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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 2922–2927
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Eigen decomposition solution to the one-dimensional time-dependent photon transport equation

Chintha C. Handapangoda, Pubudu N. Pathirana, and Malin Premaratne  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 2922-2927 (2011)
http://dx.doi.org/10.1364/OE.19.002922


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Abstract

The time-dependent one-dimensional photon transport (radiative transfer) equation is widely used to model light propagation through turbid media with a slab geometry, in a vast number of disciplines. Several numerical and semi-analytical techniques are available to accurately solve this equation. In this work we propose a novel efficient solution technique based on eigen decomposition of the vectorized version of the photon transport equation. Using clever transformations, the four variable integro-differential equation is reduced to a set of first order ordinary differential equations using a combination of a spectral method and the discrete ordinates method. An eigen decomposition approach is then utilized to obtain the closed-form solution of this reduced set of ordinary differential equations.

© 2011 Optical Society of America

1. Introduction

A number of models for solving the steady state (time-independent) PTE has been developed over the past four decades [4

4. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988). [CrossRef] [PubMed]

, 5

5. J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989). [CrossRef]

], ignoring the time dependency of the intensity profile. Some of these models include the discrete ordinates method [4

4. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988). [CrossRef] [PubMed]

], integral transformation techniques and the FN method [6

6. C. E. Siewert and J. R. Thomas, “Radiative transfer calculations in spheres and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 34, 59–64 (1985). [CrossRef]

]. Siewert [7

7. C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157–160 (1994). [CrossRef]

] and Larsen et al. [8

8. E. W. Larsen, “The inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 15, 1–5 (1975). [CrossRef]

] have worked on the inverse-source problem where the source term is determined from the known angular distribution of radiation that exits the surface. Pomraning et al. [9

9. G. C. Pomraning and G. M. Foglesong, “Transport-diffusion interfaces in radiative transfer,” J. Comput. Phys. 32, 420–436 (1979). [CrossRef]

] have coupled the diffusion approximation and the radiative transfer equation for boundary treatment for the steady state case.

With the advent of short laser pulses that are applied in various different disciplines, researchers have started developing techniques to solve the transient (time-dependent) PTE. Tan and Hsu [10

10. Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123, 466–475 (2001). [CrossRef]

] developed an integral equation formulation to treat the general transient PTE, and, Fleck used the Monte Carlo simulations [11

11. J. A. Fleck, “The calculation of nonlinear radiation transport by a Monte Carlo method,” Methods Comput. Phys. 1, 43–65 (1963).

]. Kim, Ishimaru and Moscoso have applied Chebyshev collocation techniques for solving the PTE [12

12. A. D. Kim and A. Ishimaru, “A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in discrete random media,” J. Comput. Phys. 152, 264–280 (1999). [CrossRef]

, 13

13. A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002). [CrossRef]

]. Kim [13

13. A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002). [CrossRef]

] has used Chebyshev expansion for the spatial discretization and Crank-Nicholson method for time marching. Handapangoda et al. [14

14. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008). [CrossRef]

] proposed a technique to transform the four-variable transient PTE to a set of coupled first order ordinary differential equations. However, in that work they solved this set of equations using the Runge-Kutta-Fehlberg method, which is a numerical technique. In this paper we propose an improvement to this technique, which gives a closed-form solution to the transformed PTE based on eigen decomposition. This approach improves the computational efficiency significantly as shown later.

This paper is organized as follows. In section 2 we present the eigen decomposition method to solve the reduced vectorized version of the PTE. Section 3 presents some numerical results. A comparison of the output values and the execution time of the algorithm is carried out against a those of the standard solution, which is obtained using the Laguerre Runge-Kutta-Fehlberg (LRKF) method [14

14. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008). [CrossRef]

]. Section 4 carries some concluding remarks.

