Reconstruction of Gaussian and log-normal fields with spectral smoothness

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.

💡 Research Summary

The paper presents a Bayesian framework for reconstructing log‑normal random fields from noisy observations, where the noise is assumed Gaussian. The authors adopt the minimum Gibbs free energy principle to derive a set of filter equations that simultaneously estimate the underlying field and its correlation structure, i.e., the power spectrum, directly from the data.

Starting from a linear measurement model d = R s + n with Gaussian noise covariance N, the signal s is modeled as the exponential of a Gaussian field φ (s = exp φ). The prior for φ is a zero‑mean Gaussian with covariance S, which is unknown a priori. By marginalising over φ and approximating the resulting posterior with a Gaussian, the Gibbs free energy functional can be minimized analytically. This yields the familiar Wiener‑filter‑like expression for the posterior mean m = D j, where D = (S⁻¹ + RᵀN⁻¹R)⁻¹ and j = RᵀN⁻¹d.

To infer the unknown power spectrum Pₖ (the diagonal of S in the harmonic basis), each spectral component is assigned an independent inverse‑Gamma prior IG(αₖ, qₖ). In the limit αₖ→1, qₖ→0 this becomes Jeffreys’ non‑informative prior, flat in log‑space. The authors then introduce a smoothness prior on the logarithmic spectrum pₖ = log Pₖ, penalising large second differences (or Laplacian) across neighboring k‑modes: exp