Open Access Publications from the University of California

## Spectral Methods For Solving the Radiative Transport Equation in Single and Double Spherical Harmonics and Their Application to Optical Imaging

• Author(s): Horan, Sean Thomas
Mathematical models of light propagation in turbid media are integral components of many optical imaging modalities. The radiative transport equation is a principal model and is commonly used to describe the behavior of light transport at distances larger than the average scattering length of light in a medium. Current deterministic methods tend to be computationally inexpensive but either do not accurately recreate scattered radiance in layered media. However, these methods are sufficient to obtain functionals of radiance such as fluence and reflectance, or only represent them for certain optical properties and at low ($\leq 0.1/l^*$) spatial frequencies. Stochastic methods are capable of higher degrees of accuracy but are often cumbersome to compute. I present a novel deterministic spectral method for solving the Radiative Transport Equation, based on double spherical harmonic functions, which is capable of accurate reconstructions of scattered radiance and is more robust to changes in spatial frequency. It is provides accurate reconstructions of radiance and various useful functionals thereof at much higher spatial frequencies than current best practices. I demonstrate both theory and MATLab implementation for homogeneous as well as layered media, and present a staged inversion method for the recovery of optical properties from layered media using the method of (single) spherical harmonic expansion upon which my proposed double spherical harmonic approach is based. This inversion technique may be applied to any solution method for the Radiative Transport Equation.