2. Eigen decomposition method

The one dimensional transient photon transport equation for a medium without any internal sources is given by [14

14. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008). [CrossRef]

, 15

15. A. D. Kim and M. Moscoso, “Chebyshev spectral methods fro radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002). [CrossRef]

]
1vtI(z,u,ϕ,t)+uzI(z,u,ϕ,t)σs4π02π11P(u,ϕ;u,ϕ)I(z,u,ϕ,t)dudϕ+σtI(z,u,ϕ,t)=0,
(1)
where I (z, u, ϕ, t) is the light intensity (radiance), (z, θ,ϕ) are the standard spherical coordinates, u = cosθ, t denotes time, σt and σs are attenuation and scattering coefficients, respectively. The speed of light in the medium is denoted by v and P (u′,ϕ′; u,ϕ) is the phase function.

Suppose Eq. (1) is subject to the boundary condition I (z = 0, u0, ϕ0, t) = f (t)δ(uu0)δ(ϕϕ0), where δ(x) is the Dirac delta function [16

16. A. B. Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication (McGraw-Hill, 1986).

] and f (t) is the temporal profile of the incident pulse. Using the transformation
τ=tzvu
(2)
which maps Eq. (1) to a moving reference frame with the pulse, we get
uzI(z,u,ϕ,τ)+σtI(z,u,ϕ,τ)σs4π02π11P(u,ϕ;u,ϕ)I(z,u,ϕ,τ)dudϕ=0,
(3)
subject to I (z = 0, u, ϕ,τ) = f (τ)δ(uu0)δ(ϕϕ0). Use of Eq. (2) in Eq. (1) eliminates the time derivative term in Eq. (1).

Then, the azimuthal angle, ϕ, can be discretized using the discrete ordinates method [4

4. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988). [CrossRef] [PubMed]

,17

17. S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).

,18

18. K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981). [CrossRef]

] by applying the Gaussian quadrature rule [19

19. W. H. Press, S. A. Teukolsk, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge University Press, 2003).

] for the integral that corresponds to ϕ.

Gaussian quadrature [19

19. W. H. Press, S. A. Teukolsk, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge University Press, 2003).

] is used to approximate an integral of the form abW(x)f(x)dx, where W (x) is the weight function and f (x) is any arbitrary function, by a summation such that abW(x)f(x)dxj=0N1wjf(xj).

Application of the discrete ordinates method to the azimuthal angle of Eq. (3) will result in a set of uncoupled equations
uzI(z,u,ϕr,τ)+σtI(z,u,ϕr,τ)σs4πj=1Lwjϕ11P(u,ϕj;u,ϕr)I(z,u,ϕj,τ)du=0,
(4)
where r = 1, . . . , L, ϕj is the jth quadrature point of ϕ and wjϕ is the corresponding Gaussian weight. We then discretize the cosine of the zenith angle, u, of Eq. (4) using the discrete ordinates method, which results in
uizI(z,ui,ϕr,τ)+σtI(z,ui,ϕr,τ)σs4πj=1Lwjϕk=1KwkuP(uk,ϕj;ui,ϕr)I(z,uk,ϕj,τ)=0,
(5)
where i = 1, . . . , K, r = 1, . . . , L, uk is the kth quadrature point of u and wku is the corresponding Gaussian weight.

