ArticleScientific reports2019
X-ray Fokker-Planck equation for paraxial imaging.
Article in Scientific reports, 2019. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 12 papers.
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Who cites it
12 citing papers in PubMed.
- Effects of Quantum Noise and Source Blurring on Dark-Field Signal Retrieval in X-ray Speckle-Based Imaging.ArXiv · 2026Article
- High-speed quantitative X-ray multi-contrast imaging with deep learning based modulated pattern analysis.Journal of synchrotron radiation · 2026Article
- Demonstration of a family of X-ray dark-field retrieval approaches on a common set of samples.Journal of synchrotron radiation · 2026Article
- Review and experimental comparison of speckle-tracking algorithms for X-ray phase contrast imaging.Journal of synchrotron radiation · 2025Article
- Laboratory-based x-ray dark-field microscopy.Physical review applied · 2023Article
- On the quantification of sample microstructure using single-exposure x-ray dark-field imaging via a single-grid setup.Scientific reports · 2023Article
- Multimodal intrinsic speckle-tracking (MIST) to extract images of rapidly-varying diffuse X-ray dark-field.Scientific reports · 2023Article
- Dark-field tomography of an attenuating object using intrinsic x-ray speckle tracking.Journal of medical imaging (Bellingham, Wash.) · 2022Article
- Semi-classical Monte Carlo algorithm for the simulation of X-ray grating interferometry.Scientific reports · 2022Article
- Arbitrary-Order Finite-Time Corrections for the Kramers-Moyal Operator.Entropy (Basel, Switzerland) · 2021Article
- Emphysema quantified: mapping regional airway dimensions using 2D phase contrast X-ray imaging.Biomedical optics express · 2020Article
- Applying the Fokker-Planck equation to grating-based x-ray phase and dark-field imaging.Scientific reports · 2019Article
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2 authors.
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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.
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