Abstract

We use blind deconvolution methods in optical diffusion tomography to reconstruct images of objects imbedded in or located behind turbid media from continuous-wave measurements of the scattered light transmitted through the media. In particular, we use a blind deconvolution imaging algorithm to determine both a deblurred image of the object and the depth of the object inside the turbid medium. Preliminary results indicate that blind deconvolution produces better reconstructions than can be obtained using backpropagation techniques. Moreover, it does so without requiring prior knowledge of the characteristics of the turbid medium or of what the blur-free target should look like: important advances over backpropagation.

©2002 Optical Society of America

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References

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  1. E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image transfer through a scattering medium, (Springer-Verlag, Berlin-Heidelberg, 1991), Chap. 6.
    [Crossref]
  2. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997) http://www.opticsexpress.org/oearchive/source/1884.htm.
    [Crossref] [PubMed]
  3. C. L. Matson and H. Liu, “Backpropagation in turbid media,” J. Opt. Soc. Am. A 16, 1254–1265 (1999).
    [Crossref]
  4. C. L. Matson, “Deconvolution-based spatial resolution in optical diffusion tomography,” Appl. Opt. 40, 5791–5801 (2001).
    [Crossref]
  5. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, 1996), Chap. 3.
  6. R. K. Pina and R. C. Puetter, “Incorporation of spatial information in Bayesian image reconstruction: the maximum residual likelihood criterion,” PASP 104, 1096–1103 (1992).
    [Crossref]
  7. M. Lloyd-Hart, S. M. Jefferies, E. K. Hege, and J. R. P. Angel, “Wave Front Sensing with Time-of-flight Phase Diversity,” Opt. Lett. 26, 402–404 (2001).
    [Crossref]
  8. D. G. Sheppard, B. R. Hunt, and M.W. Marcellen, “Iterative multi-frame super-resolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A 15, 978–992 (1998).
    [Crossref]
  9. C. L. Matson, N. Clark, L. McMackin, and J. S. Fender, “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves,” Appl. Opt. 36, 214–220 (1997).
    [Crossref] [PubMed]
  10. X. Cheng and D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 3, 118–123 (1998), http://www.opticsexpress.org/oearchive/source/5663.htm.
    [Crossref] [PubMed]
  11. C. L. Matson and H. Liu, “Resolved object imaging and localization with the use of a backpropagation algorithm,” Opt. Express 6, 168–174 (2000), http://www.opticsexpress.org/oearchive/source/19570.htm.
    [Crossref] [PubMed]
  12. A. S. Carasso, “Direct blind deconvolution,” SIAM J. Appl. Math. 61, 1980–2007 (2001).
    [Crossref]
  13. S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astro. Phys. J. 415, 862–864 (1993).
    [Crossref]

2001 (3)

2000 (1)

1999 (1)

1998 (2)

1997 (2)

1993 (1)

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astro. Phys. J. 415, 862–864 (1993).
[Crossref]

1992 (1)

R. K. Pina and R. C. Puetter, “Incorporation of spatial information in Bayesian image reconstruction: the maximum residual likelihood criterion,” PASP 104, 1096–1103 (1992).
[Crossref]

Angel, J. R. P.

Boas, D. A.

Carasso, A. S.

A. S. Carasso, “Direct blind deconvolution,” SIAM J. Appl. Math. 61, 1980–2007 (2001).
[Crossref]

Cheng, X.

Christou, J. C.

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astro. Phys. J. 415, 862–864 (1993).
[Crossref]

Clark, N.

Fender, J. S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, 1996), Chap. 3.

Hege, E. K.

Hunt, B. R.

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image transfer through a scattering medium, (Springer-Verlag, Berlin-Heidelberg, 1991), Chap. 6.
[Crossref]

Jefferies, S. M.

M. Lloyd-Hart, S. M. Jefferies, E. K. Hege, and J. R. P. Angel, “Wave Front Sensing with Time-of-flight Phase Diversity,” Opt. Lett. 26, 402–404 (2001).
[Crossref]

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astro. Phys. J. 415, 862–864 (1993).
[Crossref]

Katsev, I. L.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image transfer through a scattering medium, (Springer-Verlag, Berlin-Heidelberg, 1991), Chap. 6.
[Crossref]

Liu, H.

