On the peak height distribution of non-stationary Gaussian random fields: 1D general covariance and scale space

TL;DR

This work derives an explicit peak-height density for centered, smooth Gaussian processes with general (non-stationary) covariance in 1D, showing the density is governed by two clearly interpretable quantities $\rho(t)$ and $\tilde{\sigma}(t)$. It then extends the analysis to multidimensional scale-space Gaussian fields, proving that the peak-height distribution is invariant to location and scale, which streamlines both theory and application. To enable practical computation, the authors introduce two numerical Kac–Rice algorithms that evaluate conditional Gaussian expectations rather than relying on brute-force field simulations, with validations showing substantial efficiency gains. The results provide a principled framework for peak detection in non-stationary Gaussian fields across scales, with direct implications for fields like neuroimaging and image analysis, and offer practical tools for implementing peak-inference in nonstationary settings.

Abstract

We study the peak height distribution of certain non-stationary Gaussian random fields. The explicit peak height distribution of smooth, non-stationary Gaussian processes in 1D with general covariance is derived. The formula is determined by two parameters, each of which has a clear statistical meaning. For multidimensional non-stationary Gaussian random fields, we generalize these results to the setting of scale space fields, which play an important role in peak detection by helping to handle peaks of different spatial extents. We demonstrate that these properties not only offer a better interpretation of the scale space field but also simplify the computation of the peak height distribution. Finally, two efficient numerical algorithms are proposed as a general solution for computing the peak height distribution of smooth multidimensional Gaussian random fields in applications.

Figures (10)

• Figure 1: The peak height density \ref{['eq:peak_height_1D']} with different parameters. Left panel: Fixing $\tilde{\sigma}(t)$ at 1, the effect of $\rho(t)$ on the peak height density. Right panel: Fixing $\rho(t)$ at $-1/\sqrt{3}$, the effect of $\tilde{\sigma}(t)$ on the peak height density.
• Figure 2: Simulated instances of the Gaussian process with non-constant standard deviation function $\sigma(t) = t + 0.1$, its spatial correlation, and the parameters $\sigma(t)$, $\tilde{\sigma}(t)$, and $\rho(t)$.
• Figure 3: Simulated instances of the Cosine process in Example \ref{['ex:cos']}, $\sigma(t)$, $\tilde{\sigma}(t)$ and the peak height distribution at $t = \pi/4$ (large $\tilde{\sigma}(t)$) versus $t = \pi/2$ (small $\tilde{\sigma}(t)$).
• Figure 4: Simulated instances of the Gaussian process with non-constant bandwidth (Gaussian kernel) described in Example \ref{['ex:non_const_band']}, its spatial correlation, and the parameters $\sigma(t)$, $\tilde{\sigma}(t)$, and $\rho(t)$.
• Figure 5: Simulated instances of the Gaussian process with non-constant bandwidth and variance described in Example \ref{['ex:non_const_var_band']}, and the parameters $\sigma(t)$, $\tilde{\sigma}(t)$, and $\rho(t)$.

Theorems & Definitions (15)

• Lemma 1
• Theorem 1
• Example 1
• Proposition 1
• Example 2
• Theorem 2
• Example 3
• Example 4
• Theorem 3
• proof : Proof of Lemma \ref{['lem:X_cov']}