Signal inference in radio astronomy

Diese Dissertation befasst sich mit dem Rekonstruieren von unvollständig gemessenen Signalen in der Radioastronomie. Es werden zwei bildgebende Algorithmen entwickelt, die im Formalismus der Informationsfeldtheorie hergeleitet werden. Beide basieren auf dem Prinzip der Bayesischen Analyse, die Informationen aus der unvollständigen Messung werden dabei durch a priori Informationen ergänzt. Hierfür werden beide Informationsquellen in Form von Wahrscheinlichkeitsdichten formuliert und zu einer a posteriori Wahrscheinlichkeitsdichte zusammengeführt. Die a priori Informationen werden dabei minimal gehalten und beschränken sich auf die Annahme, dass das ursprüngliche Signal bezüglich des Ortes nicht beliebig stark fluktuiert. Dies erlaubt eine statistische Abschätzung des ursprünglichen Signales auf allen Größenskalen. Der erste bildgebende Algorithmus errechnet eine Abschätzung der dreidimensionalen freien Elektronendichte im interstellaren Medium der Milchstraße aus Dispersionsmessungen von Pulsaren. Die Dispersion der Radiostrahlung eines Pulsars ist proportional zu der Gesamtanzahl der freien Eletronen auf der Sichtlinie zwischen Pulsar und Beobachter. Jede gemessene Sichtlinie enthält somit Informationen über die Verteilung von freien Elektronen im Raum. Das Rekonstruktionsproblem ist damit ein Tomographieproblem ähnlich dem in der medizinischen Bildgebung. Anhand einer Simulation wird untersucht, wie detailliert die Elektronendichte mit Daten des noch im Bau befindlichen Square Kilometre Array (SKA) rekonstruiert werden kann. Die Ergebnisse zeigen, dass die großen Strukturen der freien Elektronendichte der Milchstraße mit den Daten des SKA rekonstruiert werden können. Der zweite bildgebende Algorithmus trägt den Namen fastResolve und rekonstruiert die Intensität von Radiostrahlung anhand von Messdaten eines Radiointerferometers. fastResolve baut auf dem bestehenden Algorithmus Resolve auf. fastResolve erweitert dessen Funktionalität um die separate Abschätzung von Punktquellen und rekonstruiert simultan auch die Messunsicherheit. Gleichzeitig ist fastResolve etwa 100 mal schneller. Ein Vergleich des Algorithmus’ mit CLEAN, dem Standardalgorithmus in der Radioastronomie, wird anhand von Beobachtungsdaten des Galaxienhaufens Abell 2199, aufgenommen mit dem Very Large Array, durchgeführt. fastResolve kann feinere Details des Intensitätsverlaufs rekonstruieren als CLEAN. Gleichzeitig erzeugt fastResolve weniger Artefakte wie negative Intensität. Außerdem liefert fastResolve eine Abschätzung der Rekonstruktionsunsicherheit. Diese ist wichtig für die wissenschaftliche Weiterverarbeitung und kann mit CLEAN nicht errechnet werden. Weiterhin wird ein Verfahren entwickelt, mit dem die Leistungsspektren von Gaußschen Feldern und die von log-normal Feldern ineinander umgewandelt werden können. Dieses ermöglicht die Verjüngung des Leistungsspektrums der großskaligen Dichtestruktur des Universums, was durch Vergleiche mit einer störungstheoretischen Methode und einem kosmischen Emulator validiert wird.This dissertation addresses the problem of inferring a signal from an incomplete measurement in the field of radio astronomy. Two imaging algorithms are developed within the framework of information field theory. Both are based on Bayesian analysis; information from the incomplete measurement is complemented by a priori information. To that end both sources of information are formulated as probability distributions and merged to an a posteriori probability distribution. The a priori information is kept minimal. It reduces to the assumption that the real signal does not fluctuate arbitrarily strong with respect to position. This construction allows for a statistical estimation of the original signal on all scales. The first imaging algorithm calculates a three-dimensional map of the Galactic free electron density using dispersion measure data from pulsars. The dispersion of electromagnetic waves in the radio spectrum that a pulsar emits is proportional to the total number of free electrons on the line of sight between pulsar and observer. Therefore, each measured line of sight contains information about the distribution of free electrons in space. The reconstruction problem is a tomography problem similar to the one in medical imaging. We investigate which level of detail of the free electron density can be reconstructed with data of the upcoming Square Kilometre Array (SKA) by setting up a simulation. The results show that the large-scale features free electron density of the Milky Way will be reconstructible with the SKA. The second imaging algorithm is named fastResolve. It reconstructs the radio intensity of the sky from interferometric data. fastResolve is based on Resolve, but adds the capability to separate point sources and to estimate the measurement uncertainty. Most importantly, it is 100 times faster. A comparison of the algorithm with CLEAN, the standard imaging method for interferometric data in radio astronomy, is performed using observational data of the galaxy cluster Abell 2199 recorded with the Very Large Array. fastResolve reconstructs finer details than CLEAN while introducing fewer artifacts such as negative intensity. Furthermore, fastResolve provides an uncertainty map. This quantity is important for proper scientific use of the result, but is not available using CLEAN. Furthermore, a formalism is developed, which allows the conversion of power spectra of Gaussian fields into the power spectra of log-normal fields and vice versa. This allows the rejuvenation of the power spectrum of the large-scale matter distribution of the Universe. We validate the approach by comparison with a perturbative method and a cosmic emulator

