Abstract

We propose an estimation-theoretic approach to the inference of an incoherent 3D scattering density from 2D scattered speckle field measurements. The object density is derived from the covariance of the speckle field. The inference is performed by a constrained optimization technique inspired by compressive sensing theory. Experimental results demonstrate and verify the performance of our estimates.

© 2010 Optical Society of America

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  1. R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
    [CrossRef]
  2. J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
    [CrossRef]
  3. T. Schulz, “Penalized maximum-likelihood estimation of covariance matrices with linear structure,” IEEE Trans. Signal Process. 45, 3027–3038 (1997).
    [CrossRef]
  4. A. D. Lanterman, “Statistical radar imaging of diffuse and specular targets using an expectation-maximization algorithm,” Proc. SPIE 4053, 20–31 (2000).
    [CrossRef]
  5. R. G. Dantas and E. T. Costa, “Ultrasound speckle reduction using modified gabor filters,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 530–538 (2007).
    [CrossRef] [PubMed]
  6. D. L. Marks, T. S. Ralston, and S. A. Boppart, “Speckle reduction by I-divergence regularization in optical coherence tomography,” J. Opt. Soc. Am. A 22, 2366–2371 (2005).
    [CrossRef]
  7. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049(2009).
    [CrossRef] [PubMed]
  8. J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2006).
  9. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  10. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [CrossRef]
  11. Y. K. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17, 12285–12292 (2009).
    [CrossRef] [PubMed]
  12. Z. Yaqoob, D. Psaltis, M. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
    [CrossRef]
  13. T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).
  14. J. W. Goodman, Statistical Optics (Wiley Interscience, 2000).
  15. P. Stoica and R. L. Moses, Spectral Analysis of Signals (Prentice-Hall, 2005).
  16. E. Candés, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
    [CrossRef]
  17. R. G. Baraniuk, E. Candes, R. Nowak, and M. Vetterli, “Compressive sampling,” IEEE Signal Process. Mag. 25, 12–13(2008).
    [CrossRef]
  18. R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Inf. Theory 49, 3320–3325 (2003).
    [CrossRef]
  19. J. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
    [CrossRef]
  20. T. F. Chan and J. A. Olkin, “Circulant preconditioners for toeplitz-block matrices,” Numer. Algorithms 6, 89–101 (1994).
    [CrossRef]
  21. B. Fischer and J. Modersitzki, “Fast inversion of matrices arising in image processing,” Numer. Algorithms 22, 1–11(1999).
    [CrossRef]

2009 (2)

2008 (2)

Z. Yaqoob, D. Psaltis, M. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[CrossRef]

R. G. Baraniuk, E. Candes, R. Nowak, and M. Vetterli, “Compressive sampling,” IEEE Signal Process. Mag. 25, 12–13(2008).
[CrossRef]

2007 (1)

R. G. Dantas and E. T. Costa, “Ultrasound speckle reduction using modified gabor filters,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 530–538 (2007).
[CrossRef] [PubMed]

2006 (2)

J. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

E. Candés, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

2005 (1)

2003 (1)

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Inf. Theory 49, 3320–3325 (2003).
[CrossRef]

2000 (1)

A. D. Lanterman, “Statistical radar imaging of diffuse and specular targets using an expectation-maximization algorithm,” Proc. SPIE 4053, 20–31 (2000).
[CrossRef]

1999 (2)

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

B. Fischer and J. Modersitzki, “Fast inversion of matrices arising in image processing,” Numer. Algorithms 22, 1–11(1999).
[CrossRef]

1997 (1)

T. Schulz, “Penalized maximum-likelihood estimation of covariance matrices with linear structure,” IEEE Trans. Signal Process. 45, 3027–3038 (1997).
[CrossRef]

1994 (1)

T. F. Chan and J. A. Olkin, “Circulant preconditioners for toeplitz-block matrices,” Numer. Algorithms 6, 89–101 (1994).
[CrossRef]

1982 (1)

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

1962 (1)

Badizadegan, K.

Baraniuk, R. G.