Lloyd-Hart, M.

Marcellen, M.W.

Matson, C. L.

McMackin, L.

Pina, R. K.

R. K. Pina and R. C. Puetter, “Incorporation of spatial information in Bayesian image reconstruction: the maximum residual likelihood criterion,” PASP 104, 1096–1103 (1992).
[Crossref]

Puetter, R. C.

R. K. Pina and R. C. Puetter, “Incorporation of spatial information in Bayesian image reconstruction: the maximum residual likelihood criterion,” PASP 104, 1096–1103 (1992).
[Crossref]

Sheppard, D. G.

Zege, E. P.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image transfer through a scattering medium, (Springer-Verlag, Berlin-Heidelberg, 1991), Chap. 6.
[Crossref]

Appl. Opt. (2)

Astro. Phys. J. (1)

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astro. Phys. J. 415, 862–864 (1993).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Opt. Lett. (1)

PASP (1)

R. K. Pina and R. C. Puetter, “Incorporation of spatial information in Bayesian image reconstruction: the maximum residual likelihood criterion,” PASP 104, 1096–1103 (1992).
[Crossref]

SIAM J. Appl. Math. (1)

A. S. Carasso, “Direct blind deconvolution,” SIAM J. Appl. Math. 61, 1980–2007 (2001).
[Crossref]

Other (2)

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image transfer through a scattering medium, (Springer-Verlag, Berlin-Heidelberg, 1991), Chap. 6.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, 1996), Chap. 3.

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Figures (4)

Fig. 1.
Fig. 1.

Schematic showing the geometry of the optical diffusion tomography system for the development in the paper. The imbedded object is represented by the cube.

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Fig. 2.
Fig. 2.

Schematic of the experimental setup used for the data acquisition.

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Fig. 3.
Fig. 3.

The different targets used to generate the experimental data. The first two targets, from left to right, were positioned behind a 2.5 cm thick phantom. The third target (far right) was imbedded in the center of a 5.5 cm thick phantom. From left to right: F-111 model aircraft, F-18 model aircraft, and Boeing 747 model aircraft. The F-111 and F-18 aircraft models, when illuminated and viewed through the turbid phantom, produce similar images (see Fig. 4). This provides a stringent test of the resolving capability of the two restoration algorithms considered in this work.

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Fig. 4.
Fig. 4.

Columns, left to right: raw data, reconstructed object, o2D(x,y;z1), using BP, and BD respectively. Top row: F-111 model airplane, second row: F-18 model airplane, bottom row: 747 model.

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Equations (11)

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ϕ ( x , y , z ) = ϕ hom ( x , y , z ) + ϕ pert ( x , y , z )
ϕ pert ( x , y , z ) = ∫ ∫ ∫ ϕ hom ( x ′ , y ′ , z ′ ) o ( x ′ , y ′ , z ′ ) h ( x − x ′ , y − y ′ , z − z ′ ) dx ′ dy ′ dz ′
ϕ pert ( x , y , z o ) = o 2 D ( x , y , z 1 ) * h ( x , y , Δz )
o 2 D ( x , y , z 1 ) = − ∫∫∫ object support ϕ hom ( x ′ , y ′ , z′ ) o ( x ′ , y ′ , z′ ) h ( x − x ′ , y − y ′ , z 1 − z′ ) dx ′ dy ′ dz′
ε = ∑ x , y ≠ 0,0 ( r ⊗ r ) x , y 2
r ( x , y ) = [ d ( x , y ) − ( o 2 D * h ) x , y ] s d ( x , y ) ( o 2 D * h ) x , y + n ( x , y ) 2
o 2 D ( x , y ) = θ 2 ( x , y ) s o ( x , y )
h ( x , y ) = ( ψ 2 * τ ) x , y s h ( x , y )
H ( u , v ) = exp [ − Δz ( 2 πu ) 2 + ( 2 πv ) 2 + α ]
h ( x , y ) = N − 2 ∑ u , v T ( u , v ) H ( u , v ) exp ( i 2 π [ ux + vy ] / N )
ε p = N − 2 ∑ u , v > cutoff ∣ H ( u , v ) ∣ 2 + N − 2 ∑ u , v > cutoff ∣ F ( u , v ) ∣ 2

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