Abstract

A signal recovery technique is motivated and derived for the recovery of several nonnegative signals from measurements of their autocorrelation and cross-correlation functions. The iterative technique is shown to preserve nonnegativity of the signal estimates and to produce a sequence of estimates whose correlations better approximate the measured correlations as the iterations proceed. The method is demonstrated on simulated data for active imaging with dual-frequency or dual-polarization illumination.

© 2005 Optical Society of America

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  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  2. T. J. Schulz, D. L. Snyder, “Image recovery from correlations,” J. Opt. Soc. Am. A 9, 1266–1272 (1992).
    [Crossref]
  3. G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its application,” Opt. Lett. 13, 547–549 (1988).
    [Crossref]
  4. T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
    [Crossref] [PubMed]
  5. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [Crossref]
  6. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  7. T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Image reconstruction from Fourier transform magnitude with applications to synthetic aperture radar imaging,” J. Opt. Soc. Am. A 13, 922–934 (1996).
    [Crossref]
  8. B. R. Hunt, T. L. Overman, P. Gough, “Image reconstruction from pairs of Fourier-transform magnitude,” Opt. Lett. 23, 1123–1125 (1998).
    [Crossref]
  9. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  10. R. B. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imagingtechnique utilizing two speckle fields: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (2002).
    [Crossref]
  11. R. B. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458–471 (2002).
    [Crossref]
  12. D. G. Voelz, J. F. Belsher, L. Ulibarri, V. Gamiz, “Ground-to-space laser imaging: review 2001,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 35–47 (2002).
    [Crossref]
  13. J. R. Fienup, P. S. Idell, “Imaging correlography with sparse arrays of detectors,” Opt. Eng. 27, 778–784 (1988).
    [Crossref]
  14. D. G. Voelz, J. D. Gonglewski, P. S. Idell, “Image synthesis from nonimaged laser-speckle patterns: comparison of theory, computer simulation, and laboratory results,” Appl. Opt. 30, 3333–3344 (1991).
    [Crossref] [PubMed]
  15. L. K. Jones, C. L. Byrne, “General entropy criteria for inverse problems, with applications to data compression, pattern classification, and cluster analysis,” IEEE Trans. Inf. Theory 36, 23–30 (1990).
    [Crossref]
  16. I. Csiszar, “Why least squares and maximum entropy?—An axiomatic approach to inverse problems,” Ann. Stat. 19, 2033–2066 (1991).
    [Crossref]
  17. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B Methodol. 39, 1–37 (1977) (with discussion).
  18. G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions (Wiley, New York, 1997).
  19. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Heidelberg, Germany, 1984).
  20. J. M. Geary, Introduction to Wavefront Sensors, Vol. TT 18 of Tutorial Texts in Optical Engineering (SPIE, Bellingham, Wash., 1995).
  21. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  22. D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).

2002 (2)

1998 (1)

1996 (1)

1993 (1)

1992 (2)

1991 (2)

1990 (1)

L. K. Jones, C. L. Byrne, “General entropy criteria for inverse problems, with applications to data compression, pattern classification, and cluster analysis,” IEEE Trans. Inf. Theory 36, 23–30 (1990).
[Crossref]

1988 (2)

J. R. Fienup, P. S. Idell, “Imaging correlography with sparse arrays of detectors,” Opt. Eng. 27, 778–784 (1988).
[Crossref]

G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its application,” Opt. Lett. 13, 547–549 (1988).
[Crossref]

1982 (1)

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B Methodol. 39, 1–37 (1977) (with discussion).

Ayers, G. R.

Belsher, J. F.

D. G. Voelz, J. F. Belsher, L. Ulibarri, V. Gamiz, “Ground-to-space laser imaging: review 2001,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 35–47 (2002).
[Crossref]

Byrne, C. L.

L. K. Jones, C. L. Byrne, “General entropy criteria for inverse problems, with applications to data compression, pattern classification, and cluster analysis,” IEEE Trans. Inf. Theory 36, 23–30 (1990).
[Crossref]

Csiszar, I.

I. Csiszar, “Why least squares and maximum entropy?—An axiomatic approach to inverse problems,” Ann. Stat. 19, 2033–2066 (1991).
[Crossref]

Dainty, J. C.

G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its application,” Opt. Lett. 13, 547–549 (1988).
[Crossref]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B Methodol. 39, 1–37 (1977) (with discussion).

Fairchild, P.

Fienup, J. R.

J. R. Fienup, P. S. Idell, “Imaging correlography with sparse arrays of detectors,” Opt. Eng. 27, 778–784 (1988).
[Crossref]

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

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Gamiz, V.

