of image recovery using noniterative phase retrieval is proposed and investigated by simulation. This method adapts the Cauchy–Riemann equations to evaluate derivatives of phase based on derivatives of magnitude. The noise sensitivity of the approach is reduced by employing a least-mean-squares fit. This method uses the analytic properties of the Fourier transform of an object, the magnitude of which is measured with an intensity interferometer. The solution exhibits the degree of nonuniqueness expected from root-flipping arguments for the one-dimensional case, but a simple assumption that restricts translational ambiguity also restricts the space of solutions and permits essentially perfect reconstructions for a number of nonsymmetric one-dimensional objects of interest. Very good reconstructions are obtained for a large fraction of random objects, within an overall image flip, which may be acceptable in many applications. Results for the retrieved phase and recovered images are presented for some one-dimensional objects and for different noise levels. Extensions to objects of two dimensions are discussed. Requirements for signal-to-noise ratio are derived for intensity interferometry with use of the proposed processing.

© 2004 Optical Society of America

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References

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  1. 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).
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  3. H. B. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
    [CrossRef] [PubMed]
  4. D. Israelevitz, 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]
  5. 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]
  6. R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
    [CrossRef]
  7. P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampleddata,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
    [CrossRef]
  8. P. J. Bones, C. R. Parker, B. L. Satherley, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995).
    [CrossRef]
  9. D. C. Ghiglia, L. A. Romero, G. A. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024–1036 (1993).
    [CrossRef]
  10. R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
    [CrossRef]
  11. S. M. Ebstein, “High-light-level variance of estimators for intensity interferometry and fourth-order correlation interferometry,” J. Opt. Soc. Am. A 8, 1450–1456 (1991).
    [CrossRef]
  12. 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]
  13. K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), p. 231.
  14. See, for example, J. Mathews, R. L. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin, Reading, Mass., 1970), p. 477.
  15. Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 8, 304–308 (1979).
    [CrossRef]
  16. See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 76.
  17. A. J. Noushin, M. A. Fiddy, J. Graham-Eagle, “Some new findings on the zeros of band-limited functions,” J. Opt. Soc. Am. A 16, 1857–1863 (1999), and references therein.
    [CrossRef]
  18. T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Role of support information and zero locations in phase retrieval by a quadratic approach,” J. Opt. Soc. Am. A 16, 1845–1856 (1999).
    [CrossRef]
  19. R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (2002).
    [CrossRef]
  20. E. M. Vartiainen, K. Peiponen, T. Asakura, “Phase retrieval in optical spectroscopy: resolving optical constants from power spectra,” Appl. Spectrosc. 50, 1283–1289 (1996), and references therein.
    [CrossRef]
  21. W. C. Danchi, C. H. Townes, Space Sciences Laboratory, University of California, Berkeley, Calif. 94720 (personal communication, 1999).
  22. M. S. Belen’kii, “Enhanced backscatter and turbulence effects on pupil plane speckle imaging system,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds, Proc. SPIE4489, 66–75 (2001).
    [CrossRef]
  23. R. Holmes, S. Ma, A. Bhowmik, C. Greninger, “Analysis and simulation of a synthetic-aperture technique for imaging through a turbulent medium,” J. Opt. Soc. Am. A 13, 351–364 (1996).
    [CrossRef]

2002 (1)

1999 (2)

1996 (2)

1995 (1)

1993 (1)

1991 (1)

1987 (3)

D. Israelevitz, 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]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

1985 (2)

1982 (1)

1979 (1)

Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 8, 304–308 (1979).
[CrossRef]

1972 (1)

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 (1)

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
[CrossRef]

Asakura, T.

Atkinson, K. E.

K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), p. 231.

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]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

Belen’kii, M. S.

M. S. Belen’kii, “Enhanced backscatter and turbulence effects on pupil plane speckle imaging system,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds, Proc. SPIE4489, 66–75 (2001).
[CrossRef]

Bhowmik, A.

Bones, P. J.

Bruck, Yu. M.

Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 8, 304–308 (1979).
[CrossRef]

Chen, P.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampleddata,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
[CrossRef]

Danchi, W. C.

