Linear processing of X-ray scattering patterns with missing pixels

TL;DR

This paper introduces linear methods for processing X-ray scattering patterns with missing pixels, enabling accurate analysis and intensity retrieval without nonlinear distortions, improving upon traditional phase retrieval techniques.

Contribution

It derives linear formulations of Fourier transforms and basis functions that explicitly account for missing pixels, allowing linear enforcement of constraints and intensity retrieval.

Findings

Linear formulations exclude nonlinear distortions caused by masks.

Methods enable full constraint enforcement and intensity retrieval.

Applications demonstrate effectiveness in typical cases.

Abstract

X-ray scattering patterns from emerging single particle experiments have commonly many missing or contaminated pixels. This complicates different analyses including projections on Fourier or other basis functions (for noise suppression, compression, feature extraction, or retrieval of real-space patterns), as they require integration over all pixels. Here, we derive alternative formulations for Discrete Fourier Transform and a common orthogonal basis by explicit consideration of missing pixels and finite size. Such linear formulations exclude the nonlinear distortion that would be caused by multiplication of the complete scattering pattern with the mask function. Contrary to nonlinear and non-convex phase retrieval optimizations, such reduced-dimension formulations can be used to fully enforce the constraints and to retrieve unknown intensities in a linear fashion. Applications are demonstrated for some typical cases, and extensions to more general cases are discussed.