D. G. Voelz, J. F. Belsher, L. Ulibarri, V. Gamiz, “Ground-to-space laser imaging: review 2001,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 35–47 (2002).
[Crossref]

Geary, J. M.

J. M. Geary, Introduction to Wavefront Sensors, Vol. TT 18 of Tutorial Texts in Optical Engineering (SPIE, Bellingham, Wash., 1995).

Gonglewski, J. D.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Heidelberg, Germany, 1984).

Gough, P.

Holmes, R. B.

Holmes, T. J.

Hughes, K.

Hunt, B. R.

Idell, P. S.

Isernia, T.

Jones, L. K.

L. K. Jones, C. L. Byrne, “General entropy criteria for inverse problems, with applications to data compression, pattern classification, and cluster analysis,” IEEE Trans. Inf. Theory 36, 23–30 (1990).
[Crossref]

Krishnan, T.

G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions (Wiley, New York, 1997).

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B Methodol. 39, 1–37 (1977) (with discussion).

McLachlan, G. J.

G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions (Wiley, New York, 1997).

Miller, M. I.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).

Overman, T. L.

Pascazio, V.

Pierri, R.

Roggemann, M. C.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B Methodol. 39, 1–37 (1977) (with discussion).

Schirinzi, G.

Schulz, T. J.

Smith, A.

Snyder, D. L.

T. J. Schulz, D. L. Snyder, “Image recovery from correlations,” J. Opt. Soc. Am. A 9, 1266–1272 (1992).
[Crossref]

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).

Spivey, B.

Ulibarri, L.

D. G. Voelz, J. F. Belsher, L. Ulibarri, V. Gamiz, “Ground-to-space laser imaging: review 2001,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 35–47 (2002).
[Crossref]

Voelz, D. G.

D. G. Voelz, J. D. Gonglewski, P. S. Idell, “Image synthesis from nonimaged laser-speckle patterns: comparison of theory, computer simulation, and laboratory results,” Appl. Opt. 30, 3333–3344 (1991).
[Crossref] [PubMed]

D. G. Voelz, J. F. Belsher, L. Ulibarri, V. Gamiz, “Ground-to-space laser imaging: review 2001,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 35–47 (2002).
[Crossref]

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Ann. Stat. (1)

I. Csiszar, “Why least squares and maximum entropy?—An axiomatic approach to inverse problems,” Ann. Stat. 19, 2033–2066 (1991).
[Crossref]

Appl. Opt. (2)

IEEE Trans. Inf. Theory (1)

L. K. Jones, C. L. Byrne, “General entropy criteria for inverse problems, with applications to data compression, pattern classification, and cluster analysis,” IEEE Trans. Inf. Theory 36, 23–30 (1990).
[Crossref]

J. Opt. Soc. Am. A (6)

J. R. Stat. Soc. Ser. B Methodol. (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B Methodol. 39, 1–37 (1977) (with discussion).

Opt. Eng. (1)

J. R. Fienup, P. S. Idell, “Imaging correlography with sparse arrays of detectors,” Opt. Eng. 27, 778–784 (1988).
[Crossref]

Opt. Lett. (2)

Other (8)

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

D. G. Voelz, J. F. Belsher, L. Ulibarri, V. Gamiz, “Ground-to-space laser imaging: review 2001,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 35–47 (2002).
[Crossref]

G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions (Wiley, New York, 1997).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Heidelberg, Germany, 1984).

J. M. Geary, Introduction to Wavefront Sensors, Vol. TT 18 of Tutorial Texts in Optical Engineering (SPIE, Bellingham, Wash., 1995).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).

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Figures (4)

Fig. 1
Fig. 1

Incoherent reflectivity for (a) horizontal, o 1 , and (b) vertical, o 2 , polarizations.

Fig. 2
Fig. 2

(a) Autocorrelation for horizontal polarization reflectivity, r 11 , (b) autocorrelation for vertical polarization reflectivity, r 22 , and (c) cross correlation for horizontal and vertical polarization reflectivities, r 12 .

Fig. 3
Fig. 3

Reflectance estimates for noise-free data: (a) horizontal polarization reflectance, (b) estimate of horizontal reflectance, (c) vertical polarization reflectance, and (d) estimate of vertical reflectance.

Fig. 4
Fig. 4

Reflectance estimates for 700 speckle realizations: (a) horizontal polarization reflectance, (b) estimate of horizontal reflectance, (c) vertical polarization reflectance, and (d) estimate of vertical reflectance.

