The Green's function for the three-dimensional linear Boltzmann equation via Fourier transform

Abstract

The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering.

Light transport in homogeneous tissue with m-dependent anisotropic scattering II: Fourier transform solution and consistent relations

“…In the complex plane, is a Legendre orthogonal polynomial [ 20 ] and is defined as [ 20 – 22 ] and satisfies the Eq. (2) with initial terms [ 17 ] Because and are polynomials in , they are analytic in the complex plane.…”

“…However, in our research we usually use the following definition where is not on the cut line −1 to 1. Similarly, we define the associated Legendre function of the second kind as [ 17 , 20 ] where is in the complex plane with a cut along the interval on the real axis. From Eq.…”

“…From Eq. (6) , we can get the three-term recurrence for with initial terms [ 17 ] where is the Kronecker delta, for , and otherwise .…”

“…The CSEs solutions are always seen as the exact solutions of integro-differential transport equations and naturally suitable for the light transport in turbid tissue [ 1 – 9 ]. Fourier transform method is a useful method and applied successfully to the transport equations in the regular space [ 10 – 17 ]. Ganapol [ 12 – 16 ] derived the Fourier transform solution for the infinite medium with azimuthal symmetric scattering kernel and showed the consistent theory in a more mathematically unified manner.…”

“…Ganapol [ 12 – 16 ] derived the Fourier transform solution for the infinite medium with azimuthal symmetric scattering kernel and showed the consistent theory in a more mathematically unified manner. Machida [ 17 ] used the Fourier transform and rotated reference frames technique to compute three-dimension Green’s function for three-dimension linear Boltzmann equation. Their outstanding work opened up a new scope to solve the exact radiance distributions that we are seeking for the light transport in biological tissue.…”