Abstract

The reconstruction problem in diffuse optical tomography can be formulated as an optimization problem, in which an objective function has to be minimized. Current model-based iterative image reconstruction schemes commonly use information about the gradient of the objective function to locate the minimum. These gradient-based search algorithms often find local minima close to an initial guess, or do not converge if the gradient is very small. If the initial guess is too far from the solution, gradient-based schemes prove inefficient for finding the global minimum. In this work we introduce evolution-strategy (ES) algorithms for diffuse optical tomography. These algorithms seek to find the global minimum and are less sensitive to initial guesses and regions with small gradients. We illustrate the fundamental concepts by comparing the performance of gradient-based schemes and ES algorithms in finding optical properties (absorption coefficient µa, scattering coefficient µs, and anisotropy factor g) of a homogenous medium.

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  2. Optical Tomography and Spectroscopy of Tissue II: Theory, Instrumentation, Model, and Human Studies, B. Chance, R.R. Alfano, A. Katzir, eds. Proc. of the SPIE Vol. 2979 (1997).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. R. Roy and E.M. Sevick-Muraca, "Truncated Newton's optimization scheme for absorption and fluorescence optical tomography: Part I Theory and formulation," Opt. Express 4, 353-371 (1999). http://www.opticsexpress.org/oearchive/source/9268.htm
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    [CrossRef]
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    [CrossRef] [PubMed]
  29. H.J. van Staveren, C.J.M. Moes, J. van Marle, S.A. Prahl, M.J.C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 30, 4507-4514 (1991).
    [CrossRef] [PubMed]
  30. S.T. Flock, S.L. Jacques, B.C. Wilson, W.M. Star, M.J.C. van Gemert, "Optical properties of Intralipid: A phantom medium for light propagation studies," Lasers Surg. Med. 12, 510-519 (1992).
    [CrossRef] [PubMed]
  31. A.J. Welch, M.J.C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue, (New York, NY, Plenum Press 1995).
  32. D.A. Pierre, Optimization Theory with Applications (Mineola, NY, Dover Publication 1986).
  33. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, (New York, NY Cambridge University Press 1992),pp. 420-425.
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    [CrossRef]
  35. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, (New York, NY Cambridge University Press 1992), (pp. 288-290).

Other (35)

Optical Tomography and Spectroscopy of Tissue III, B. Chance, R.R. Alfano, B. J. Tromberg, eds. Proc. of the SPIE Vol. 3597 (1999).

Optical Tomography and Spectroscopy of Tissue II: Theory, Instrumentation, Model, and Human Studies, B. Chance, R.R. Alfano, A. Katzir, eds. Proc. of the SPIE Vol. 2979 (1997).

Optical Tomography, Photon Migration, and Spectroscopy of tissue and Model MediaI: Theory, human Studies, and Instrumentation, B. Chance, R.R. Alfano, A. Katzir, eds. Proc. of the SPIE Vol. 2389 (1995).

K. Wells, J.C. Hebden. F.E.W. Schmidt, and D.T. Delpy, " The UCL multichannel time-resolved system for optical tomography", in Optical Tomograpgy and Spectroscopy of Tissue, B. Chance and R.R. Alfano, Eds., Proc. of the SPIE Vol. 2979, pp. 599-607 (1997).

J.P. Vanhouten, D.A. Benaron, S. Spilman, and D.K. Stevenson, "Imaging brain injury using time-resolved near-infrared light scanning," Pediatric Research 39, 470-476 (1996).
[CrossRef]

M. Miwa and Y. Ueda, "Development of time-resolved spectroscopy system for quantitative noninvasive tissue measurement," in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media, B. Chance and R.R. Alfano, eds., Proc. of the SPIE Vol. 2389, pp.142-149 (1995).