Equations (54)

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r nm ( y ) = x o n ( x ) o m ( x + y ) , n = 1 ,   2 , ,   M ;
m = 1 ,   2 , ,   M .
μ nm ( y ) = z h nm ( y - z ) r nm ( z ) = z h nm ( y - z ) x o n ( x ) o m ( x + z ) ,
d nm ( y ) = N { μ nm ( y ) + b nm ( y ) } , ( n ,   m ) M ,
o ˆ = arg   min o 0   D ( o ) ,
D ( o ) = n = 1 M m = 1 M a nm y d nm ( y ) ln   d nm ( y ) μ nm ( y ) + b nm ( y ) + y [ μ nm ( y ) + b nm ( y ) - d nm ( y ) ] ,
a nm = 1 ( n ,   m ) M 0 ( n ,   m ) M
D ( o new ) D ( o old ) .
F 1 ( u ;   t ) = f 1 ( x ;   t ) exp j   2 π λ d   u x d x ,
E [ f 1 ( x ;   t ) ] = 0 ,
E [ f 1 ( x ;   t ) f 1 * ( y ;   t ) ] = o 1 ( x ) δ ( x - y ) ,
E [ F 1 ( u ;   t ) F 1 * ( v ;   t ) ] = O 1 u - v λ d ,
O 1 ( ν ) = o 1 ( x ) exp ( j 2 π ν x ) d x .
A ( u ) = 1 inside of aperture 0 outside of aperture .
d 11 ( y ;   t ) = | G 1 ( u ;   t ) | 2   exp ( j 2 π u y ) d u = A 2 ( u ) | F 1 ( u ;   t ) | 2   exp ( j 2 π u y ) d u .
| d 11 ( y ;   t ) | 2 E [ | d 11 ( y ;   t ) | 2 ] = A 2 ( u ) A 2 ( v ) ×   E [ | F 1 ( u ;   t ) | 2 | F 1 ( v ;   t ) | 2 ] × exp [ j 2 π y ( u - v ) ] d u d v = A 2 ( u ) A 2 ( v ) × O 1 2 ( 0 ) + O 1 u - v λ d 2 × exp [ j 2 π y ( u - v ) ] d u d v = O 1 2 ( 0 ) A 2 ( u ) exp ( j 2 π u y ) d u 2 + A 2 ( w + v ) A 2 ( v ) d v × O 1 w λ d 2   exp ( j 2 π w y ) d w = O 1 2 ( 0 ) | r a ( y ) | 2 + ( λ d ) 2 | r a ( y ) | 2   *   r 11 ( λ dy ) ,
r a ( y ) = A 2 ( u ) exp ( j 2 π u y ) d u .
| d 22 ( y ;   t ) | 2 O 2 2 ( 0 ) | r a ( y ) | 2 + ( λ d ) 2 | r a ( y ) | 2   *   r 22 ( λ dy ) ,
d 22 ( y ;   t ) = | G 2 ( u ;   t ) | 2   exp ( j 2 π u y ) d u = A 2 ( u ) | F 2 ( u ;   t ) | 2   exp ( j 2 π u y ) d u .
d 12 ( y ;   t ) = G 1 ( u ;   t ) G 2 * ( u ;   t ) exp ( j 2 π u y ) d u = A 2 ( u ) F 1 ( u ;   t ) F 2 * ( u ;   t ) exp ( j 2 π u y ) d u .
| d 12 ( y ;   t ) | 2 E [ | d 12 ( y ;   t ) | 2 ] = A 2 ( u ) A 2 ( v ) × E [ F 1 ( u ;   t ) F 2 * ( u ;   t ) F 1 * ( v ;   t ) F 2 ( v ;   t ) ] × exp [ j 2 π y ( u - v ) ] d u d v = A 2 ( u ) A 2 ( v ) O 1 u - v λ d O 2 * u - v λ d × exp [ j 2 π y ( u - v ) ] d u d v = A 2 ( w + v ) A 2 ( v ) d v × O 1 w λ d O 2 * w λ d exp ( j 2 π w y ) d w = ( λ d ) 2 | r a ( y ) | 2   *   o 1 ( λ dy + x ) o 2 ( x ) d x = ( λ d ) 2 | r a ( y ) | 2   *   r 21 ( λ dy ) ,
f ( x ) = o ( x ) 2 1 / 2 g ( x ) ,
E [ g ( x ) g * ( y ) ] = 1 x = y 0 x y .
Δ u = λ d N Δ x ,
L ( o ) = nm a nm y d nm ( y ) ln [ μ nm ( y ) + b nm ( y ) ] - y [ μ nm ( y ) + b nm ( y ) ] = - D ( o ) + nm a nm y { d nm ( y ) ln   d nm ( y ) - d nm ( y ) } = - D ( o ) + O . T . ,
L ( o new ) L ( o old )
