# X-ray Fokker--Planck equation for paraxial imaging

## TL;DR

This paper introduces a Fokker--Planck equation framework for modeling paraxial x-ray imaging, capturing both phase contrast and scattering effects, and extends to inverse problems for sample characterization.

## Contribution

It derives and generalizes the Fokker--Planck equation for x-ray imaging, enabling modeling of complex intensity evolution including scattering and phase effects.

## Key findings

- Provides two derivations of the x-ray Fokker--Planck equation.
- Extends the equation to a Kramers--Moyal generalization.
- Applicable to forward and inverse imaging problems.

## Abstract

The Fokker--Planck Equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This forms a natural generalisation of the transport-of-intensity equation. The x-ray Fokker--Planck equation can simultaneously account for both propagation-based phase contrast, and the diffusive effects of sample-induced small-angle x-ray scattering, when forming an x-ray image of a thin sample. Two derivations are given for the Fokker--Planck equation associated with x-ray imaging, together with a Kramers--Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample micro-structure. These equations may be applied to the forward problem of modelling image formation in the presence of both coherent and diffusive energy transport. They may also be used to formulate associated inverse problems of retrieving the phase shifts due to a sample placed in an x-ray beam, together with the diffusive properties of the sample. The domain of applicability for the Fokker--Planck and Kramers--Moyal equations for paraxial imaging is at least as broad as that of the transport-of-intensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons.