Abstract

Time-resolved Fourier optical diffuse tomography is a novel approach for imaging of objects in a highly scattering turbid medium with use of an incident (near) plane wave. The theory of the propagation of spatial Fourier components of the scattered wave field is presented, along with a fast algorithm for three-dimensional reconstruction in a parallel planar geometry. Examples of successful reconstructions of simulated hidden absorptive or scattering objects embedded inside a human-tissue-like semi-infinite turbid medium are provided.

© 2001 Optical Society of America

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  1. B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389 (1995).
  2. R. R. Alfano, S. G. Demos, S. K. Gayen, “Advances in optical imaging of biomedical media,” Ann. N.Y. Acad. Sci. 820, 248–271 (1997).
    [CrossRef] [PubMed]
  3. J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
    [CrossRef] [PubMed]
  4. S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  5. W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
    [CrossRef] [PubMed]
  6. X. D. Li, T. Durduran, A. G. Yodh, “Diffraction tomography for biomedical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
    [CrossRef] [PubMed]
  7. X. Cheng, D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 4, 118–123 (1998), http://www.opticsexpress.org .
    [CrossRef]
  8. W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
    [CrossRef]
  9. T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Express 4, 247–262 (1999), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  10. S. R. Arridge, “Photon-measurement density functions. Part I. Analytic forms,” Appl. Opt. 34, 7395–7409 (1995).
    [CrossRef] [PubMed]
  11. P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
    [CrossRef]
  12. P. C. Hansen, D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1053 (1993).
    [CrossRef]

1999

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Express 4, 247–262 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

1998

X. Cheng, D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 4, 118–123 (1998), http://www.opticsexpress.org .
[CrossRef]

1997

R. R. Alfano, S. G. Demos, S. K. Gayen, “Advances in optical imaging of biomedical media,” Ann. N.Y. Acad. Sci. 820, 248–271 (1997).
[CrossRef] [PubMed]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

X. D. Li, T. Durduran, A. G. Yodh, “Diffraction tomography for biomedical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef] [PubMed]

1996

W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
[CrossRef] [PubMed]

1995

1993

P. C. Hansen, D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1053 (1993).
[CrossRef]

1992

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Alfano, R. R.

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

R. R. Alfano, S. G. Demos, S. K. Gayen, “Advances in optical imaging of biomedical media,” Ann. N.Y. Acad. Sci. 820, 248–271 (1997).
[CrossRef] [PubMed]

W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
[CrossRef] [PubMed]

Alrubaiee, M.

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

Arridge, S. R.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, “Photon-measurement density functions. Part I. Analytic forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

Boas, D. A.

X. Cheng, D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 4, 118–123 (1998), http://www.opticsexpress.org .
[CrossRef]

Cai, W.

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
[CrossRef] [PubMed]

Cheng, X.

X. Cheng, D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 4, 118–123 (1998), http://www.opticsexpress.org .
[CrossRef]

Culver, J. P.

Das, B. B.

W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
[CrossRef] [PubMed]

Delpy, D. T.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

Demos, S. G.

R. R. Alfano, S. G. Demos, S. K. Gayen, “Advances in optical imaging of biomedical media,” Ann. N.Y. Acad. Sci. 820, 248–271 (1997).
[CrossRef] [PubMed]

Durduran, T.

Gayen, S. K.

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

R. R. Alfano, S. G. Demos, S. K. Gayen, “Advances in optical imaging of biomedical media,” Ann. N.Y. Acad. Sci. 820, 248–271 (1997).
[CrossRef] [PubMed]

Hansen, P. C.

P. C. Hansen, D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1053 (1993).
[CrossRef]

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Hebden, J. C.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Holboke, M. J.

Lax, M.

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
[CrossRef] [PubMed]

Li, X. D.

Liu, F.

W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
[CrossRef] [PubMed]

O’Leary, D. P.

P. C. Hansen, D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1053 (1993).
[CrossRef]

Xu, M.

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

Yodh, A. G.

Zevallos, M.

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
[CrossRef] [PubMed]

Zubkov, L.

Ann. N.Y. Acad. Sci.