In order to remove the dependency on τ, we can expand I (z, ui, ϕr, τ) and f (τ) using Laguerre polynomials with respect to τ, such that I(z,ui,ϕr,τ)k=0NBk(z,ui,ϕr)Lk(τ) and f(τ)k=0NFkLk(τ) where Bk and Fk are the coefficients corresponding to the Laguerre polynomial Lk (τ). Expanding the transformed one-dimensional PTE, Eq. (5) and the boundary condition using a Laguerre basis and taking moments we get,
uizBn(z,ui,ϕr)+σtBn(z,ui,ϕr)σs4πj=1Lwjϕk=1KwkuP(uk,ϕj;ui,ϕr)Bn(z,uk,ϕj)=0,
(6)
subject to
Bn(z=0,ui,ϕr)=Fnδ(uiu0)δ(ϕrϕ0),
(7)
where n = 1, . . . , N, i = 1, . . . , K and r = 1, . . . , L. Thus, there are K ×L coupled equations for each Laguerre coefficient, Bn and Eq. (6) corresponds to the ith discrete ordinate of u and rth discrete ordinate of ϕ. We can write this set of equations in matrix form as;
ddzABn+σtBnσs4πPWBn=0,
(8)
where Bn = [Bn (z, ui, ϕr)]K×L,1, P = [P(uk, ϕj; ui, ϕr)]K×L,K×L, and, A is a (K × L) by (K × L) diagonal matrix with diagonal elements u1 to uK repeating L times. The matrix W is also a (K × L) by (K × L) diagonal matrix with diagonal elements wrϕ×wku with the pattern w1ϕ×w1u, w1ϕ×w2u, . . . , w1ϕ×wKu, w2ϕ×w1u, . . . , wLϕ×wKu.

Rearranging, Eq. (8) can be written as
ddzBn=YBn,
(9)
where Y=A1[σs4πPWσtI]and I denotes the identity matrix. Hence, the original transient PTE is reduced to a one-variable ordinary differential equation. The boundary condition given by Eq. (7) can be decomposed into a set of values on the boundary corresponding to each discrete ordinate, which can be represented in a column matrix as Bn0=[Bn(z=0,ui,ϕr)]K×L,1.

An explicit solution to Eq. (9) can be found using eigen decomposition. The square matrix Y can be written as a product of three matrices composed of its eigen values and eigen vectors, as follows.
Y=VDV1,
(10)
where V = [v1 v2 . . . vm]. vi is the ith eigen vector of Y and D is a diagonal matrix with its ith diagonal element equal to the ith eigen value, λi, of Y. Using Eq. (10) in Eq. (9) we get
ddzBn=YBn=VDV1BnV1ddzBn=DV1BnddzX=DX
(11)
where
X=V1Bn.
(12)
Since D in Eq. (11) is a diagonal matrix, each row of Eq. (11) can be solved independently resulting in
X=(x1x2xm)=(x10eλ1zx20eλ2zxm0eλmz).
(13)
The column matrix X0=V1Bn0 is composed of the boundary values x10 to xm0. Once X is evaluated using Eq. (13), the solution to Eq. (9) can be found using the transformation in Eq. (12). That is, Bn = VX. Thus, the explicit solution to Eq. (9) is given by
Bn=x10eλ1zv1+x20eλ2zv2++xm0eλmzvm.
Without loss of generality, here we have assumed that the eigen values of matrix Y are real and distinct, which is the case for most of the widely used phase functions in practice. However, if for a particular application the eigen values and eigen vectors contain complex values (in the form of complex conjugate pairs), a real solution to Eq. (9) can be obtained as illustrated in [20

20. C. H. Edwards and D. E. Penney, Differential Equations: Computing and Modeling, 3rd ed. (Prentice Hall, 2004).

]. For example, suppose λi = p + qi and λj = pqi are two complex eigen values and vi = a + bi and vj = abi are their corresponding complex eigen vectors. It can be shown that we can then obtain two real valued solutions to Eq. (9), wi = epz(a cos qzb sin qz) and wj = epz(b cos qz + a sin qz) [20

20. C. H. Edwards and D. E. Penney, Differential Equations: Computing and Modeling, 3rd ed. (Prentice Hall, 2004).

]. The general solution to Eq. (9) then becomes
Bn=c1eλ1zv1+c2eλ2zv2++ciwi+cjwj++cmeλmzvm.
(14)
Equation (14) can be written in matrix notation as
Bn=VC,
(15)
where V = [eλ1zv1 eλ2zv2 . . . wi wj . . . eλmzvm] and C is a column matrix with its ith element equal to ci. The constants in C can be found using Eq. (15) (ie. Cinv(V)Bn0).

For the case of repeated eigen values, the general solution to Eq. (9) will be of the form [20

20. C. H. Edwards and D. E. Penney, Differential Equations: Computing and Modeling, 3rd ed. (Prentice Hall, 2004).

]
Bn=c1eλ1zv1+c2eλ2zv2++cieλizvi+ci+1eλiz(viz+vi+1)+ci+2eλiz(12viz2+vi+1z+vi+2)++cmeλmzvm,
where the eigen value λi is repeated 3 times and the corresponding generalized eigenvectors, vi, vi+1 and vi+2 are found such that (YλiI)2 vi+2 = 0, (YλiI) vi+2 = vi+1, (YλiI) vi+1 = vi and (YλiI) vi = 0 as illustrated in [20

20. C. H. Edwards and D. E. Penney, Differential Equations: Computing and Modeling, 3rd ed. (Prentice Hall, 2004).

].

3. Results and discussion

Figure 1 shows a comparison of irradiance profiles obtained from the work proposed in this paper (Eigen method) and the standard solution (obtained using the LRKF method [14

14. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008). [CrossRef]

]), at the exit plane of a layer of a human skin specimen of 1 mm thickness, illuminated by a Gaussian pulse of I0 intensity (of arbitrary units). The Henyey-Greenstein phase function with an asymmetry factor of 0.7 together with σa = 0.5 mm−1, σs = 0.8 mm−1, v = 0.215 mm/ps and T = 1.5 was used for this simulation. The results produced by the two methods for the same number of discrete ordinates were exactly the same as shown by Fig. 1.

Fig. 1 Irradiance profile transmitted from a human skin specimen

In the proposed algorithm numerical errors occur due to the truncation of the Laguerre series and due to the finite number of discrete ordinates chosen for θ and ϕ. However, as discussed in detail in [21

21. C. C. Handapangoda and M. Premaratne, “Implicitly causality enforced solution of multidimensional transient photon transport equation,” Opt. Express 17, 23423–23442 (2009). [CrossRef]

] and [14

14. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008). [CrossRef]

], it is possible to obtain a very accurate Laguerre fit, in the observation window, with only 63 terms for a Gaussian pulse with an arbitrary width by using a suitable scaling factor. Hence, we can consider that the numerical errors are introduced only due to the discretization of the zenith and azimuthal angles. Figure 2 shows a comparison of execution times of the Eigen method proposed in this paper with the standard solution. The rate of increase of the execution time with the number of discrete ordinates is significantly low for the Eigen method compared to the standard solution.

Fig. 2 A comparision of execution times for different numbers of directions

4. Conclusion

From the simulation results it can be concluded that the proposed algorithm is accurate to the same degree compared to the standard solution which is based on a numerical solution to the reduced vectorized version of the PTE. However, the proposed algorithm is significantly faster in execution time compared to the standard solution method. For example, increasing the number of directions from 25 to 196 increased the execution time of the proposed algorithm by only about 11 seconds (6.4%) where as that of the standard method was increased by about 166 seconds (97.0%).

References and links

1.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005). [CrossRef]

2.

N. Y. Gnedin, “Multi-dimensional cosmological radiative transfer with a variable Eddington tensor formalism,” New Astron. 6, 437–455 (2001). [CrossRef]

3.

D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41–59 (1992). [CrossRef]

4.

K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988). [CrossRef] [PubMed]

5.

J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989). [CrossRef]

6.

C. E. Siewert and J. R. Thomas, “Radiative transfer calculations in spheres and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 34, 59–64 (1985). [CrossRef]

7.

C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157–160 (1994). [CrossRef]

8.

E. W. Larsen, “The inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 15, 1–5 (1975). [CrossRef]

9.

G. C. Pomraning and G. M. Foglesong, “Transport-diffusion interfaces in radiative transfer,” J. Comput. Phys. 32, 420–436 (1979). [CrossRef]

10.

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123, 466–475 (2001). [CrossRef]

11.

