Reconstructing signals from noisy data with unknown signal and noise covariance

TL;DR

This paper introduces a novel algorithm for reconstructing Gaussian signals from noisy measurements, effectively handling uncertainties in both signal and noise covariances, with demonstrated superior performance in various scenarios.

Contribution

It extends existing signal reconstruction methods by incorporating simultaneous uncertainties in signal and noise covariances using a Gibbs free energy framework.

Findings

The algorithm performs well across different applications.

It outperforms methods that do not account for noise covariance uncertainty.

Demonstrates robustness in astrophysical data scenarios.

Abstract

We derive a method to reconstruct Gaussian signals from linear measurements with Gaussian noise. This new algorithm is intended for applications in astrophysics and other sciences. The starting point of our considerations is the principle of minimum Gibbs free energy which was previously used to derive a signal reconstruction algorithm handling uncertainties in the signal covariance. We extend this algorithm to simultaneously uncertain noise and signal covariances using the same principles in the derivation. The resulting equations are general enough to be applied in many different contexts. We demonstrate the performance of the algorithm by applying it to specific example situations and compare it to algorithms not allowing for uncertainties in the noise covariance. The results show that the method we suggest performs very well under a variety of circumstances and is indeed qualitatively superior to the other methods in cases where uncertainty in the noise covariance is present.