M.A. Franceschini, K.T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W.W. Mantulin, M. Seeber, P.M. Schlag, and M. Kaschke, "Frequency-domain techniques enhance optical mammography: initial clinical results," Proc. of the National Academy of Sciences of the USA, vol. 94, pp. 6468-647 (1997).
[CrossRef]

S.R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, pp. R41-R93 (1999).
[CrossRef]

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

R.L. Barbour, H.L. Graber, J.W. Chang, S.L. S. Barbour, P.C. Koo, R. Aronson, "MRI-guided optical tomography: Prospects and computation for a new imaging method," IEEE Computational Science & Engineering 2, pp. 63-77 (1995).
[CrossRef]

Y.Q. Yao, Y. Wang, Y.L. Pei, W. W. Zhu, R.L. Barbour, " Frequency-domain optical imaging of absorption and scattering distrinbutions by Born iterative method," J. Opt. Soc. Am. A 14, pp. 325-342 (1997).
[CrossRef]

K.D. Paulsen and H. Jiang, "Enhanced frequency domain optical image reconstruction in tissues through total variation minimization," Appl. Opt. 35, pp. 3447-3458 (1996).
[CrossRef] [PubMed]

B.W. Pogue, T.O. McBride, J. Prewitt, U.L. Osterberg, K.D. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, 2950-2961 (1999).
[CrossRef]

H. Jiang, K.D. Paulsen, and U.L. �sterberg, "Optical image reconstruction using DC data: simulations and experiments," Phys. Med. Biol. 41, pp.1483-1498 (1996).
[CrossRef] [PubMed]

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

D.Y. Paithankar, A.U. Chen, B.W. Pogue, M.S. Patterson, and E.M. Sevick-Muraca, "Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media," Appl. Opt. 36, pp. 2260-2272 (1997).
[CrossRef] [PubMed]

M.V. Klibanov, T.R. Lucas, and R.M. Frank, "A fast and accurate imaging algorithm in optical diffusion tomography," Inverse Problems 13, 1341-1361, 1997.
[CrossRef]

O. Dorn,"A transport-backtransport method for optical tomography," Inverse Problems 14,1107-1130 (1998).
[CrossRef]

A.D. Klose and A.H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Medical Physics 26, 1698-1707 (1999).
[CrossRef] [PubMed]

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, "Model-based image reconstruction from time-resolved diffusion data," in Medical Imaging: Image Processing, Proc. of the SPIE Vol. 3034, pp. 369-380, 1997.

S.R. Arridge and M. Schweiger, "A gradient-based optimisation scheme for optical tomography," Opt. Express 2, 213-226 (1998). http://www.opticsexpress.org/oearchive/source/4014.htm
[CrossRef] [PubMed]

A.H. Hielscher, A.D. Klose, K.M. Hanson, "Gradient-based iterative image reconstruction scheme for time-resolved optical tomography," IEEE Transactions on Medical Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

R. Roy and E.M. Sevick-Muraca, "Truncated Newton's optimization scheme for absorption and fluorescence optical tomography: Part I Theory and formulation," Opt. Express 4, 353-371 (1999). http://www.opticsexpress.org/oearchive/source/9268.htm
[CrossRef] [PubMed]

H.P. Schwefel, Evolution and Optimum Seeking, (John Wiley & Sons, New York, NY 1995).

T. B�ck, H.P. Schwefel, "An overview of evolutionary algorithms for parameter optimization," Evolutionary Computation 1, 1-23 1993.
[CrossRef]

Z. Michlewicz, Genetic Algorithms + Data Structures = Evolution Programs, (Springer, New York, NY 1999).

W.M. Star, J.P.A. Marijnissen, H. Jansen, M. Keijzer, M.J.C. van Gemert, "Light dosimetry for photodynamic therapy by whole bladder wall irradiation," Photchem. Photobiol. 46, 619-624 (1987).
[CrossRef]

C.J.M. Moes, M.J.C. van Gemert, W.M. Star, J.P.A. Marijnissen, S.A. Prahl, "Measurements and calculations of the energy fluence rate in a scattering and absorbing phantom at 633 nm," Appl. Opt. 28, 2292-2296 (1989).
[CrossRef] [PubMed]

H.J. van Staveren, C.J.M. Moes, J. van Marle, S.A. Prahl, M.J.C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 30, 4507-4514 (1991).
[CrossRef] [PubMed]

S.T. Flock, S.L. Jacques, B.C. Wilson, W.M. Star, M.J.C. van Gemert, "Optical properties of Intralipid: A phantom medium for light propagation studies," Lasers Surg. Med. 12, 510-519 (1992).
[CrossRef] [PubMed]

A.J. Welch, M.J.C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue, (New York, NY, Plenum Press 1995).

D.A. Pierre, Optimization Theory with Applications (Mineola, NY, Dover Publication 1986).

