X-ray multi-modal intrinsic-speckle-tracking

Abstract

We develop x-ray multi-modal intrinsic-speckle-tracking (MIST), a form of x-ray speckle-tracking that is able to recover both the position-dependent phase shift and the position-dependent small-angle x-ray scattering (SAXS) signal of a phase object. MIST is based on combining a Fokker–Planck description of paraxial x-ray optics, with an optical-flow formalism for x-ray speckle-tracking. Only two images need to be taken in the presence of the sample, corresponding to two different transverse positions of the speckle-generating membrane, in order to recover both the refractive and local-SAXS properties of the sample. Like the optical-flow x-ray phase-retrieval method which it generalises, the MIST method implicitly rather than explicitly tracks both the transverse motion and the diffusion of speckles that is induced by the presence of a sample. Application to x-ray synchrotron data shows the method to be efficient, rapid and stable.

Directional dark-field implicit x-ray speckle tracking using an anisotropic-diffusion Fokker-Planck equation

“…This enables us to write down a directional-darkfield generalization of Eq. ( 1), namely the following anisotropic-diffusion forward-finite-difference Fokker-Planck speckle-tracking equation due to Pavlov et al [58]:…”

“…More recently, another random-mask speckle-tracking approach was developed. This third approach is the Optical Flow (OF) method [56], together with its Fokker-Planck generalization (Multi-modal Intrinsic Speckle Tracking (MIST)) [57,58]. We now focus attention on this third method, which implicitly rather than explicitly tracks speckles.…”

“…This simplicity is obtained at the cost of being significantly less general than the XSVT and UMPA approaches. In the MIST extension of OF, a Fokker-Planck [60] generalization of OF speckle tracking is employed [57,58,61]. MIST implicitly tracks the transverse motion, lensing and diffusion of speckles.…”

“…MIST implicitly tracks the transverse motion, lensing and diffusion of speckles. Its associated partial differential equation [57], which is of Fokker-Planck form, is amenable to closed-form solution [58]. The latter fact is the core motivation for pursuing this particular approach.…”

X-ray dark-field and phase retrieval without optics, via the Fokker-Planck equation

“…The Fokker-Planck equation may thus be viewed as the natural diffusive generalization of the TIE, which simultaneously models attenuation, phase and dark-field effects in PBI settings. The Fokker-Planck equation has been applied to grid/grating-based imaging in the context of the forward problem as seen in Morgan & Paganin [14], and has also been used to retrieve phase and dark-field signals in the context of the inverse problem applied to x-ray speckle-tracking [20], [21]. Both of these dark-field imaging techniques, as well as analyzerbased dark-field imaging, have the disadvantage of requiring extra hardware in the form of optical elements to extract phase and dark-field contrast.…”