R. R. Alfano, S. G. Demos, S. K. Gayen, “Advances in optical imaging of biomedical media,” Ann. N.Y. Acad. Sci. 820, 248–271 (1997).
[CrossRef] [PubMed]

Appl. Opt.

W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, R. R. Alfano, “Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements,” Appl. Opt. 38, 1–10 (1999).
[CrossRef]

S. R. Arridge, “Photon-measurement density functions. Part I. Analytic forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A.

W. Cai, B. B. Das, F. Liu, M. Zevallos, M. Lax, R. R. Alfano, “Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media,” Proc. Natl. Acad. Sci. U.S.A. 93, 13561–13564 (1996).
[CrossRef] [PubMed]

SIAM J. Sci. Comput.

P. C. Hansen, D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1053 (1993).
[CrossRef]

SIAM Rev.

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Other

B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389 (1995).

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of image reconstruction.

Fig. 2
Fig. 2

Geometry for time-resolved Fourier optical diffuse tomography with use of backscattered photons. The source is a picosecond (near-) plane-wave pulse and a series of snapshots of a 10×10 cm2 area on the surface are computed as the input to image reconstruction. Absorptive objects A (-2.5, -1.875, -0.75) cm, B (-1.25, -0.31, -0.75) cm, C (0.94, 1.56, -1.95) cm, and D (0.94, -0.625, -1.95) cm or scattering objects E (-2.5, -1.875, -0.75) cm, F (-1.25, -0.31, -0.75) cm, G (0.94, 1.56, -1.35) cm, and H (0.94, -0.625, -1.35) cm are used in the simulation.

Fig. 3
Fig. 3

Absorption depth profile for (a) with 1% noise, (b) 5% noise, and (c) 10% noise.

Fig. 4
Fig. 4

Layer reconstruction at a noise level of 1%: (a) resolved objects A (left) and B (right) at z=0.75 cm (layer 3); (b) resolved objects C (upper) and D (lower) at z=1.95 cm (layer 7). The darkness of the pixel represents the resolved absorption coefficient in units of inverse millimeters.

Fig. 5
Fig. 5

Layer reconstruction at a noise level of 5%: (a) resolved objects A (left) and B (right) at z=0.75 cm (layer 3); (b) resolved objects C (upper) and D (lower) at z=1.95 cm (layer 7). The darkness of the pixel represents the resolved absorption coefficient in units of inverse millimeters.

Fig. 6
Fig. 6

Layer reconstruction at a noise level of 10%: (a) resolved objects A (left) and B (right) at z=0.75 cm (layer 3); (b) resolved objects C (upper) and D (lower) at z=1.95 cm (layer 7). The darkness of the pixel represents the resolved absorption coefficient in units of inverse millimeters.

Fig. 7
Fig. 7

Scattering depth profiles for (a) with 1% noise, (b) 5% noise, and (c) 10% noise.

Fig. 8
Fig. 8

Layer reconstruction at a noise level of 1%: (a) resolved objects E (left) and F (right) at z=0.75 cm (layer 3); (b) resolved objects G (upper) and H (lower) at z=1.35 cm (layer 5). The darkness of the pixel represents the resolved reduced scattering coefficient in units of inverse millimeters.

Fig. 9
Fig. 9

Layer reconstruction at a noise level of 5%: (a) resolved objects E (left) and F (right) at z=0.75 cm (layer 3); (b) resolved objects G (upper) and H (lower) at z=1.35 cm (layer 5). The darkness of the pixel represents the resolved reduced scattering coefficient in units of inverse millimeters.

Fig. 10
Fig. 10

Layer reconstruction at a noise level of 10%: (a) resolved objects E (left) and F (right) at z=0.75 cm (layer 3); (b) resolved objects G (upper) and H (lower) at z=1.35 cm (layer 5). The darkness of the pixel represents the resolved reduced scattering coefficient in units of inverse millimeters.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