D ( o new ) D ( o old ) .
E [ d ˜ nm ( x ,   y ,   z ) ] = h nm ( y - z ) o n ( x ) o m ( x + z ) ,
E [ b ˜ nm ( y ) ] = b nm ( y ) .
d nm ( y ) = x z d ˜ nm ( x ,   y ,   z ) + b ˜ nm ( y ) ,
E [ d nm ( y ) ] = x z E [ d ˜ nm ( x ,   y ,   z ) ] + E [ b ˜ nm ( y ) ] = x z h nm ( y - z ) o n ( x ) o m ( x + z ) + b nm ( y ) = μ nm ( y ) + b nm ( y ) .
L cd ( o ) = n , m a nm x , y , z { d ˜ nm ( x ,   y ,   z ) × ln [ h nm ( y - z ) o n ( x ) o m ( x + z ) ] - h nm ( y - z ) o n ( x ) o m ( x + z ) } .
Q ( o ;   o old ) = E [ L cd ( o ) | d ,   o = o old ] ,
Q ( o ;   o old ) = n , m a nm x , y , z { E [ d ˜ nm ( x ,   y ,   z ) | d ,   o old ] × ln [ h nm ( y - z ) o n ( x ) o m ( x + z ) ] - h nm ( y - z ) o n ( x ) o m ( x + z ) } .
o new = arg   max o O   Q ( o ;   o old ) ,
Q ( o ;   o old ) o n ( x ) = m   a nm o n ( x )   y , z { E [ d ˜ nm ( x ,   y ,   z ) | d ,   o old ] + E [ d ˜ mn ( x - z ,   y ,   z ) | d ,   o old ] } - m ( a nm H nm + a mn H mn ) O m = 0 ,
Q ( o ;   o old ) o n ( x ) = m   a nm o n ( x )   y , z { E [ d ˜ nm ( x ,   y ,   z ) | d ,   o old ] + E [ d ˜ mn ( x - z ,   y ,   z ) | d ,   o old ] } - m ( a nm H nm + a mn H mn ) O m 0 ,
H nm = y h nm ( y ) ,
O m = x o m ( x ) .
o n new ( x ) = m a nm y , z { E [ d ˜ nm ( x ,   y ,   z ) | d ,   o old ] + E [ d ˜ mn ( x - z ,   y ,   z ) | d ,   o old ] } m ( a nm H nm + a mn H mn ) O m new ,
O m new = x o m new ( x ) .
E [ d ˜ nm ( x ,   y ,   z ) | d ,   o old ]
= h nm ( y - z ) o n old ( x ) o m old ( x + z )   d nm ( y ) μ nm old ( y ) + b nm ( y ) = def d nm old ( x ,   y ,   z ) ,
μ nm old ( y ) = z h nm ( y - z ) x o n old ( x ) o m old ( x + z ) .
E [ d ˜ nm ( x - z ,   y ,   z ) | d ,   o old ]
= h nm ( y - z ) o n old ( x - z ) o m old ( x )   d nm ( y ) μ nm old ( y ) + b nm ( y ) = d nm old ( x - z ,   y ,   z ) .
O n new = m [ a nm D nm old + a mn D mn old ] m ( a nm H nm + a mn H mn ) O m new ,
D nm old = x d nm old ( x ,   y ,   z ) = x d nm old ( x - z ,   y ,   z ) = y   μ nm old ( y ) μ nm old ( y ) + b nm ( y )   d nm ( y ) .
O n new = m [ a nm D nm old + a mn D mn old ] m ( a nm H nm + a mn H mn ) O m new ,
D nm old = y μ nm old ( y ) μ nm old ( y ) + b nm ( y )   d nm ( y ) ,
H nm = y h nm ( y ) ,
μ nm old ( y ) = z h nm ( y - z ) x o n old ( x ) o m old ( x + z ) .
o n new ( x ) = m y z [ a nm d nm old ( x ,   y ,   z ) + a mn d mn old ( x - z ,   y ,   z ) ] m ( H nm a nm + H mn a mn ) O m new ,
d nm old ( x ,   y ,   z ) = h nm ( y - z ) o n old ( x ) o m old × ( x + z )   d nm ( y ) μ nm old ( y ) + b nm .

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