On a family of differential approximations of the radiative transfer equation

Han, Weimin; Eichholz, Joseph; Wang, Ge

doi:10.1007/s10910-011-9916-2

Cited by 12 publications

(12 citation statements)

References 23 publications

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“…Recently, there is much interest in analysis and numerical simulation of the RTE and its related inverse problems, motivated by applications in biomedical optics ([2, 3, 5, 11, 12, 13, 14, 15, 24, 26]).…”

Section: Introductionmentioning

confidence: 99%

Radiative transfer with delta-Eddington-type phase functions

Han

Long

Cong

et al. 2017

Applied Mathematics and Computation

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The radiative transfer equation (RTE) arises in a wide variety of applications, in particular, in biomedical imaging applications associated with the propagation of light through the biological tissue. However, highly forward-peaked scattering feature in a biological medium makes it very challenging to numerically solve the RTE problem accurately. One idea to overcome the difficulty associated with the highly forward-peaked scattering is through the use of a delta-Eddington phase function. This paper is devoted to an RTE framework with a family of delta-Eddington-type phase functions. Significance in biomedical imaging applications of the RTE with delta-Eddington-type phase functions are explained. Mathematical studies of the problems include solution existence, uniqueness, and continuous dependence on the problem data: the inflow boundary value, the source function, the absorption coefficient, and the scattering coefficient. Numerical results are presented to show that employing a delta-Eddington-type phase function with properly chosen parameters provides accurate simulation results for light propagation within highly forward-peaked scattering media.

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Section: Introductionmentioning

confidence: 99%

Radiative transfer with delta-Eddington-type phase functions

Han

Long

Cong

et al. 2017

Applied Mathematics and Computation

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“…Also, the RTE has a scattering integral of the light intensity multiplied by the phase function over the whole directions, which expresses the energy incoming to the position of interest from all the directions by scattering. Because analytical solutions of the RTE are obtained for the limited cases of simple geometries such as infinite and homogeneous media [22], the RTE is approximated to simpler formulations like the simplified spherical harmonics system [26,12] or numerically solved for complicated geometries of organs and tissue volumes. In the numerical calculation of the RTE, the independent variables (position, direction, and time) are discretized by numerical schemes.…”

Section: Introductionmentioning

confidence: 99%

Renormalization of the highly forward-peaked phase function using the double exponential formula for radiative transfer

et al. 2016

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Numerical calculation of photon migration in biological tissue using the radiative transfer equation (RTE) has attracted great interests in biomedical optics and imaging. Because biological tissue is a highly forward-peaked scattering medium, renormalization of the phase function in numerical calculation of the RTE is crucial. This paper proposes a simple approach of renormalizing the phase function by the double exponential formula, which was heuristically modified from the original one. Firstly, the validity of the proposed approach was tested by comparing numerical results for an average cosine of the polar scattering angle calculated by the proposed approach with those by the conventional approach in highly forward-peaked scattering. The results show that calculation of the average cosine converged faster using the proposed approach than using the conventional one as a total number of discrete angular directions increases. Next, the accuracy of the numerical solutions of the RTE using the proposed approach was examined by comparing the numerical solutions with the analytical solutions of the RTE in a homogeneous medium with highly forward-peaked scattering. It was found that the proposed approach reduced the errors of the numerical solutions from those using the conventional one especially at a small value of the total number of the directions. This result suggests that the proposed approach has a possibility to

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“…For these reasons, various approximations of RTE have been proposed in the literature, e.g., the delta-Eddington approximation ( [25]), the Fokker-Planck approximation ( [39,40]), the Boltzmann-Fokker-Planck approximation ( [41,8]), the generalized Fokker-Planck approximation ( [27]), the Fokker-Planck-Eddington approximation and the generalized Fokker-Planck-Eddington approximation ( [16]). In [21] a family of differential approximations of the RTE(RT/DA) was proposed.…”

Section: List Of Figuresmentioning

confidence: 99%

“…In [20], a preliminary study of a family of differential approximations of the RTE (RT/DA) was provided. In this chapter, we give a mathematical study of the RT/DA equations, as well as numerical experiments on the corresponding inverse problems.…”

Section: Chapter 3 Differential Approximations and Their Applicationsmentioning

confidence: 99%

“…The idea of the derivation of the RT/DA equations is based on the approximation of the integral operator S by a sequence of linear combinations of the inverse of linear elliptic differential operators on the unit sphere ( [20]). The point of departure of the approach is the knowledge of eigenvalues and eigenfunctions of the operator S. Specifically, for a spherical harmonic of order n, Y n (ω) (cf.…”

Section: Differential Approximation Of the Integral Operatormentioning

confidence: 99%

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Some Approximations to the Radiative Transfer Equation and Their Applications

Sheng¹

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On a family of differential approximations of the radiative transfer equation

Cited by 12 publications

References 23 publications

Radiative transfer with delta-Eddington-type phase functions

Radiative transfer with delta-Eddington-type phase functions

Renormalization of the highly forward-peaked phase function using the double exponential formula for radiative transfer

Some Approximations to the Radiative Transfer Equation and Their Applications

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