Astrophysical data analysis with information field theory

Abstract

Abstract. Non-parametric imaging and data analysis in astrophysics and cosmology can be addressed by information field theory (IFT), a means of Bayesian, data based inference on spatially distributed signal fields. IFT is a statistical field theory, which permits the construction of optimal signal recovery algorithms. It exploits spatial correlations of the signal fields even for nonlinear and non-Gaussian signal inference problems. The alleviation of a perception threshold for recovering signals of unknown correlation structure by using IFT will be discussed in particular as well as a novel improvement on instrumental selfcalibration schemes. IFT can be applied to many areas. Here, applications in in cosmology (cosmic microwave background, large-scale structure) and astrophysics (galactic magnetism, radio interferometry) are presented. INFORMATION FIELD THEORYInformation field theory (IFT) is information theory for fields, describing in mathematical language how information on spatially distributed quantities following some physical laws can optimally be extracted from data. IFT exploits known or inferred correlation structures of the field of interest s (s is regarded as a fuction s = s(x) and as a vector s = (s x ) x∈Ω in a function space) over some domain Ω = {x} in order to regularize the otherwise ill-posed inverse problem of determining virtually infinitely many field degrees of freedom from a finite datasetWhat distinguishes it from many non-parametric inference methods is that an IFT is defined over continuous spaces, and any pixelization of the field used in actual computations must preserve this continuum limit and recover it for infinitely small pixels. A concise introduction into IFT can be found Ref.[1], exhaustive ones in Refs. [2,3], and the numerical issues of properly discretized fields are addressed in Refs. [4,5].Signal field estimation in IFT relies on Bayes theorem that gets recast into a (statistical) field theoretical language:This defines the information Hamiltonian and its partition function to beand permits the usage of any appropriate field theoretical technique in order to do our signal inference. These include, among others, classical solutions (or Maximum A Posteriori (MAP) approximation), Feynman diagrams, renormalization and re-summation techniques, Gibbs free energy minimization (or Maximum Entropy), and mean field approaches. See Refs. [1,3] for an overview. IFT can be applied in scientific areas that rely on imaging, like astronomy and astrophysics, and examples of such applications can be found at https://www.mpa-garching.mpg.de/ift. Useful IFT apporaches to some generic astrophysical signal inference problems are reviewed below. THE MEASUREMENT PROBLEMIn a generic linear measurement the data depend linearly on the signal,

Denoising, deconvolving, and decomposing photon observations

“…A signal field, s = s(x), is a function of a continuous position x in some position space Ω. In order to avoid a dependence of the reconstruction on the partition of Ω, the according calculus regarding fields is geared to preserve the continuum limit (Enßlin 2013(Enßlin , 2014Selig et al 2013). In general, we are interested in the a posteriori mean estimate m of the signal field given the data, and its (uncertainty) covariance D, defined as…”

“…The fluxes from diffuse and point-like sources contribute equally to the observed photon counts, but their morphological imprints are very different. The proposed algorithm, derived in the framework of information field theory (IFT) (Enßlin et al 2009;Enßlin 2013Enßlin , 2014, therefore incorporates prior assumptions in form of a hierarchical parameter model. The fundamentally different morphologies of diffuse and point-like contributions reflected in different prior correlations and statistics.…”

The Denoised, Deconvolved, and Decomposed Fermi gamma-ray sky

“…For the analysis of X-ray images, which pose the same challenges as γ-ray images, a Bayesian background-source separation technique was proposed by Guglielmetti et al (2009). We deploy the D 3 PO inference algorithm (Selig & Enßlin 2013) derived within the framework of information field theory (IFT, Enßlin et al 2009;Enßlin 2013Enßlin , 2014. It simultaneously provides non-parametric estimates for the diffuse and the point-like photon flux given a photon count map.…”

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

“…Both are challenging to compute due to their inhomogeneous behaviour in terms of space and energy. In order to efficiently perform statistical field inference, for example, by using NIFTy (Numerical Information Field Theory) [ 3 , 4 , 5 ], a software package for the numerical application of information field theory [ 6 , 7 , 8 , 9 ], these responses must be represented numerically in a way that is fast and differentiable. One promising candidate for the efficient representation of instrument responses are butterfly transforms, a linear neural network structure inspired by the structure of the fast Fourier transform (FFT) algorithm, whose size scales with , where is the number of pixels.…”