A study of the radiative transfer equation using a spherical harmonics-nodal collocation method

Abstract

Optical tomography has found many medical applications that need to know how the photons interact with the different tissues. The majority of the photon transport simulations are done using the diffusion approximation, but this approximation has a limited validity when optical properties of the different tissues present large gradients, when structures near the photons source are studied or when anisotropic scattering has to be taken into account. As an alternative to the diffusion model, the P L equations for the radiative transfer problem are studied. These equations are discretized in a rectangular mesh using a nodal collocation method. The performance of this model is studied by solving different 1D and 2D benchmark problems of light propagation in tissue having media with isotropic and anisotropic scattering.

High-order spherical harmonics–nodal collocation scheme for the numerical solution of the time-dependent radiative transfer equation

“…The SHNC code was developed initially for neutron transport problems [16,17], and later adapted to solve external fixed source stationary problems in the field of optical tomography [28]. In this work, we have incorporated the time evolution and the treatment of semitransparent boundary conditions between media with different refractive index and optical properties.…”

“…2 shows the photon intensity (28) as a function of the distance z from the source. The P L solutions for L = 1, 3, 5, 7, are compared to the analytical solution (28), obtained by numerical integration using QUADPACK [24]. The P 1 approximation can not reproduce the exact solution of the intensity, while the P 3 , P 5 and P 7 solutions are very close to the exact solution.…”

Improved Models of Imaging of Skylight Polarization Through a Fisheye Lens

“…In addition, it can solve the vector radiation transfer equation (VRTE) for the construction of the skylight polarization pattern. This modeling can cover the matrix operator theory [19], the discrete ordinate theory [20], the spherical harmonics theory [21], and the multiple scattering theory [22]. From a computational point of view, this type of model is extremely complex and depends on the initial conditions.…”