Abstract
This paper is the first of two deriving the analytical solutions for light transport in infinite homogeneous tissue with an azimuth-dependent (m-dependent) anisotropic scattering kernel by two approaches, Case’s singular eigenfuncions expansion and Fourier transform, as well as proving the consistence of the two solutions. In this paper, Case’s method was applied and extended to the general m-dependent anisotropic scattering case. The explicit Green’s function of radiance distributions, which was regarded as the comparative standard for the equivalent solution via Fourier transform and inversion in our second accompanying paper, was expanded into a complete set of the discrete and continuous eigenfunctions. Considering that the two kinds of m-dependent Chandrasekhar orthogonal polynomials that play vital roles in these analytical solutions are very sensitive to the typical optical parameters of biological tissue as well as the degrees or orders, four numerical evaluation methods were benchmarked to find the stable, reliable and feasible numerical evaluation methods in high degrees and high orders.
Highlights
Though for decades many approximation methods, such as the spherical harmonics ( PN ) [1, 2], the simplified spherical harmonics ( SPN ) [3,4,5,6], the diffusion equation ( DE ) [7,8,9,10], FN method [11, 12], etc., are applied to deal with the light transport equations and make great success, the traditional but effective Case’s singular eigenfunctions (CSEs) expansion solutions [13] are still regarded as the exact analytical solutions and as the criteria for testing the approximate methods
The Case’s method is used to find the radiance distributions of the light transport in the infinite homogeneous tissue with m-dependent anisotropic scattering kernel, and the final analytical solution is expanded by the contributions of the discrete eigenfunctions and the continuous singular eigenfunctions
In our second accompanying paper, we will apply the Fourier transform to obtain another analytical solution of the light transport equation and prove the consistence between the two solutions in a mathematical manner
Summary
Though for decades many approximation methods, such as the spherical harmonics ( PN ) [1, 2], the simplified spherical harmonics ( SPN ) [3,4,5,6], the diffusion equation ( DE ) [7,8,9,10], FN method [11, 12], etc., are applied to deal with the light transport equations and make great success, the traditional but effective Case’s singular eigenfunctions (CSEs) expansion solutions [13] are still regarded as the exact analytical solutions and as the criteria for testing the approximate methods. In our first paper, we firstly derived the Green’s function based on the CSEs expansion for the light transport in infinite homogeneous tissue with m-dependent anisotropic scattering kernel and made a comparative standard for the Fourier transform solution which is to be discussed in our second accompanying paper. Our numerical evaluation experiments showed that in biological optics, unlike the neutron transport domain or the extremely critical testing parameters, the Chandrasekhar orthogonal polynomials are very sensitive to the optical parameters of biological tissue, such as single particle albedo π and anisotropy factor g. We extended Ganapol’s methods in the case m = 0 to benchmark the m-dependent Chandrasekhar polynomials with the typical optical parameters of biological tissue to find the stable, reliable, feasible numerical evaluation methods in high degrees and high orders
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