Efficient Representations of Spatially Variant Point Spread Functions with Butterfly Transforms in Bayesian Imaging Algorithms

: 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 this reason, 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 subquadratically 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 respresentation of a synthetic spatially variant point spread function up to a 1% error.