X-ray Fokker--Planck equation for paraxial imaging

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.

References in corpus (4)

• Single-Image Geometric-Flow X-Ray Speckle Tracking
• Phase-and-amplitude recovery from a single phase contrast image using partially spatially coherent X-ray radiation
• Propagation of fully-coherent and partially-coherent complex scalar fields in aberration space
• Speckled cross-spectral densities and their associated correlation singularities for a modern source of partially coherent x rays

Cited by in corpus (11)

• Single-shot x-ray speckle-based imaging of a single-material object
• Dark-field signal extraction in propagation-based phase-contrast imaging
• Boosting spatial resolution by incorporating periodic boundary conditions into single-distance hard-x-ray phase retrieval
• X-ray Multimodal Intrinsic-Speckle-Tracking
• X-ray phase-contrast imaging: a broad overview of some fundamentals
• High-speed processing of X-ray wavefront marking data with the Unified Modulated Pattern Analysis (UMPA) model
• Arbitrary-order finite-time corrections for the Kramers-Moyal operator
• Single-exposure x-ray dark-field imaging: quantifying sample microstructure using a single-grid setup
• Paraxial diffusion-field retrieval
• Directional dark-field implicit x-ray speckle tracking using an anisotropic-diffusion Fokker-Planck equation
• Applying the Fokker--Planck equation to grating-based x-ray phase and dark-field imaging