Paraxial diffusion-field retrieval

TL;DR

This paper develops a theoretical framework for retrieving diffusion fields caused by unresolved microstructure in samples, using inverse PDE solutions for various illumination schemes, applicable to multiple types of paraxial radiation.

Contribution

It introduces a closed-form solution approach for inverse diffusion-field retrieval from intensity images, linking microstructure statistics to observable diffusion fields.

Findings

Closed-form inverse solutions for diffusion fields.

Applicable to various radiation types like light, X-rays, neutrons, electrons.

Links microstructure properties to diffusion field characteristics.

Abstract

Unresolved spatially-random microstructure, in an illuminated sample, can lead to position-dependent blur when an image of that sample is formed. For a small propagation distance, between the exit surface of the sample and the entrance surface of a position-sensitive detector, the paraxial approximation implies that the blurring influence of the sample may be modeled using an anomalous-diffusion field. This diffusion field may have a scalar or tensor character, depending on whether the random microstructure has an autocorrelation function that is rotationally isotropic or anisotropic, respectively. Partial differential equations are written down and then solved, in a closed-form manner, for several variants of the inverse problem of diffusion-field retrieval given suitable intensity images. Both uniform-illumination and structured-illumination schemes are considered. Links are made, between the recovered diffusion field and certain statistical properties of the unresolved microstructure. The developed theory -- which may be viewed as a crudely parallel form of small-angle scattering under the Guinier approximation -- is applicable to a range of paraxial radiation and matter fields, such as visible light, x rays, neutrons, and electrons.