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Applied mathematics, Optics
Mathematical models for the scattering of light (and other radiation) tend to fall into two categories based on a weak field condition (single scattering processes) or a strong field condition (multiple scattering processes). In the latter case, the complexity of deterministic models coupled with available solutions often fail to provide results that are of value to engineering systems (e.g. imaging systems). For this reason, multiple scattering problems are often approached using stochastic modelling methods whose foundations lie in random walk theory where the amplitude and phase of the scattered field are taken to conform to an appropriate statistical process and distribution. In the case of intermediate scattering processes (scattering by a tenuous random medium), where the scattered field is neither ‘weak’ or ‘strong’, the problem is reduced to finding a suitable approach for constructing and solving a mathematical model that is, ideally, of value to an engineering system.
In this paper, we consider the basis for describing strong scattering in terms of diffusive processes based on the diffusion equation. Intermediate strength scattering is then considered in terms of an (inhomogeneous) fractional diffusion equation which is studied using results from fractional calculus. The diffusion equation for modelling intermediate strength scattering is based on a generalization of the diffusion equation to fractional form. This equation can be justified in terms of the generalization of a random walk model with no statistical bias in the phase to a random walk that has a phase bias and is thus, only ‘partially’ or ‘fractionally’ diffusive. Green’s function solutions to the fractional diffusion equation are studied and results derived that provide a model for an incoherent image obtained from light scattered by a tenuous medium. Applications include image enhancement of star fields and other cosmological bodies imaged through interstellar dust clouds, an example of which is provided in this paper.
Blackledge, J.: Scattering from a Tenuous Random Medium with Applications in Optics. ISAST Transactions on Electronics and Signal Processing, vol: 4, issue: 1, pages: 1-19, 2009.