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

Abstract

Emerging methods of x-ray imaging that capture phase and dark-field effects are equipping medicine with complementary sensitivity to conventional radiography. These methods are being applied over a wide range of scales, from virtual histology to clinical chest imaging, and typically require the introduction of optics such as gratings. Here, we consider extracting x-ray phase and dark-field signals from bright-field images collected using nothing more than a coherent x-ray source and detector. Our approach is based on the Fokker-Planck equation for paraxial imaging, which is the diffusive generalization of the transport-of-intensity equation. Specifically, we utilize the Fokker-Planck equation in the context of propagation-based phase-contrast imaging, where we show that two intensity images are sufficient for successful retrieval of the projected thickness and dark-field signals associated with the sample. We show the results of our algorithm using both a simulated dataset and an experimental dataset. These demonstrate that the x-ray darkfield signal can be extracted from propagation-based images, and that x-ray phase can be retrieved with better spatial resolution when dark-field effects are taken into account. We anticipate the proposed algorithm will be of benefit in biomedical imaging, industrial settings, and other non-invasive imaging applications.

Paraxial diffusion-field retrieval

“…Rather than hardware optical elements being used to generate a darkfield image, such as the use of a knife edge in Schlieren imaging [129], the filtration is instead performed in the second (i.e., reconstruction stage) of a two-step process. This enables computational diffusive dark-field images to be obtained using bright-field data [16,[130][131][132].…”