OSA's Digital Library

Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 1, Iss. 1 — Jul. 7, 1997
  • pp: 6–11
« Show journal navigation

A diffraction tomographic model of the forward problem using diffuse photon density waves

Charles L. Matson  »View Author Affiliations


Optics Express, Vol. 1, Issue 1, pp. 6-11 (1997)
http://dx.doi.org/10.1364/OE.1.000006


View Full Text Article

Acrobat PDF (98 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We present a model of the forward problem for diffuse photon density waves in turbid medium using a diffraction tomographic problem formulation. We consider a spatially-varying inhomogeneous structure whose absorption properties satisfy the Born approximation and whose scattering properties are identical to the homogeneous turbid media in which it is imbedded. The two-dimensional Fourier transform of the scattered field, measured in a plane, is shown to be related to the three-dimensional Fourier transform of the object evaluated on a surface which in many cases is approximately a plane.

© Optical Society of America

For our problem formulation, we allow an arbitrary structure for the wavefront but assume that our measurements are made in a plane. We start our analysis by assuming that the inhomogeneities in the turbid media are small, purely absorptive, perturbations when compared to the background media, enabling us to employ the first Born solution [11

11 . A. Kak and M. Slaney , Principles of Computerized Tomographic Imaging ( IEEE Press, New York , 1988 )

] to the wave equation

(2+k2)uB(r)=o(r)uo(r)
(1)

which is denoted by uB(r⃗) and is given by

uB(r)=o(r)uo(r)g(rr)dr′
(2)

The Green’s function in Eq.(2) can be expressed as [12

12 . A. Baños Jr. , Dipole Radiation in the Presence of a Conducting Half-Space ( Pergamon Press, Oxford , 1966 )

]

g(r)=exp[ikr]4πr
=18π21αx2+αy2k2exp{zαx2+αy2k2+ixαx+iyαy}xy
(3)

where the second equality is the angular spectrum representation of the Green’s function and the square root is determined by requiring the real part to be greater than zero. An interesting property of Eq.(3) is that the Green’s function has an analytic Fourier transform even though the Green’s function is singular at r⃗ = (0,0,0). This is a consequence of the nonzero imaginary component of k, which keeps the denominator from becoming zero because the spatial frequencies are real. Substituting Eq.(3) into Eq.(2) gives us

uB(r)=18π2o(r)uo(r)1αx2+αy2k2exp{zz′
×αx2+αy2k2+i(xx′)αx+i(yy′)αy}xydr′
=18π2o(r)uo(r)1γexp{zz′(γr+iγi)
+i(xx′)αx+i(yy′)αy}xydr′
(4)

where

γγr+iγi
=Re{αx2+αy2k2}+iIm{αx2+αy2k2}
(5)

and where Re() denotes the real part and Im() denotes the imaginary part of a complex number. We desire to rewrite Eq.(4) so that the integration over r⃗′ can be viewed as a Fourier transform of all the functions which depend upon r⃗′. To this end, we reexpress Eq.(4) as

uB(x,y,zo)=18π2exp{ixαx+iyαyizoγi}γ
×o(x′,y′,z′)uo(x′,y′,z′)exp{zoz′γr}
×exp{ix′αxiyαyiz′(γi)}dxdydz′xy
(6)

where we have explicitly written the integration over r⃗′ in terms of x’, y’, and z’, and we have interchanged the order of integration, integrating over the spatial coordinates before the spatial frequency coordinates. We have lumped the exponential term which depends upon γr with the object and source terms, and have explicitly displayed our measurement geometry by indicating that the scattered field is measured in the z=zo plane. We also assume an observation geometry where all the sources and inhomogeneities are at z values less than zo. This enables us to replace |zo - z′| with its argument. We have included this simplification for the terms multiplying γi but have retained the absolute value function for the terms multiplying γr in order to more closely match Fourier transform pairs as listed in standard tables. [13

13 . K.B. Howell , “ Fourier transforms ,” in The Transforms and Applications Handbook , A.D. Poularikas , ed. ( CRC Press, Boca Raton , 1996 )

]

It can be seen in Eq.(6) that the inner integral term is the three-dimensional Fourier transform of the object function multiplied by both the illumination function and an exponential term which results from the complex nature of k. Denoting the Fourier transform of this product by O, we can rewrite Eq.(6) as

uB(x,y,zo)=18π2exp{ixαx+iyαyizoγi}γOuγ(αx,αy,γi)xy
(7)