Bayesian weak lensing tomography: Reconstructing the 3D large-scale distribution of matter with a lognormal prior

We present a Bayesian reconstruction algorithm that infers the three-dimensional large-scale matter distribution from the weak gravitational lensing effects measured in the image shapes of galaxies. The algorithm is designed to also work with non-Gaussian posterior distributions which arise, for example, from a non-Gaussian prior distribution. In this work, we use a lognormal prior and compare the reconstruction results to a Gaussian prior in a suite of increasingly realistic tests on mock data. We find that in cases of high noise levels (i.e. for low source galaxy densities and/or high shape measurement uncertainties), both normal and lognormal priors lead to reconstructions of comparable quality, but with the lognormal reconstruction being prone to mass-sheet degeneracy. In the low-noise regime and on small scales, the lognormal model produces better reconstructions than the normal model: The lognormal model 1) enforces non-negative densities, while negative densities are present when a normal prior is employed, 2) better traces the extremal values and the skewness of the true underlying distribution, and 3) yields a higher pixel-wise correlation between the reconstruction and the true density.Comment: 23 pages, 12 figures; updated to match version accepted for publication in PR

Dynamic system classifier

Stochastic differential equations describe well many physical, biological and sociological systems, despite the simplification often made in their derivation. Here the usage of simple stochastic differential equations to characterize and classify complex dynamical systems is proposed within a Bayesian framework. To this end, we develop a dynamic system classifier (DSC). The DSC first abstracts training data of a system in terms of time dependent coefficients of the descriptive stochastic differential equation. Thereby the DSC identifies unique correlation structures within the training data. For definiteness we restrict the presentation of DSC to oscillation processes with a time dependent frequency {\omega}(t) and damping factor {\gamma}(t). Although real systems might be more complex, this simple oscillator captures many characteristic features. The {\omega} and {\gamma} timelines represent the abstract system characterization and permit the construction of efficient signal classifiers. Numerical experiments show that such classifiers perform well even in the low signal-to-noise regime.Comment: 11 pages, 8 figure

Signal inference with unknown response: Calibration-uncertainty renormalized estimator

The calibration of a measurement device is crucial for every scientific experiment, where a signal has to be inferred from data. We present CURE, the calibration uncertainty renormalized estimator, to reconstruct a signal and simultaneously the instrument's calibration from the same data without knowing the exact calibration, but its covariance structure. The idea of CURE, developed in the framework of information field theory, is starting with an assumed calibration to successively include more and more portions of calibration uncertainty into the signal inference equations and to absorb the resulting corrections into renormalized signal (and calibration) solutions. Thereby, the signal inference and calibration problem turns into solving a single system of ordinary differential equations and can be identified with common resummation techniques used in field theories. We verify CURE by applying it to a simplistic toy example and compare it against existent self-calibration schemes, Wiener filter solutions, and Markov Chain Monte Carlo sampling. We conclude that the method is able to keep up in accuracy with the best self-calibration methods and serves as a non-iterative alternative to it