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, (New York, NY Cambridge University Press 1992),pp. 420-425.

G. Schneider, J. Schuchhardt, P.Wrede, "Evolutionary optimization in multimodal space," Biological Cybernetics 74, pp. 203-207 (1996).
[CrossRef]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, (New York, NY Cambridge University Press 1992), (pp. 288-290).

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Figures (8)

Fig. 1.
Fig. 1.

Basic scheme of the evolution strategy employed in this work. The parent population is initialized by randomly choosing three parents, p1, p2, p3, each with different optical properties µa , µs , and g. During the recombination process 12 parent pairs (pi,pj)1 to (pi,pj)12 are randomly generated. For each pair the average optical properties are calculated (Eq. 5 recombination rule) and one child (c1 to c12) is produced by mutation in which optical properties are randomly drawn from a Gauss distribution centered at the average optical properties of the parents (Eq. 6). In the selection process the fittest members of the offspring population are determined by evaluating the objective function for each child (Eq. 2). The three children with the lowest objective function become the three new parents for the next iteration.

Fig. 2.
Fig. 2.

Comparison of gradient-based reconstruction algorithms (a-left) and evolution-strategy algorithms (b-right) for diffuse optical tomography. The lower index k indicates different updates and generations. The upper indices n and m indicate different parents and children.

Fig. 3.
Fig. 3.

Setup and measurement geometry for the numerical model. Synthetic data were generated for 2 sources (open circles) and 32 detectors (gray circles) with a time-independent transport-theory-based algorithm.

Fig. 4.
Fig. 4.

Contour-map representations of 3-dimensional slices through the 4-dimensional objective function (Eq. 2). Slices are taken for different g values, which results in Log(Φ) versus µa and µs maps. The minimum of Log(Φ) is at µa =0.6 cm-1, µs =6.0 cm-1, g=0.5. Note that the x-axis is chosen so that the reduced scattering coefficient µ′s =(1-g) µs ranges from 1 to 6 cm-1 in all figures. It can be seen that many combinations of µa , µs , g lead to similar low values in the log(Φ).

Fig. 5.
Fig. 5.

Objective function (Eq. 2) as a function of outer iterations (conjugate-gradient scheme) and number of generations (evolution-strategy scheme). For the evolution algorithm only the member with the smallest objective function for each generation is shown. The time required to complete all calculations for one generation equals the time it takes to complete one outer iteration within ±10%.

Fig. 6.
Fig. 6.

Three-dimensional depiction of objective function Φ(see Fig. 4c). The µa -axis has been stretched by a factor of 10. Open circles indicate values of Φ for outer iterations during the gradient-based algorithm starting from (µa =0.2 cm-1, µs =4 cm-1), circles with a line show values of Φ starting from (µa =0.8 cm-1, µs =7 cm-1), circles with a cross show values of Φ starting from (µa =0.4 cm-1, µs =10 cm-1). The diamonds show sampling points during the ES algorithms.

Fig. 7.
Fig. 7.

Same as Fig. 6 and Fig. 4c, only displayed as a 2-dimensional map and without a stretched µa -axis. The minimum is located at (µa =0.6 cm-1, µs =6.0 cm-1).

Fig.8: .
Fig.8: .

alues of the objective function at the bottom of the elongated, curved (blue) valley in Fig. 6. See also blue valleys in Fig. 4c and 7. Note that in Fig. 4c, 6, and 7 Log(Φ) is displayed while here Φ is plotted. It can be seen that the gradient of the objective function is much steeper to the left of the minimum at µs =6 cm-2 than to the right of the minimum

Equations (7)

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

Φ ( ξ ) = χ 2 ( ξ ) Σ s Σ d ( M s , d P s , d ( ξ ) ) 2 2 η s , d 2
p ( cos θ ) = 1 g 2 2 ( 1 + g 2 2 g cos θ ) 3 / 2 ,
ξ k + 1 = ξ k + α · d k
d k + 1 = γ k + 1 + γ k + 1 T ( γ k + 1 γ k ) γ k 2 d k
ξ i new = ( ξ i k 1 + ξ i k 2 ) / 2
ξ i new = ξ i new * + σ i y for i { 1 , 2 , 3 } ,
y = 2 log ( ν 1 ) sin ( 2 πν 2 )

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