Our interest is in the two-dimensional Fourier-domain properties of uB(x,y,zo). Specifically, we desire to know the relationship between the two-dimensional Fourier transform of uB(x,y,zo), UBxy,zo), and the three-dimensional Fourier transform Oxyz). Therefore, we Fourier transform Eq.(7) and interchange the order of integration, resulting in

uB(ωx,ωy,zo)=18π2exp{izoγi}γOuγ(αx,αy,γi)
×exp{i(xαx+yαy)}exp{i(x+y)}dxdydαxdαy
(8)

The inner integrals are separable and evaluate to 4π2δ(ωxx)δ(ωyy). Thus Eq.(8) becomes

UB(ωx,ωy,zo)=12exp{izoγi}γOuγ(ωx,ωy,γi)
(9)

which is the main result of the paper. To reconstruct the object spectrum from the measured data as given in Eq.(9), it is necessary to either use a filtered backpropagation algorithm modified from the standard diffraction tomographic approach [7

7 . A.J. Devaney , “ Reconstructive tomography with diffracting wavefields ,” Inverse Problems 2 , 161 – 183 ( 1986 ) [CrossRef]

] or use Fourier interpolation after deconvolving the illumination and attenuation functions (see Eq.(6)). We are developing both approaches at this time.

From Eqs.(5) and (9) it can be seen that Oxy ,-γi ) is a portion of Oxyz) evaluated in a two-dimensional region. We must analyze the γi term to characterize this region further. From Eq.(5) we have

γi=Im{ωx2+ωy2k2}
(10)

For DPDWs, the functional form of the k2 term is given by [14

14 . D.A. Boas , M.A. O’Leary , B. Chance , and A.G. Yodh , “ Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications ,” Proc. Natl. Acad. Sci. USA 91 , 4887 – 4891 ( 1994 ) [CrossRef] [PubMed]

]

k2=(vμa+i2πft)3μ′sv
(11)

where v is the speed of light in the turbid medium, ft is the temporal frequency of the DPDW, μa is the absorption coefficient and μ′s is the reduced scattering coefficient. We see from Eq.(11) that the real part of k2 is always negative and its imaginary part is always positive. As a result, the expression ωx2+ωy2k2 has a negative imaginary part whose absolute value is always less than its real part and which is dominated by the real part as spatial frequency becomes arbitrarily large. It is instructive to compare the standard diffraction tomographic curve in two-dimensional Fourier space with the curve generated by γi. We generate a two-dimensional curve by setting ωx=0 in the expression for γi. The results are shown in Figure 1, where we have plotted three curves: one where Re{k2}<<Im{k2}, one where Re{k2}>>Im{k2}, and one where k is real and positive, as it is in standard diffraction tomography. All three curves have been normalized so that |k| = 1. Notice the familiar band-limited circular shape of the curve for real and positive k. It does not intersect zero spatial frequency because we have not included plane-wave illumination effects for the plots. Notice that the two curves corresponding to DPDWs are not bandlimited. The curve where Re{k2}<<Im{k2} is more flattened than the curve for real k, and the curve where Re{k2}>>Im{k2} is essentially a straight line. From Eq.(11), we see that the third curve results when either the absorption coefficient of the medium is large or when the temporal frequency of the DPDW is small. For many types of human tissue, [15

15 . W.F. Cheong , S.A. Prahl , and A.J. Welch , “ A review of the optical properties of biological tissues ,” IEEE J. Quantum Electronics 26 , 2166 – 2185 ( 1990 ) [CrossRef]

] μa is on the order of 0.5 cm-1, μ′s is on the order of 10 cm-1, and v is on the order of 2×1010 cm/s. For these values, k2 is given by

Figure 1 Plots of the two-dimensional projection of the surface on which the Fourier transform of the convolved object function is obtained with diffraction tomography. The dotted line is for real values of k, the dashed line is for Re{k2}<<Im{k2}, and the solid line is for Re{k2}>>Im{k2}. The horizontal axis is ωx and the vertical axis is ωy.
k2=15+i9ft
(12)

where the temporal frequency ft in Eq.(12) is to be expressed in GHz units. Therefore, for DPDWs which are generated using a few hundreds of megahertz, we have Re{k2}>>Im{k2}. We must have temporal frequencies of many gigahertz before Re{k2}<<Im{k2}. Recent results [16

16 . H. Heusmann , J. Koelzer , and G. Mitic , “ Characterization of female breasts in vivo by time-resolved and spectroscopic measurements in near infrared spectroscopy ,” J.Biomed.Opt. 1 , 425 – 434 ( 1996 ) [CrossRef]

] for human breast tissue show that typical values of μa are around 0.05cm-1, which indicates that Im{k2} becomes significant at temporal frequencies of tens of megahertz instead of hundreds of megahertz for the larger values of μa.