Improving self-calibration

Response calibration is the process of inferring how much the measured data depend on the signal one is interested in. It is essential for any quantitative signal estimation on the basis of the data. Here, we investigate self-calibration methods for linear signal measurements and linear dependence of the response on the calibration parameters. The common practice is to augment an external calibration solution using a known reference signal with an internal calibration on the unknown measurement signal itself. Contemporary self-calibration schemes try to find a self-consistent solution for signal and calibration by exploiting redundancies in the measurements. This can be understood in terms of maximizing the joint probability of signal and calibration. However, the full uncertainty structure of this joint probability around its maximum is thereby not taken into account by these schemes. Therefore better schemes -- in sense of minimal square error -- can be designed by accounting for asymmetries in the uncertainty of signal and calibration. We argue that at least a systematic correction of the common self-calibration scheme should be applied in many measurement situations in order to properly treat uncertainties of the signal on which one calibrates. Otherwise the calibration solutions suffer from a systematic bias, which consequently distorts the signal reconstruction. Furthermore, we argue that non-parametric, signal-to-noise filtered calibration should provide more accurate reconstructions than the common bin averages and provide a new, improved self-calibration scheme. We illustrate our findings with a simplistic numerical example.Comment: 17 pages, 3 figures, revised version, title change

Signal inference in radio astronomy

Diese Dissertation befasst sich mit dem Rekonstruieren von unvollständig gemessenen Signalen in der Radioastronomie. Es werden zwei bildgebende Algorithmen entwickelt, die im Formalismus der Informationsfeldtheorie hergeleitet werden. Beide basieren auf dem Prinzip der Bayesischen Analyse, die Informationen aus der unvollständigen Messung werden dabei durch a priori Informationen ergänzt. Hierfür werden beide Informationsquellen in Form von Wahrscheinlichkeitsdichten formuliert und zu einer a posteriori Wahrscheinlichkeitsdichte zusammengeführt. Die a priori Informationen werden dabei minimal gehalten und beschränken sich auf die Annahme, dass das ursprüngliche Signal bezüglich des Ortes nicht beliebig stark fluktuiert. Dies erlaubt eine statistische Abschätzung des ursprünglichen Signales auf allen Größenskalen. Der erste bildgebende Algorithmus errechnet eine Abschätzung der dreidimensionalen freien Elektronendichte im interstellaren Medium der Milchstraße aus Dispersionsmessungen von Pulsaren. Die Dispersion der Radiostrahlung eines Pulsars ist proportional zu der Gesamtanzahl der freien Eletronen auf der Sichtlinie zwischen Pulsar und Beobachter. Jede gemessene Sichtlinie enthält somit Informationen über die Verteilung von freien Elektronen im Raum. Das Rekonstruktionsproblem ist damit ein Tomographieproblem ähnlich dem in der medizinischen Bildgebung. Anhand einer Simulation wird untersucht, wie detailliert die Elektronendichte mit Daten des noch im Bau befindlichen Square Kilometre Array (SKA) rekonstruiert werden kann. Die Ergebnisse zeigen, dass die großen Strukturen der freien Elektronendichte der Milchstraße mit den Daten des SKA rekonstruiert werden können. Der zweite bildgebende Algorithmus trägt den Namen fastResolve und rekonstruiert die Intensität von Radiostrahlung anhand von Messdaten eines Radiointerferometers. fastResolve baut auf dem bestehenden Algorithmus Resolve auf. fastResolve erweitert dessen Funktionalität um die separate Abschätzung von Punktquellen und rekonstruiert simultan auch die Messunsicherheit. Gleichzeitig ist fastResolve etwa 100 mal schneller. Ein Vergleich des Algorithmus’ mit CLEAN, dem Standardalgorithmus in der Radioastronomie, wird anhand von Beobachtungsdaten des Galaxienhaufens Abell 2199, aufgenommen mit dem Very Large Array, durchgeführt. fastResolve kann feinere Details des Intensitätsverlaufs rekonstruieren als CLEAN. Gleichzeitig erzeugt fastResolve weniger Artefakte wie negative Intensität. Außerdem liefert fastResolve eine Abschätzung der Rekonstruktionsunsicherheit. Diese ist wichtig für die wissenschaftliche Weiterverarbeitung und kann mit CLEAN nicht errechnet werden. Weiterhin wird ein Verfahren entwickelt, mit dem die Leistungsspektren von Gaußschen Feldern und die von log-normal Feldern ineinander umgewandelt werden können. Dieses ermöglicht die Verjüngung des Leistungsspektrums der großskaligen Dichtestruktur des Universums, was durch Vergleiche mit einer störungstheoretischen Methode und einem kosmischen Emulator validiert wird.This dissertation addresses the problem of inferring a signal from an incomplete measurement in the field of radio astronomy. Two imaging algorithms are developed within the framework of information field theory. Both are based on Bayesian analysis; information from the incomplete measurement is complemented by a priori information. To that end both sources of information are formulated as probability distributions and merged to an a posteriori probability distribution. The a priori information is kept minimal. It reduces to the assumption that the real signal does not fluctuate arbitrarily strong with respect to position. This construction allows for a statistical estimation of the original signal on all scales. The first imaging algorithm calculates a three-dimensional map of the Galactic free electron density using dispersion measure data from pulsars. The dispersion of electromagnetic waves in the radio spectrum that a pulsar emits is proportional to the total number of free electrons on the line of sight between pulsar and observer. Therefore, each measured line of sight contains information about the distribution of free electrons in space. The reconstruction problem is a tomography problem similar to the one in medical imaging. We investigate which level of detail of the free electron density can be reconstructed with data of the upcoming Square Kilometre Array (SKA) by setting up a simulation. The results show that the large-scale features free electron density of the Milky Way will be reconstructible with the SKA. The second imaging algorithm is named fastResolve. It reconstructs the radio intensity of the sky from interferometric data. fastResolve is based on Resolve, but adds the capability to separate point sources and to estimate the measurement uncertainty. Most importantly, it is 100 times faster. A comparison of the algorithm with CLEAN, the standard imaging method for interferometric data in radio astronomy, is performed using observational data of the galaxy cluster Abell 2199 recorded with the Very Large Array. fastResolve reconstructs finer details than CLEAN while introducing fewer artifacts such as negative intensity. Furthermore, fastResolve provides an uncertainty map. This quantity is important for proper scientific use of the result, but is not available using CLEAN. Furthermore, a formalism is developed, which allows the conversion of power spectra of Gaussian fields into the power spectra of log-normal fields and vice versa. This allows the rejuvenation of the power spectrum of the large-scale matter distribution of the Universe. We validate the approach by comparison with a perturbative method and a cosmic emulator

