Applying the Fokker–Planck equation to grating-based x-ray phase and dark-field imaging

Abstract

X-ray imaging has conventionally relied upon attenuation to provide contrast. In recent years, two complementary modalities have been added; (a) phase contrast, which can capture low-density samples that are difficult to see using attenuation, and (b) dark-field x-ray imaging, which reveals the presence of sub-pixel sample structures. 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.

Quantitative analysis of speckle-based X-ray dark-field imaging using numerical wave-optics simulations

“…In this work, we only focused on dark-field and transmission signals due to the particular sample design; however, in general, speckle displacement might also occur due to extended phase-gradients. While current digital tracking algorithms can obtain all signals simultaneously, theoretical studies have shown that there can be cross-talk between the various signals 47 . To what extent X-ray interactions with heterogeneous anatomy influence the dark-field signal extraction needs to be investigated (e.g., for lung imaging the scattering caused by the rib cage and spine).…”

Modeling classical wavefront sensors

“…(3) to include minor effects, e.g., dark fields from indirect scattering. This will yield straightforward extension to existing scattering models in propagation-based wavefront sensing, for example the Fokker-Planck equation in paraxial X-ray imaging [49,50]. Another direction is to impose stronger assumptions on samples, e.g., weak phase that φ 1.…”

Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator

“…The Kramers–Moyal equation serves as a stepping stone to adequately describe time-series data with both diffusive and discontinuous characteristics, but it is nevertheless challenged by finite-time sampling in real-world data. Recent applications of the Kramers–Moyal equation include brain [ 3 , 4 ] and heart dynamics [ 5 ], stochastic harmonic oscillators [ 6 ], renewable-energy generation [ 7 ], solar irradiance [ 8 ], turbulence [ 9 ], nano-scale friction [ 10 ], and X-ray imaging [ 11 , 12 ].…”