R. G. Baraniuk, E. Candes, R. Nowak, and M. Vetterli, “Compressive sampling,” IEEE Signal Process. Mag. 25, 12–13(2008).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

Boppart, S. A.

Brady, D. J.

Candes, E.

R. G. Baraniuk, E. Candes, R. Nowak, and M. Vetterli, “Compressive sampling,” IEEE Signal Process. Mag. 25, 12–13(2008).
[CrossRef]

Candés, E.

E. Candés, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Chan, T. F.

T. F. Chan and J. A. Olkin, “Circulant preconditioners for toeplitz-block matrices,” Numer. Algorithms 6, 89–101 (1994).
[CrossRef]

Choi, K.

Choi, W.

Costa, E. T.

R. G. Dantas and E. T. Costa, “Ultrasound speckle reduction using modified gabor filters,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 530–538 (2007).
[CrossRef] [PubMed]

Dantas, R. G.

R. G. Dantas and E. T. Costa, “Ultrasound speckle reduction using modified gabor filters,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 530–538 (2007).
[CrossRef] [PubMed]

Dasari, R.

Feld, M.

Y. K. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17, 12285–12292 (2009).
[CrossRef] [PubMed]

Z. Yaqoob, D. Psaltis, M. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[CrossRef]

Fienup, J. R.

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

Fischer, B.

B. Fischer and J. Modersitzki, “Fast inversion of matrices arising in image processing,” Numer. Algorithms 22, 1–11(1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2006).

J. W. Goodman, Statistical Optics (Wiley Interscience, 2000).

Gribonval, R.

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Inf. Theory 49, 3320–3325 (2003).
[CrossRef]

Horisaki, R.

Lanterman, A. D.

A. D. Lanterman, “Statistical radar imaging of diffuse and specular targets using an expectation-maximization algorithm,” Proc. SPIE 4053, 20–31 (2000).
[CrossRef]

Leith, E. N.

Lim, S.

Marks, D. L.

Modersitzki, J.

B. Fischer and J. Modersitzki, “Fast inversion of matrices arising in image processing,” Numer. Algorithms 22, 1–11(1999).
[CrossRef]

Moon, T. K.

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

Moses, R. L.

P. Stoica and R. L. Moses, Spectral Analysis of Signals (Prentice-Hall, 2005).

Nielsen, M.

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Inf. Theory 49, 3320–3325 (2003).
[CrossRef]

Nowak, R.

R. G. Baraniuk, E. Candes, R. Nowak, and M. Vetterli, “Compressive sampling,” IEEE Signal Process. Mag. 25, 12–13(2008).
[CrossRef]

Olkin, J. A.

T. F. Chan and J. A. Olkin, “Circulant preconditioners for toeplitz-block matrices,” Numer. Algorithms 6, 89–101 (1994).
[CrossRef]

Park, Y. K.

Paxman, R. G.

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

Psaltis, D.

Z. Yaqoob, D. Psaltis, M. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[CrossRef]

Ralston, T. S.

Reiley, M. F.

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

Romberg, J. K.

E. Candés, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Schulz, T.

T. Schulz, “Penalized maximum-likelihood estimation of covariance matrices with linear structure,” IEEE Trans. Signal Process. 45, 3027–3038 (1997).
[CrossRef]

Stirling, W. C.

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

Stoica, P.

P. Stoica and R. L. Moses, Spectral Analysis of Signals (Prentice-Hall, 2005).

Tao, T.

E. Candés, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Thelen, B. J.

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

Tropp, J.

J. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

Upatnieks, J.

Vetterli, M.

R. G. Baraniuk, E. Candes, R. Nowak, and M. Vetterli, “Compressive sampling,” IEEE Signal Process. Mag. 25, 12–13(2008).
[CrossRef]

Yang, C.

Z. Yaqoob, D. Psaltis, M. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[CrossRef]

Yaqoob, Z.

