Butterfly Transforms for Efficient Representation of Spatially Variant Point Spread Functions in Bayesian Imaging

Abstract

Bayesian imaging algorithms are becoming increasingly important in, e.g., astronomy, medicine and biology. Given that many of these algorithms compute iterative solutions to high-dimensional inverse problems, the efficiency and accuracy of the instrument response representation are of high importance for the imaging process. For efficiency reasons, point spread functions, which make up a large fraction of the response functions of telescopes and microscopes, are usually assumed to be spatially invariant in a given field of view and can thus be represented by a convolution. For many instruments, this assumption does not hold and degrades the accuracy of the instrument representation. Here, we discuss the application of butterfly transforms, which are linear neural network structures whose sizes scale sub-quadratically with the number of data points. Butterfly transforms are efficient by design, since they are inspired by the structure of the Cooley–Tukey fast Fourier transform. In this work, we combine them in several ways into butterfly networks, compare the different architectures with respect to their performance and identify a representation that is suitable for the efficient representation of a synthetic spatially variant point spread function up to a 1% error. Furthermore, we show its application in a short synthetic example.

The first spatio-spectral Bayesian imaging of SN1006 in X-rays

“…However, this is not a trivial task, as an invariant PSF can be applied via multiplication in Fourier space, whereas a spatially varying PSF cannot. Methods are currently being developed to solve this problem in the language of IFT, including a neural network approach recently presented by Eberle et al (2023). In addition, a line model capable of modelling lines in the thermal emission will help to further resolve the energy direction.…”

Re-Envisioning Numerical Information Field Theory (NIFTy.re): A Library for Gaussian Processes and Variational Inference

“…We expect NIFTy.re to be highly useful for many imaging applications and envision many applications within and outside of astrophysics Arras et al, 2022;Eberle et al, 2022Eberle et al, , 2023Frank et al, 2017;S. Hutschenreuter et al, 2022;Sebastian Hutschenreuter et al, 2023;Leike et al, 2020;Mertsch & Phan, 2023;J.…”