Analysis and interpretation of the Cramer-Rao lower-bound in Astrometry.

 

This line explores the interplay between information and estimation theory and inverse problems in astronomy, in the context of characterizing fundamental performance bounds and deriving optimal image acquisition and analysis techniques.  The short-term focus is in the problem of optimal joint flux + location determination for photometric + high-precision relative astrometry of point sources from a set of measurements obtained with solid-state detectors on a pixel array (CCDs).  A fundamental question here is to precisely evaluate the accuracy of standard estimation techniques.  This has usually been made empirically based on heuristic knowledge. We propose to attack this problem rigorously through the use of classical results in estimation theory: the so-called Cramér-Rao lower bound and its extensions.  We expect that these studies will lead to a number of practical consequences, ranging from optimal experimental design (detectors) and observational planning (required signal, required image quality), to the benchmarking of data analysis packages.
Contribution: 
  1. Rene Mendez, Jorge F. Silva, Rodrigo Orostica, and Rodrigo Lobos, “Analysis of the Cramer-Rao lower-bound in the joint estimation of astrometry and photometry,” Publications of the Astronomical Society of the Pacific (PASP), vol. 126, August, 2014.
  2. Rene Mendez, Jorge F. Silva and Rodrigo Lobos, ”Analysis and interpretation of the Cramer-Rao lower-bound in astrometry: One dimensional case, Publications of the Astronomical Society of the Pacific, vol. 125, pp. 580594, May, 2013.

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Bayesian Cramér-Rao Bound in joint Photometry and Astrometry

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The determination of the best precision that can be achieved to jointly determine the intensity (photometry) and location (astrometry)of a stellar-like object have been topics of permanent interest in the astronomic community. In this work, we want to extend the precision analysis for joint photometry and astrometry, transiting from the classical parametric setting, where the intensity and  position of an stellar object are parameters (fixed but unknown), to the richer Bayesian setting, where the intensity and position are random variables equipped with a prior distribution.  This changes the inference problem from a parametric context in which we are estimating parameters from a set of observations to a Bayesian setting in which we estimate a random object from observations that are statistically dependent with the underlying intensity and  position. The first objective is to develop closed-form expressions for the Bayesian Cramer-Rao (BRC) in photometry and astrometry. The second objetive is to derive expressions to quantify the gain in performance with respect to the parametric case. Finally, we propose to introduce formal definitions for the prior information and the information attributed to the observations, and with these concepts we can determine when the information of the prior distribution is relevant or irrelevant with respect to the information provided by the observations. This last distinction will define concrete conditions where a significant gain in photometric and astrometric precision can be achieved from the prior distribution.  These findings will be significant for observational planning, as prior information could impact the quality of the inference task, and consequently, the resources needed to achieve a given precision in photometry and astrometry.

Contribution: 

  1. Alex Echeverria, Jorge F. Silva, Rene A. Mendez, and Marcos E. Orchard , “Analy- sis of the Bayesian Cramer-Rao lower bound in astrometry: Studying the impact of prior information in the location of an object,” Astronomy & Astrophysics (A&A), 2016.
 

Performance Analysis of the Least-Squares, Weighted Least-Squares  and Maximum Likelihood Estimator in Astrometry

The focus of this research line is to characterize the performance of the widely used least-squares estimator and maximum likelihood estimator  in astrometry in terms of a comparison with the Cramér–Rao lower bound. It is well-known that the performance of these optimization driven estimators do not offer a closed-form expression. Then, the technical challenge is to derive results to bound the performance of these estimators and with these bounds analyze regimes where these estimators achieve the optimal performance bounds, if any.  At IDS we have shown that the least-squares (LS) estimator is near-optimal with respect to the CR bound in the low SNR regime, and that there is a significant difference between the the performance of LS method and the CR bound at high SNR.  In this context, we plan to further enrich this analysis in the parametric case, revisiting  the maximum likelihood (ML) estimator as well as adaptive variations of the weighted LS (WLS) approach.

Contribution: 

  1. Rodrigo Lobos, Jorge F. Silva, Rene A. Mendez, and Marcos E. Orchard , “Per- formance analysis of the Least-Squares estimator in Astrometry,” Publications of the Astronomical Society of the Pacific (PASP), vol. 127: pp. 1166-1182, Nov. 2015.

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Compressed Sensing Applied to Radio Interferometry

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Interferometry has enrich the way to acquire observations from the sky and nowadays it is one of the most advanced techniques used for high resolution imaging in radio astronomy.  Radio interferometry offers  high-resolution information (sampling the sky in the Fourier domain) and it has provided the mean to observe complex extended objects. This Fourier samples are non-exhaustive, which rises the  sampling  aspect of the problem. The conventional way to treat this ill-post image synthesis problem has been by the use of deconvolution driven algorithms. Departing from this angle, this research line considers a sampling strategy and in particular the use of the signal structure of astronomical objects as a relevant prior knowledge to do image synthesis. On this context, Compressed Sensing (CS) offers a rich machinery of theoretical results and algorithms to address this problem. The basic idea of CS is to model the compressible structure of astronomical objects in a transform domain (Wavelets bases, DCT) to promote reconstructions that are sparse given the measurements. The main goal is to formalize the problem of basis selection and sampling design in the context of CS-based algorithms, with the objective to find the optimal tradeoff between the number of samples and the reconstruction errors.