Y. K. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17, 12285–12292 (2009).
[CrossRef] [PubMed]

Z. Yaqoob, D. Psaltis, M. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[CrossRef]

Commun. Pure Appl. Math. (1)

E. Candés, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

IEEE Signal Process. Mag. (1)

R. G. Baraniuk, E. Candes, R. Nowak, and M. Vetterli, “Compressive sampling,” IEEE Signal Process. Mag. 25, 12–13(2008).
[CrossRef]

IEEE Trans. Inf. Theory (2)

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Inf. Theory 49, 3320–3325 (2003).
[CrossRef]

J. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

IEEE Trans. Signal Process. (1)

T. Schulz, “Penalized maximum-likelihood estimation of covariance matrices with linear structure,” IEEE Trans. Signal Process. 45, 3027–3038 (1997).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

R. G. Dantas and E. T. Costa, “Ultrasound speckle reduction using modified gabor filters,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 530–538 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nat. Photon. (1)

Z. Yaqoob, D. Psaltis, M. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[CrossRef]

Numer. Algorithms (2)

T. F. Chan and J. A. Olkin, “Circulant preconditioners for toeplitz-block matrices,” Numer. Algorithms 6, 89–101 (1994).
[CrossRef]

B. Fischer and J. Modersitzki, “Fast inversion of matrices arising in image processing,” Numer. Algorithms 22, 1–11(1999).
[CrossRef]

Opt. Express (2)

Phys. Rep. (1)

R. H. T. Bates, “Astronomical speckle imaging,” Phys. Rep. 90, 203–297 (1982).
[CrossRef]

Proc. SPIE (2)

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

A. D. Lanterman, “Statistical radar imaging of diffuse and specular targets using an expectation-maximization algorithm,” Proc. SPIE 4053, 20–31 (2000).
[CrossRef]

Other (5)

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

J. W. Goodman, Statistical Optics (Wiley Interscience, 2000).

P. Stoica and R. L. Moses, Spectral Analysis of Signals (Prentice-Hall, 2005).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2006).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

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Figures (6)

Fig. 1
Fig. 1

(a) Singular value spectra of a 100 × 1000 Gaussian random matrix A , its Gram matrix A H A , and its associated B matrix and (b) singular value spectra of two examples of , denoted by B 1 and B 2 , associated with two examples of H created with two different axial sample spacings.

Fig. 2
Fig. 2

(a) Photograph of experimental setup: M, mirror; BS, beam splitter; and FT lens, Fourier transform lens. (b) Photograph of a 3D diffuse object “DISP.”

Fig. 3
Fig. 3

(a) Intensity of the backpropagation with a single speckle field, (b) the average of the backpropagation intensities of 50 speckle fields [see Eq. (10)], and (c) an image obtained by applying a 5 × 5 median filter to the image in (b).

Fig. 4
Fig. 4

(a) Estimate obtained by solving Eq. (15), (b) a Tikhonov-regularized pseudoinverse (i.e., d ˜ = ˜ α + σ 2 ω ˜ ) with λ t = 10 8 using 50 speckle fields both for (a), (b) and (c), (d) reconstructions (with β = 0.3 ) obtained by solving the preconditioned formulation in Eq. (19) with (c) a single speckle field and (d) 50 speckle fields.

Fig. 5
Fig. 5

Reconstructions with (a) β = 0.1 and (b) β = 0.2 .

Fig. 6
Fig. 6

(a) Photograph of the objects: a real ladybug in front and the letters “LADYBUG” in the back with their dimensions; the yellow rectangle shows the location of the in-focus plane, (b) the average image of the backpropagation reconstruction intensities of 20 speckle fields, and (c) our reconstruction obtained by solving Eq. (19).