W. C. Danchi, C. H. Townes, Space Sciences Laboratory, University of California, Berkeley, Calif. 94720 (personal communication, 1999).

Deighton, H. B.

Ebstein, S. M.

Fairchild, P.

Fiddy, M. A.

Fienup, J. R.

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).

Ghiglia, D. C.

Graham-Eagle, J.

Greenaway, A. H.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampleddata,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
[CrossRef]

Greninger, C.

Hanbury Brown, R.

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
[CrossRef]

Holmes, R.

Hughes, K.

Isernia, T.

Israelevitz, D.

D. Israelevitz, 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]

Jackson, J. D.

See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 76.

Lane, R. G.

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (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]

Leone, G.

Lim, J. S.

D. Israelevitz, 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]

Ma, S.

Mastin, G. A.

Mathews, J.

See, for example, J. Mathews, R. L. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin, Reading, Mass., 1970), p. 477.

Noushin, A. J.

Parker, C. R.

Peiponen, K.

Pierri, R.

Romero, L. A.

Satherley, B. L.

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.

Smith, A.

Sodin, L. G.

Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 8, 304–308 (1979).
[CrossRef]

Soldovieri, F.

Spivey, B.

Townes, C. H.

W. C. Danchi, C. H. Townes, Space Sciences Laboratory, University of California, Berkeley, Calif. 94720 (personal communication, 1999).

Twiss, R. Q.

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
[CrossRef]

Vartiainen, E. M.

Walker, R. L.

See, for example, J. Mathews, R. L. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin, Reading, Mass., 1970), p. 477.

Wang, Y.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampleddata,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
[CrossRef]

Watson, R. W.

Appl. Opt. (1)

Appl. Spectrosc. (1)

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

D. Israelevitz, 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. Opt. Soc. Am. A (9)

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

P. J. Bones, C. R. Parker, B. L. Satherley, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995).
[CrossRef]

D. C. Ghiglia, L. A. Romero, G. A. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024–1036 (1993).
[CrossRef]

S. M. Ebstein, “High-light-level variance of estimators for intensity interferometry and fourth-order correlation interferometry,” J. Opt. Soc. Am. A 8, 1450–1456 (1991).
[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]

A. J. Noushin, M. A. Fiddy, J. Graham-Eagle, “Some new findings on the zeros of band-limited functions,” J. Opt. Soc. Am. A 16, 1857–1863 (1999), and references therein.
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Role of support information and zero locations in phase retrieval by a quadratic approach,” J. Opt. Soc. Am. A 16, 1845–1856 (1999).
[CrossRef]

R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (2002).
[CrossRef]

R. Holmes, S. Ma, A. Bhowmik, C. Greninger, “Analysis and simulation of a synthetic-aperture technique for imaging through a turbulent medium,” J. Opt. Soc. Am. A 13, 351–364 (1996).
[CrossRef]

Nature (London) (1)

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
[CrossRef]

Opt. Commun. (1)

Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 8, 304–308 (1979).
[CrossRef]

Opt. Lett. (1)

Optik (1)

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).

Other (6)

W. C. Danchi, C. H. Townes, Space Sciences Laboratory, University of California, Berkeley, Calif. 94720 (personal communication, 1999).

M. S. Belen’kii, “Enhanced backscatter and turbulence effects on pupil plane speckle imaging system,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds, Proc. SPIE4489, 66–75 (2001).
[CrossRef]

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampleddata,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
[CrossRef]

See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 76.

K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), p. 231.

See, for example, J. Mathews, R. L. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin, Reading, Mass., 1970), p. 477.

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Figures (7)

Fig. 1
Fig. 1

Schematic of an intensity interferometer.

Fig. 2
Fig. 2

Path of calculation in the complex (ξ, ψ) plane.

Fig. 3
Fig. 3

Relationship between log-amplitude and phase differences in the (ξ, ψ) plane.

Fig. 4
Fig. 4

Flattop object and reconstruction: (a) true (black) and reconstructed (gray) flattop object, (b) magnitude of the Fourier transform of the object, (c) true (black) and reconstructed (gray) phase of the Fourier transform of the object.