Because the surface which is defined by -γi does not intersect the zero spatial frequency point, a tomographic reconstruction using multiple views will result in a band-limited high-pass version of the object. In standard diffraction tomography, this problem is overcome by using plane wave illumination. The convolution of the Fourier transform of the plane wave with the object (see Eq.(6)) shifts the surface along the z-axis in Fourier space [11

11 . A. Kak and M. Slaney , Principles of Computerized Tomographic Imaging ( IEEE Press, New York , 1988 )

] to include the zero spatial frequency point, permitting a complete low-pass-filtered version of the object’s Fourier transform to be reconstructed using multiple views. Conceptually, this can be done with DPDWs as well. However, DPDWs are typically created using a few point illumination sources, which do not provide this convenient Fourier-domain shift. However, looking again at Eq.(6), we see that the object function is multiplied by an exponential function whose Fourier transform G(ωxyz) is given by [13

13 . K.B. Howell , “ Fourier transforms ,” in The Transforms and Applications Handbook , A.D. Poularikas , ed. ( CRC Press, Boca Raton , 1996 )

]

G(ωx,ωy,ωz)=4π2δ(ωx)δ(ωy)2γrγr2+ωz2
(13)

where we have set zo=0 because its value depends only upon the location of our coordinate system. Because O(ωxyz) is convolved with G(ωxyz), the correlation scale of G(ωxyz) (that is, its smoothness) is imposed upon O(ωxyz). When the imaginary part of k2 is much smaller than the real part of k2, the correlation scale of O(ωxyz) convolved with G(ωxyz) along the ωz axis is much larger than γi. The implication of this fact is that we have

Ouγ(ωx,ωy,γi)Ouγ(ωx,ωy,0)
(14)

References and Links

1 .

M.A. O’Leary , D.A. Boas , B. Chance , and A.G. Yodh , “ Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography ,” Opt. Lett. 20 , 426 – 428 ( 1995 ) [CrossRef] [PubMed]

2 .

H.B. Jiang , K.D. Paulsen , U.L. Osterberg , and M.S. Patterson , “ Frequency-domain optical-image reconstruction in turbid media - an experimental study of single-target detectability ,” Appl. Opt. 36 , 52 – 63 ( 1997 ) [CrossRef] [PubMed]

3 .

S.A. Walker , S. Fantini , and E. Gratton , “ Image reconstruction using back-projection from frequency-domain optical measurements in highly scattering media ,” Appl. Opt. 36 , 170 – 179 ( 1997 ) [CrossRef] [PubMed]

4 .

S.B. Colak , D.G. Papioannou , G.W. Hooft , and M.B. van der Mark , “ Optical image reconstruction with deconvolution in light diffusing media ,” in Photon Propagation in Tissues , B. Chance , D.T. Delpy , and G.J. Mueller , eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2626 , 306 – 315 ( 1995 )

5 .

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

6 .

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]

7 .

A.J. Devaney , “ Reconstructive tomography with diffracting wavefields ,” Inverse Problems 2 , 161 – 183 ( 1986 ) [CrossRef]

8 .

A. Schatzberg and A.J. Devaney , “ Super-resolution in diffraction tomography ,” Inverse Problems 8 , 149 – 164 ( 1992 ) [CrossRef]

9 .

A.J. Devaney , “ Linearised inverse scattering in attenuating media ,” Inverse Problems 3 , 389 – 397 ( 1987 ) [CrossRef]

10 .

A.J. Devaney , “ The limited-view problem in diffraction tomography ,” Inverse Problems 5 , 501 – 521 ( 1989 ) [CrossRef]

11 .

A. Kak and M. Slaney , Principles of Computerized Tomographic Imaging ( IEEE Press, New York , 1988 )

12 .

A. Baños Jr. , Dipole Radiation in the Presence of a Conducting Half-Space ( Pergamon Press, Oxford , 1966 )

13 .

K.B. Howell , “ Fourier transforms ,” in The Transforms and Applications Handbook , A.D. Poularikas , ed. ( CRC Press, Boca Raton , 1996 )

14 .

D.A. Boas , M.A. O’Leary , B. Chance , and A.G. Yodh , “ Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications ,” Proc. Natl. Acad. Sci. USA 91 , 4887 – 4891 ( 1994 ) [CrossRef] [PubMed]

15 .