J. A. Fleck, “The calculation of nonlinear radiation transport by a Monte Carlo method,” Methods Comput. Phys. 1, 43–65 (1963).

12.

A. D. Kim and A. Ishimaru, “A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in discrete random media,” J. Comput. Phys. 152, 264–280 (1999). [CrossRef]

13.

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002). [CrossRef]

14.

C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008). [CrossRef]

15.

A. D. Kim and M. Moscoso, “Chebyshev spectral methods fro radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002). [CrossRef]

16.

A. B. Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication (McGraw-Hill, 1986).

17.

S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).

18.

K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981). [CrossRef]

19.

W. H. Press, S. A. Teukolsk, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge University Press, 2003).

20.

C. H. Edwards and D. E. Penney, Differential Equations: Computing and Modeling, 3rd ed. (Prentice Hall, 2004).

21.

C. C. Handapangoda and M. Premaratne, “Implicitly causality enforced solution of multidimensional transient photon transport equation,” Opt. Express 17, 23423–23442 (2009). [CrossRef]

OCIS Codes
(080.2720) Geometric optics : Mathematical methods (general)
(290.7050) Scattering : Turbid media
(010.5620) Atmospheric and oceanic optics : Radiative transfer

ToC Category:
Scattering

History
Original Manuscript: October 25, 2010
Revised Manuscript: January 27, 2011
Manuscript Accepted: January 28, 2011
Published: February 1, 2011

Virtual Issues
Vol. 6, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Chintha C. Handapangoda, Pubudu N. Pathirana, and Malin Premaratne, "Eigen decomposition solution to the one-dimensional time-dependent photon transport equation," Opt. Express 19, 2922-2927 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-2922


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References

  1. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345 (2005). [CrossRef]
  2. N. Y. Gnedin, “Multi-dimensional cosmological radiative transfer with a variable Eddington tensor formalism,” N. Astron. 6, 437–455 (2001). [CrossRef]
  3. D. M. O’Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41–59 (1992). [CrossRef]
  4. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinatemethod radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988). [CrossRef] [PubMed]
  5. J. L. Deuz, M. Herman, and R. Santer, “Fourier series expansion of the transfer equation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 41, 483–494 (1989). [CrossRef]
  6. C. E. Siewert and J. R. Thomas, “Radiative transfer calculations in spheres and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 34, 59–64 (1985). [CrossRef]
  7. C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157–160 (1994). [CrossRef]
  8. E. W. Larsen, “The inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 15, 1–5 (1975). [CrossRef]
  9. G. C. Pomraning and G. M. Foglesong, “Transport-diffusion interfaces in radiative transfer,” J. Comput. Phys. 32, 420–436 (1979). [CrossRef]
  10. Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123, 466–475 (2001). [CrossRef]
  11. J. A. Fleck, “The calculation of nonlinear radiation transport by a Monte Carlo method,” Methods Comput. Phys. 1, 43–65 (1963).
  12. A. D. Kim and A. Ishimaru, “A Chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in discrete random media,” J. Comput. Phys. 152, 264–280 (1999). [CrossRef]
  13. A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002). [CrossRef]
  14. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. 14, 105–112 (2008). [CrossRef]
  15. A. D. Kim and M. Moscoso, “Chebyshev spectral methods fro radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074–2094 (2002). [CrossRef]
  16. A. B. Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication (McGraw-Hill, 1986).
  17. S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).
  18. K. Stamnes and R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981). [CrossRef]
  19. W. H. Press, S. A. Teukolsk, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + +, 2nd ed. (Cambridge University Press, 2003).
  20. C. H. Edwards and D. E. Penney, Differential Equations: Computing andModeling, 3rd ed. (Prentice Hall, 2004).
  21. C. C. Handapangoda and M. Premaratne, “Implicitly causality enforced solution of multidimensional transient photon transport equation,” Opt. Express 17, 23423–23442 (2009). [CrossRef]

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