Equations (23)

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I ( x , y ) = | A e j α x + E ( x , y ) | 2 = | A | 2 + | E ( x , y ) | 2 + A e j α x E ( x , y ) + A e j α x E * ( x , y ) ,
E ( x , y ) = d x d y d z η ( x , y , z ) h ( x x , y y , z z ) ,
E n 1 n 2 = E ( n 1 Δ , n 2 Δ ) = 1 ( 2 π ) 2 d z d x d y d k x d k y d x d y η ( x , y , z ) e i ( k x x + k y y ) m 1 m 2 δ ( x m 1 Δ ) δ ( y m 2 Δ ) e i z k 2 k x 2 k y 2 δ ( z z ) e i ( k x x + k y y ) δ ( x n 1 Δ ) δ ( y n 2 Δ ) m 1 m 2 δ ( k x m 1 Δ k ) δ ( k y m 2 Δ k ) l δ ( z l Δ z ) = 1 N 2 l m 1 m 2 [ m 1 m 2 η m 1 m 2 l e i 2 π m 1 m 1 + m 2 m 2 N ] e i l Δ z k 2 m 1 2 Δ k 2 m 2 2 Δ k 2 e i 2 π n 1 m 1 + n 2 m 2 N ,
E n 1 n 2 = F 2 D 1 { l η ^ m 1 m 2 l e i l Δ z k 2 m 1 2 Δ k 2 m 2 2 Δ k 2 } ,
g = H f + w ,
g k = G 2 D Q B f k + w k = H f k + w k ,
[ S l ] m 1 m 2 = exp ( i k 0 l Δ z ) exp ( i l Δ z k 0 2 m 1 2 Δ k 2 m 2 2 Δ k 2 ) ,
p ( f ) = 1 π N det ( R f ) ( f H R f 1 f ) ,
f ^ = H H ( H H H ) 1 g = H H g = H H H f + H H w ,
s ^ n = 1 K k = 1 K | f ^ n k | 2 .
E [ s ^ ] = [ E [ s ^ 1 ] E [ s ^ 2 ] E [ s ^ N ] ] T = 1 K k = 1 K Diag ( E [ f ^ k f ^ k H ] ) = Diag ( E [ f ^ f ^ H ] ) = Diag ( E [ ( H H H f + H H w ) ( H H H f + H H w ) H ] ) = Diag ( H H H E [ f f H ] H H H + H H E [ w w H ] H ) = Diag ( H H H R f H H H + σ 2 H H H ) = Diag ( H H H R f H H H ) + σ 2 Diag ( H H H ) ,
E [ s ^ n ] = m = 1 N | h n , h m | 2 α m + σ 2 h n , h n = m = 1 N | h n H h m | 2 α m + σ 2 h n 2 ,
d = E [ s ^ ] = α + σ 2 ω ,
α * = arg min α Φ ( α ) , subject to d α 2 2 < ϵ ( σ 2 ) ,
α * = arg min α 1 2 d α 2 2 + β Φ ( α ) .
H = G 2 D Q B = [ H 1 H 2 H N z ] .
[ Y l 1 l 2 ] m 1 m 2 = [ S l 1 * S l 2 ] m 1 m 2 = exp ( i k ( l 2 l 1 ) Δ z ) exp ( i ( l 2 l 1 ) Δ z k 2 m 1 2 Δ k 2 m 2 2 Δ k 2 ) .
econd ( M ) = max { σ ( M | σ ( M ) > ϵ } min { σ ( M ) | σ ( M ) > ϵ } ,
α * = arg min α 1 2 d ˜ ˜ α 2 2 + β Φ ( α ) ,
A = circulant ( A 1 , A 2 , , A k ) = [ A 1 A 2 A 3 A k A k A 1 A 2 A k 1 A k 1 A k A 1 A k 2 A 2 A 3 A 4 A 1 ] ,
A k = circulant ( a 1 k , a 2 k , , a M k ) ,
H H H = [ H 1 H H 1 H 1 H H 2 H 1 H H 3 H 1 H H N z H 1 H H N z H 1 H H 1 H 1 H H 2 H 1 H H N z 1 H 1 H H N z 1 H 1 H H N z H 1 H H 1 H 1 H H N z 2 H 1 H H 2 H 1 H H 3 H 1 H H 4 H 1 H H 1 ] .
H l 1 H H l 2 = circulant ( vec ( W l 1 l 2 ) ) = circulant ( vec ( F 2 D 1 ( Y l 1 l 2 ) ) ) .

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