Fig. 5
Fig. 5

Sample asymmetric object and its reconstruction: (a) true (black) and reconstructed (gray) asymmetric object, (b) magnitude of the Fourier transform of the object, (c) true (black) and reconstructed (gray) phase of the Fourier transform of the object.

Fig. 6
Fig. 6

Examples of random objects and their reconstructions with the normalized correlation metric: random object (black) and reconstruction (gray) with a normalized image correlation of (a) 0.9994, (b) 0.9594, (c) 0.9118, (d) 0.8466.

Fig. 7
Fig. 7

SNR versus integration time, assuming the parameter values specified in the text.

Tables (1)

Tables Icon

Table 1 Normalized Correlation of Object and Reconstruction versus SNR and Sampling for Random Objects

Equations (24)

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M ( x 1 ,   x 2 ) = c { | I ( k = 0 ) | 2 + | I [ k = k 0 ( x 1 - x 2 ) ] | 2 } ,
I ( ( m ,   n ) k 0 Δ x )
= O ( θ x ,   θ y ) exp [ ik 0 Δ x ( m θ x + n θ y ) ] d θ x d θ y = Δ θ x Δ θ y j , k O ( j Δ θ ,   k Δ θ ) z x j z y k + ( m ,   n ) ,
I [ z x ( m Δ x ) ,   z y ( n Δ x ) ]
j , k O ( j Δ θ ,   k Δ θ ) z x j z y k = { I [ ( m ,   n ) k 0 Δ x ] - ( m ,   n ) } / Δ θ 2 .
| ( m ,   n ) | ( Δ θ 2 / 24 ) Δ θ obj 2 { max [ | 2 G ( θ ,   m ,   n ) / θ x 2 | ] + max [ | 2 G ( θ ,   m ,   n ) / θ y 2 | ] } ,
[ R   exp ( i Φ ) ] / ξ = [ R   exp ( i Φ ) ] / ( i ψ ) .
( R / ξ + Ri Φ / ξ ) exp ( i Φ )
= ( - i R / ψ + R Φ / ψ ) exp ( i Φ ) ,
Φ / ψ =   ln ( R ) / ξ s / ξ ,
Φ / ξ = -   ln ( R ) / ψ - s / ψ ,
Δ Φ = ( Δ s / Δ ξ ) Δ ψ ,
Δ Φ = ( - Δ s / Δ ψ ) Δ ξ .
Δ ξ = Δ   cos ( mk 0 Δ x Δ θ ) , m = 1 , , N - 1 ,
Δ ψ = Δ   sin ( mk 0 Δ x Δ θ ) , m = 1 , , N - 1 .
Δ s = s / ξ Δ ξ + s / ψ Δ ψ + O ( Δ ξ 2 ,   Δ ψ 2 ) ,
( s / ξ ) Δ ξ + ( s / ψ ) Δ ψ = Φ / ξ ( - Δ ψ ) + Φ / ψ Δ ξ Δ Φ .
2 Φ / ξ 2 + 2 Φ / ψ 2 = 0 .
Φ ( ρ ,   ϕ ) = a 0 + b 0 ϕ + j = 1 , , N ρ j [ ( a j cos ( j ϕ ) + b j sin ( j ϕ ) ] + B 0 ln   ρ + j = 1 , , N ρ - j [ ( a - j cos ( j ϕ ) + b - j sin ( j ϕ ) ] ,
min m = 1 , , N j = 1 , , N { [ 1 - 2   sin ( k 0 Δ x Δ θ / 2 ) ] j - 1 }
× { a j cos [ j ϕ ( m ) ]
    + b j   sin [ j ϕ ( m ) ] } - Δ Φ ( m ) 2 .
C I = max | Σ I e ( x + Δ x ) I t ( x ) | 2 / Σ | I e ( x ) | 2 Σ | I t ( x ) | 2 ,
SNR = J 1 J 2 | I ( k ) / I ( k = 0 ) | 2 ( Δ ν e / Δ ν o ) / { var [ ( J 1 + B 1 + D 1 ) ( J 2 + B 2 + D 2 ) ] } 1 / 2 ,

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