t ϕ(r, t)-cD(r)ϕ(r, t)+cμa(r)ϕ(r, t)
=S(r, t).
μa,obj(r)=μa+δμa(r),
μs,obj(r)=μs+δμs(r),
t ϕ(r, t)-Dc2ϕ(r, t)+μacϕ(r, t)
=S(r, t)+cδD(r)ϕ(r, t)-cδμa(r)ϕ(r, t).
ϕs(r, t)=ϕ(r, t)-ϕ0(r, t)
=d3r0tdtG(r, r, t-t)×[crδD(r)rϕ(r, t)-cδμa(r)ϕ(r, t)]
=-d3r0tdtG(r, r, t-t)δμa(r)cϕ(r, t)+d3r0tdtδμs(r)c3μs2 rG(r, r, t-t)rϕ(r, t),
G(r, r, t)=14πDctexp-|ρ-ρ|24Dct-μact×Gz(z, z, t),(t>0),
Gˆ(q, z, ρ, z, t)=d2ρG(ρ, z, ρ, z, t)exp (-iqρ)
=exp(-iqρ-Dctq2-μact)Gz(z, z, t)
=Gˆ(q, z, z, t)exp(-iqρ).
ϕs(ρ, z, t)=-d3rd2ρ0tdtG(r, r, t-t)δμa(r)×cS(ρ)G(r, ρ, zs, t)
ϕs(ρ, z, t)=-1(4π2)3d2ρdzd2ρ0tdt
×d2qGˆ(q, z, z, t-t)×exp[iq(ρ-ρ)]
×δμa(ρ, z)cd2qSˆ(q)exp(iqρ)
×d2qGˆ(q, z, zs, t)×exp[iq(ρ-ρ)]
=-c(4π2)3d2qd2qd2q0tdtdz
×exp(iqρ)Gˆ(q, z, z, t-t)×Sˆ(q)Gˆ(q, z, zs, t)
×d2ρδμa(ρ, z)exp[-iρ(q-q)]d2ρexp[iρ(q-q)]
=-c(4π2)2d2qd2q0tdtdz×exp(iqρ)Gˆ(q, z, z, t-t)×δμ^a(q-q, z)Sˆ(q)Gˆ(q, z, zs, t),
Sˆ(q)=Sˆ(q, zs)=d2ρS(ρ, zs)exp(-iqρ),
δμ^a(q, z)=d2ρδμa(ρ, z)exp(-iqρ)
ϕˆ(q, z, t)=-c4π2d2qdzδμ^a(q-q, z)Sˆ(q, zs)×0tdtGˆ(q, z, z, t-t)Gˆ(q, z, zs, t).
ϕ^s(q, z, t)=c12π2μs2d2qdzδμ^s(q-q, z)Sˆ(q, zs)×0tdtqqGˆ(q, z, z, t-t)Gˆ(q, z, zs, t)+Gˆ(q, z, z, t-t)zGˆ(q, z, zs, t)z.
wa(q, q, z, t; z)=0tdtGˆ(q, z, z, t-t)Gˆ(q, z, zs, t),
ws(q, q, z, t; z)=0tdtGˆ(q, z, z, t-t)z×Gˆ(q, z, zs, t)z,
ϕ^s(q, z, t)
=-c4π2d2qdzδμ^a(q-q, z)×Sˆ(q, zs)wa(q, q, z, t; z)+c12π2μs2×d2qdzδμ^s(q-q, z)Sˆ(q, zs)×[qqwa(q, q, z, t; z)+ws(q, q, z, t; z)].
ϕ^s(q, z, t)=-Scdzδμ^a(q, z)wa(q, 0, z, t; z)-δμ^s(q, z)3μs2 ws(q, 0, z, t; z).
ϕ^s(q, t)=ScΔzj=1Nz-δμ^a(q, zj)wa(q, 0, t; zj)+δμ^s(q, zj)3μs2 ws(q, 0, t; zj),
ϕ^s(0, t)=ScΔzj=1Nz-δμ^a(0, zj)wa(0, 0, t; zj)+δμ^s(0, zj)3μs2 ws(0, 0, t; zj),
A=-σ2S0c2πd2qdzδμ^a(q-q, z)×exp(-σ2q2/2)wa(q, q, z, t; z)=-σ2S0c2πd2qdzd2ρδμa(ρ, z)×exp[-i(q-q)ρ-σ2q2/2]×wa(q, q, z, t; z)=-σ2S0c2πd2ρdzδμa(ρ, z)exp(-iqρ)×0tdtGˆ(q, z, z, t-t)×Gz(z, zs, t)exp(-μact)×d2qexp(-σ2q2/2-Dct+iqρ)

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