[Paper Review] Applying the Fokker--Planck equation to grating-based x-ray phase and dark-field imaging

Kaye S. Morgan, David M. Paganin|arXiv (Cornell University)|Aug 4, 2019

Advanced X-ray Imaging Techniques62 references40 citations

TL;DR

This paper applies the finite-difference form of the Fokker–Planck equation to model the drift (phase signal) and diffusion (dark-field signal) in grating-based x-ray phase and dark-field imaging, revealing cross-talk between attenuation, phase, and dark-field signals. The approach provides a unified mathematical framework for understanding beamlet propagation, including edge-induced boundary diffraction waves and lensing effects, and offers a foundation for solving the inverse problem in multi-contrast x-ray imaging.

ABSTRACT

X-ray imaging has conventionally relied upon attenuation to provide contrast. In recent years, two complementary modalities have been added; phase contrast and dark-field x-ray imaging, capturing weakly attenuating and sub-pixel sample structures respectively. These three modalities can be accessed using a crystal analyser, a grating interferometer or by looking at a directly-resolved grid, grating or speckle pattern. Grating and grid-based methods extract a differential phase signal by measuring how far a feature in the illumination has been shifted transversely due to the presence of a sample. The dark-field signal is extracted by measuring how the visibility of the structured illumination is decreased, typically due to the presence of sub-pixel structures in a sample. The strength of the dark-field signal may depend on the grating period, the pixel size and the set-up distances, and additional dark-field signal contributions may be seen as a result of strong phase effects or other factors. In this paper we show that the finite-difference form of the Fokker--Planck Equation can be applied to describe the drift (phase signal) and diffusion (dark-field signal) of the periodic or structured illumination used in phase contrast x-ray imaging with gratings, in order to better understand any cross-talk between attenuation, phase and dark-field x-ray signals. In future work, this mathematical description could be used as a basis for new approaches to the inverse problem of recovering both phase and dark-field information.

Motivation & Objective

• To develop a unified mathematical framework for understanding the interplay between phase contrast and dark-field signals in grating-based x-ray imaging.
• To model how sample-induced beamlet drift (phase) and broadening (dark-field) arise from wavefront evolution using the Fokker–Planck equation.
• To investigate cross-talk effects between phase, dark-field, and attenuation signals, particularly near edges and due to focusing.
• To extend the Fokker–Planck model to include real-world imaging factors such as source size, polychromaticity, and detector resolution.
• To provide a theoretical basis for future inverse problem solutions that recover both phase and dark-field information simultaneously.

Proposed method

• Formulates the Fokker–Planck equation as a two-dimensional stochastic process describing the time-evolution of a probability density function for x-ray beamlets.
• Applies the finite-difference form of the Fokker–Planck equation to model transverse beamlet shifts (drift, D1) and broadening (diffusion, D2) due to phase gradients and scattering.
• Derives analytic expressions for the intensity profile at a downstream detector plane by solving the Fokker–Planck equation under various sample conditions.
• Incorporates edge effects by modeling the boundary diffraction wave (Young–Maggi–Rubinowicz wave) as a source of enhanced dark-field signal near object edges.
• Models lensing effects by treating the sample as a thin lens, introducing a position-dependent drift and diffusion term in the Fokker–Planck equation.
• Extends the model to include practical imaging factors: detector pixel size (via Gaussian convolution), extended source (via source-averaging or added diffusion term), and polychromatic illumination (via spectral integration).

Experimental results

Research questions

• RQ1 How can the Fokker–Planck equation be used to model the simultaneous evolution of phase and dark-field signals in grating-based x-ray imaging?
• RQ2 What is the origin of enhanced dark-field signals near object edges, and how can it be captured within the Fokker–Planck framework?
• RQ3 How do lensing effects from the sample influence the drift and diffusion components of the beamlet's probability density function?
• RQ4 To what extent do source size, polychromaticity, and detector resolution affect the measured phase and dark-field signals, and how can these be modeled?
• RQ5 Can the Fokker–Planck model be generalized to apply to both single-grid and interferometric imaging setups?

Key findings

• The Fokker–Planck equation successfully models both phase (drift) and dark-field (diffusion) signals in grating-based x-ray imaging as a stochastic process of beamlet propagation.
• Edge-induced dark-field enhancement is explained as a boundary diffraction wave contribution, which adds a localized diffusion term not captured by standard scattering models.
• Lensing effects from the sample introduce a position-dependent drift and diffusion, which can be modeled as spatially varying coefficients in the Fokker–Planck equation.
• The model accounts for practical imaging limitations: detector pixel size reduces visibility via convolution, extended sources cause blurring modeled by an additional diffusion term, and polychromatic illumination is handled by spectral integration.
• The framework provides a consistent mathematical description that unifies single-grid, speckle-tracking, and interferometric imaging modalities under a single stochastic model.
• The approach offers a foundation for solving the inverse problem in multi-contrast x-ray imaging by enabling simultaneous retrieval of phase and dark-field information from measured intensity profiles.