Gaussian processes for sound field reconstruction

Abstract

This study examines the use of Gaussian process (GP) regression for sound field reconstruction. GPs enable the reconstruction of a sound field from a limited set of observations based on the use of a covariance function (a kernel) that models the spatial correlation between points in the sound field. Significantly, the approach makes it possible to quantify the uncertainty on the reconstruction in a closed form. In this study, the relation between reconstruction based on GPs and classical reconstruction methods based on linear regression is examined from an acoustical perspective. Several kernels are analyzed for their potential in sound field reconstruction, and a hierarchical Bayesian parameterization is introduced, which enables the construction of a plane wave kernel of variable sparsity. The performance of the kernels is numerically studied and compared to classical reconstruction methods based on linear regression. The results demonstrate the benefits of using GPs in sound field analysis. The hierarchical parameterization shows the overall best performance, adequately reconstructing fundamentally different sound fields. The approach appears to be particularly powerful when prior knowledge of the sound field would not be available. V

Reconstruction of room impulse responses over extended domains for navigable sound field reproduction

“…where e(t, r) is measurement noise and p(t, r) is the unknown function that defines the sound field, and we use Gaussian process regression to find it [33].…”

“…If the measurement noise is also Gaussian, such that e(t, r) ∼ GP(0, κ e (t, r, t , r )) (10) the predictive sound field function p • (t, r) is the outcome of a Gaussian process given the measured pressure p [33] p…”

“…which is the spatial correlation between any two points r and r , (with r = r − r ), in a random wave field, and σ 2 is a variance factor scaling the amplitude of the kernel [33]. This kernel is the zeroth order spherical Bessel function j 0 (kr) in three-dimensions, and is often referred to as the Bessel kernel or diffuse-field kernel.…”

“…The rationale behind this choice is that the spatiotemporal structure of the room reverberation can be incorporated more explicitly via a Neumann type Green's function, leading to a better reconstruction. The model is formulated as a Gaussian process regression [33], which enables to incorporate the spatial statistics of the sound fields, and account for the expected anisotropy of the early part, in comparison to more isotropic late response [34]. Gaussian processes enable to do this conveniently, by including structure via the second order spatial statistics of the field.…”

“…Gaussian processes enable to do this conveniently, by including structure via the second order spatial statistics of the field. The framework is also convenient for defining a hierarchical Bayesian model, which is useful for the problem at hand [33], [35], [36].…”