Reconstruction of Gaussian and log-normal fields with spectral smoothness

TL;DR

This paper presents a novel method for reconstructing log-normal fields from noisy measurements by leveraging spectral smoothness priors and Gibbs free energy minimization, enabling joint inference of the field and its correlation structure.

Contribution

It introduces a new reconstruction algorithm that incorporates spectral smoothness priors and uses Gibbs free energy formalism for joint inference of fields and power spectra.

Findings

Effective reconstruction of log-normal fields demonstrated in test cases.

Spectral smoothness prior improves reconstruction accuracy.

Method handles varying noise levels and non-linearity.

Abstract

We develop a method to infer log-normal random fields from measurement data affected by Gaussian noise. The log-normal model is well suited to describe strictly positive signals with fluctuations whose amplitude varies over several orders of magnitude. We use the formalism of minimum Gibbs free energy to derive an algorithm that uses the signal's correlation structure to regularize the reconstruction. The correlation structure, described by the signal's power spectrum, is thereby reconstructed from the same data set. We show that the minimization of the Gibbs free energy, corresponding to a Gaussian approximation to the posterior marginalized over the power spectrum, is equivalent to the empirical Bayes ansatz, in which the power spectrum is fixed to its maximum a posteriori value. We further introduce a prior for the power spectrum that enforces spectral smoothness. The appropriateness of this prior in different scenarios is discussed and its effects on the reconstruction's results are demonstrated. We validate the performance of our reconstruction algorithm in a series of one- and two-dimensional test cases with varying degrees of non-linearity and different noise levels.