All-sky reconstruction of the primordial scalar potential from WMAP temperature data

An essential quantity required to understand the physics of the early Universe, in particular the inflationary epoch, is the primordial scalar potential $\Phi$ and its statistics. We present for the first time an all-sky reconstruction of $\Phi$ with corresponding $1\sigma$-uncertainty from WMAP's cosmic microwave background (CMB) temperature data -- a map of the very early Universe right after the inflationary epoch. This has been achieved by applying a Bayesian inference method that separates the whole inverse problem of the reconstruction into many independent ones, each of them solved by an optimal linear filter (Wiener filter). In this way, the three-dimensional potential $\Phi$ gets reconstructed slice by slice resulting in a thick shell of nested spheres around the comoving distance to the last scattering surface. Each slice represents the primordial scalar potential $\Phi$ projected onto a sphere with corresponding distance. Furthermore, we present an advanced method for inferring $\Phi$ and its power spectrum simultaneously from data, but argue that applying it requires polarization data with high signal-to-noise levels not available yet. Future CMB data should improve results significantly, as polarization data will fill the present $\ell-$blind gaps of the reconstruction

Sharpening up Galactic all-sky maps with complementary data - A machine learning approach

Galactic all-sky maps at very disparate frequencies, like in the radio and $\gamma$-ray regime, show similar morphological structures. This mutual information reflects the imprint of the various physical components of the interstellar medium. We want to use multifrequency all-sky observations to test resolution improvement and restoration of unobserved areas for maps in certain frequency ranges. For this we aim to reconstruct or predict from sets of other maps all-sky maps that, in their original form, lack a high resolution compared to other available all-sky surveys or are incomplete in their spatial coverage. Additionally, we want to investigate the commonalities and differences that the ISM components exhibit over the electromagnetic spectrum. We build an $n$-dimensional representation of the joint pixel-brightness distribution of $n$ maps using a Gaussian mixture model and see how predictive it is: How well can one map be reproduced based on subsets of other maps? Tests with mock data show that reconstructing the map of a certain frequency from other frequency regimes works astonishingly well, predicting reliably small-scale details well below the spatial resolution of the initially learned map. Applied to the observed multifrequency data sets of the Milky Way this technique is able to improve the resolution of, e.g., the low-resolution Fermi LAT maps as well as to recover the sky from artifact-contaminated data like the ROSAT 0.855 keV map. The predicted maps generally show less imaging artifacts compared to the original ones. A comparison of predicted and original maps highlights surprising structures, imaging artifacts (fortunately not reproduced in the prediction), and features genuine to the respective frequency range that are not present at other frequency bands. We discuss limitations of this machine learning approach and ideas how to overcome them