X-ray phase-contrast imaging: a broad overview of some fundamentals

TL;DR

This paper provides a comprehensive overview of the fundamentals of X-ray phase-contrast imaging, including theoretical models, image formation, and phase retrieval techniques, emphasizing both coherent and partially-coherent X-ray sources.

Contribution

It offers a detailed synthesis of the physical principles, mathematical models, and inverse problems involved in X-ray phase-contrast imaging, serving as a foundational reference.

Findings

Summarizes key physical concepts and equations in X-ray phase-contrast imaging.

Discusses models for image formation and the inverse problem of phase retrieval.

Highlights the importance of coherence and system transfer functions in image analysis.

Abstract

We outline some basics of imaging using both fully-coherent and partially-coherent X-ray beams, with an emphasis on phase-contrast imaging. We open with some of the basic notions of X-ray imaging, including the vacuum wave equations and the physical meaning of the intensity and phase of complex scalar fields. The projection approximation is introduced, together with the concepts of attenuation contrast and phase contrast. We also outline the multi-slice approach to X-ray propagation through thick samples or optical elements, together with the Fresnel scaling theorem. Having introduced the fundamentals, we then consider several aspects of the forward problem, of modelling the formation of phase-contrast X-ray images. Several topics related to this forward problem are considered, including the transport-of-intensity equation, arbitrary linear imaging systems, shift-invariant linear imaging systems, the transfer-function formalism, blurring induced by finite source size, the space-frequency model for partially-coherent fields, and the Fokker-Planck equation for paraxial X-ray imaging. Having considered these means for modelling the formation of X-ray phase-contrast images, we then consider aspects of the associated inverse problem of phase retrieval. This concerns how one may decode phase-contrast images to gain information regarding the sample-induced attenuation and phase shift.