W.F. Cheong , S.A. Prahl , and A.J. Welch , “ A review of the optical properties of biological tissues ,” IEEE J. Quantum Electronics 26 , 2166 – 2185 ( 1990 ) [CrossRef]

16 .

H. Heusmann , J. Koelzer , and G. Mitic , “ Characterization of female breasts in vivo by time-resolved and spectroscopic measurements in near infrared spectroscopy ,” J.Biomed.Opt. 1 , 425 – 434 ( 1996 ) [CrossRef]

17 .

D.C. Munson Jr. and J.L.C. Sanz , “ Image reconstruction from frequency-offset Fourier data ,” Proc. IEEE 72 , 661 – 669 ( 1984 ) [CrossRef]

18 .

C.L. Matson , I.A. Delarue , T.M. Gray , and I.E. Drunzer , “ Optimal Fourier spectrum estimation from the bispectrum ,” Computers Elect. Engng 18 , 485 – 497 ( 1992 ) [CrossRef]

OCIS Codes
(170.0170) Medical optics and biotechnology : Medical optics and biotechnology
(170.3660) Medical optics and biotechnology : Light propagation in tissues

ToC Category:
Research Papers

History
Original Manuscript: June 19, 1997
Revised Manuscript: June 17, 1997
Published: July 7, 1997

Citation
Charles Matson, "A diffraction tomographic model of the forward problem using diffuse photon density waves," Opt. Express 1, 6-11 (1997)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-1-6


Sort:  Journal  |  Reset  

References

  1. M.A. OLeary, D.A. Boas, B. Chance, and A.G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995) [CrossRef] [PubMed]
  2. H.B. Jiang, K.D. Paulsen, U.L. Osterberg, M.S. Patterson, "Frequency-domain optical-image reconstruction in turbid media - an experimental study of single-target detectability," Appl. Opt. 36, 52-63 (1997) [CrossRef] [PubMed]
  3. S.A. Walker, S. Fantini, and E. Gratton, " Image reconstruction using back-projection from frequency-domain optical measurements in highly scattering media," Appl. Opt. 36, 170-179 (1997) [CrossRef] [PubMed]
  4. S.B. Colak, D.G. Papioannou, G.W. Hooft, and M.B. van der Mark, "Optical image reconstruction with deconvolution in light diffusing media," in Photon Propagation in Tissues, B. Chance, D.T. Delpy, and G.J. Mueller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2626, 306-315 (1995)
  5. X.D. Li, T. Durduran, A.G. Yodh, B. Chance, and D.N. Pattanayak, "Diffraction tomography for biochemical imaging with diffuse-photon density waves," Opt. Lett. 22, 573-575 (1997) [CrossRef] [PubMed]
  6. 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]
  7. A.J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Problems 2, 161-183 (1986) [CrossRef]
  8. A. Schatzberg and A.J. Devaney, "Super-resolution in diffraction tomography," Inverse Problems 8, 149-164 (1992) [CrossRef]
  9. A.J. Devaney, "Linearised inverse scattering in attenuating media," Inverse Problems 3, 389-397 (1987) [CrossRef]
  10. A.J. Devaney, "The limited-view problem in diffraction tomography," Inverse Problems 5, 501-521 (1989) [CrossRef]
  11. A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988)
  12. A. Baños, Jr., Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon Press, Oxford, 1966)
  13. K.B. Howell, "Fourier transforms," in The Transforms and Applications Handbook, A.D. Poularikas, ed. (CRC Press, Boca Raton, 1996)
  14. D.A. Boas, M.A. OLeary, B. Chance, and A.G. Yodh, " Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994) [CrossRef] [PubMed]
  15. W.F. Cheong, S.A. Prahl, and A.J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electronics 26, 2166-2185 (1990) [CrossRef]
  16. H. Heusmann, J. Koelzer, and G. Mitic, "Characterization of female breasts in vivo by time-resolved and spectroscopic measurements in near infrared spectroscopy," J.Biomed.Opt. 1, 425-434 (1996) [CrossRef]
  17. D.C. Munson, Jr. and J.L.C. Sanz, "Image reconstruction from frequency-offset Fourier data," Proc. IEEE 72, 661-669 (1984) [CrossRef]
  18. C.L. Matson, I.A. Delarue, T.M. Gray, and I.E. Drunzer, "Optimal Fourier spectrum estimation from the bispectrum," Computers Elect. Engng 18, 485-497 (1992) [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Figure 1
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited