Diffusion and Fractional Diffusion based Models for Multiple Light Scattering and Image Analysis

Abstract

This paper considers a fractional light diffusion model as an approach to characterizing the case when interme- diate scattering processes are present, i.e. the scattering regime is neither strong nor weak. In order to introduce the basis for this approach, we revisit the elements of formal scattering theory and the classical diffusion problem in terms of solutions to the inhomogeneous wave and diffusion equations respectively. We then address the significance of these equations in terms of a random walk model for multiple scattering. This leads to the proposition of a fractional diffusion equation for modelling intermediate strength scattering that is based on a generalization of the diffusion equation to fractional form. It is shown how, by induction, the fractional diffusion equation can be justified in terms of the generalization of a random walk model to fractional form as characterized by the Hurst exponent. Image processing and analysis methods are proposed that are based on diffusion and fractional diffusion models and some application examples given.