Separating edges from microstructure in X-ray dark-field imaging: evolving and devolving perspectives via the X-ray Fokker-Planck equation

A key contribution to X-ray dark-field (XDF) contrast is the diffusion of X-rays by sample structures smaller than the imaging system's spatial resolution; this is related to position-dependent small-angle X-ray scattering. However, some experimental XDF techniques have reported that XDF contrast is also generated by resolvable sample edges. Speckle-based X-ray imaging (SBXI) extracts the XDF by analyzing sample-imposed changes to a reference speckle pattern's visibility. We present an algorithm for SBXI (a variant of our previously developed multimodal intrinsic speckle-tracking (MIST) algorithm) capable of separating these two physically different XDF contrast mechanisms. The algorithm uses what we call the <i>devolving</i> Fokker-Planck equation for paraxial X-ray imaging as its forward model and then solves the associated multimodal inverse problem to retrieve the attenuation, phase, and XDF properties of the sample. Previous MIST variants were based on the <i>evolving</i> Fokker-Planck equation, which considers how a reference-speckle image is modified by the introduction of a sample. The devolving perspective instead considers how the image collected in the presence of the sample and the speckle membrane optically flows in reverse to generate the reference-speckle image when the sample is removed from the system. We compare single- and multiple-exposure multimodal retrieval algorithms from the two Fokker-Planck perspectives. We demonstrate that the devolving perspective can distinguish between two physically different XDF contrast mechanisms, namely, unresolved microstructure- and sharp-edge-induced XDF. This was verified by applying the different retrieval algorithms to two experimental data sets - one phantom sample and one organic sample. We anticipate that this work will be useful in (1) yielding a pair of complementary XDF images that separate sharp-edge diffuse scatter from diffuse scatter due to spatially random unresolved microstructure, (2) XDF computed tomography, where the strong edge XDF signal can lead to strong contaminating streaking artefacts, and (3) sample preparation, as samples will not need to be embedded since the strong XDF edge signal seen between the sample and air can be separated out.