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Sound Field Estimation Using Optimal Transport Barycenters in the Presence of Phase Errors

Yuyang Liu, Johan Karlsson, Filip Elvander

TL;DR

This work addresses phase-perturbation errors in sound-field estimation by reframing plane-wave coefficients as lifted non-negative measures on the unit circle $\mathbb{T}$. An OT barycenter is computed to obtain a phase-consistent nominal estimate, with the ground cost $c(\psi_1,\psi_2) = |e^{i\psi_1}-e^{i\psi_2}|^2 + \gamma$, and sparsity induced by the $\gamma$ term. The estimator combines transport regularization with a data-fidelity penalty, yielding a convex optimization that recovers the phase-corrected coefficients via $\hat{\boldsymbol{\Phi}} = \int_{\mathbb{T}} e^{i\psi} d\boldsymbol{\mu}^{(0)}(\psi)$. Simulations in 2D show improved NMSE over baseline methods under additive noise and phase errors, with stronger gains as more microphones are used, indicating robust performance and practical potential for sound-field reconstruction under calibration uncertainties.

Abstract

This study introduces a novel approach for estimating plane-wave coefficients in sound field reconstruction, specifically addressing challenges posed by error-in-variable phase perturbations. Such systematic errors typically arise from sensor mis-calibration, including uncertainties in sensor positions and response characteristics, leading to measurement-induced phase shifts in plane wave coefficients. Traditional methods often result in biased estimates or non-convex solutions. To overcome these issues, we propose an optimal transport (OT) framework. This framework operates on a set of lifted non-negative measures that correspond to observation-dependent shifted coefficients relative to the unperturbed ones. By applying OT, the supports of the measures are transported toward an optimal average in the phase space, effectively morphing them into an indistinguishable state. This optimal average, known as barycenter, is linked to the estimated plane-wave coefficients using the same lifting rule. The framework addresses the ill-posed nature of the problem, due to the large number of plane waves, by adding a constant to the ground cost, ensuring the sparsity of the transport matrix. Convex consistency of the solution is maintained. Simulation results confirm that our proposed method provides more accurate coefficient estimations compared to baseline approaches in scenarios with both additive noise and phase perturbations.

Sound Field Estimation Using Optimal Transport Barycenters in the Presence of Phase Errors

TL;DR

This work addresses phase-perturbation errors in sound-field estimation by reframing plane-wave coefficients as lifted non-negative measures on the unit circle . An OT barycenter is computed to obtain a phase-consistent nominal estimate, with the ground cost , and sparsity induced by the term. The estimator combines transport regularization with a data-fidelity penalty, yielding a convex optimization that recovers the phase-corrected coefficients via . Simulations in 2D show improved NMSE over baseline methods under additive noise and phase errors, with stronger gains as more microphones are used, indicating robust performance and practical potential for sound-field reconstruction under calibration uncertainties.

Abstract

This study introduces a novel approach for estimating plane-wave coefficients in sound field reconstruction, specifically addressing challenges posed by error-in-variable phase perturbations. Such systematic errors typically arise from sensor mis-calibration, including uncertainties in sensor positions and response characteristics, leading to measurement-induced phase shifts in plane wave coefficients. Traditional methods often result in biased estimates or non-convex solutions. To overcome these issues, we propose an optimal transport (OT) framework. This framework operates on a set of lifted non-negative measures that correspond to observation-dependent shifted coefficients relative to the unperturbed ones. By applying OT, the supports of the measures are transported toward an optimal average in the phase space, effectively morphing them into an indistinguishable state. This optimal average, known as barycenter, is linked to the estimated plane-wave coefficients using the same lifting rule. The framework addresses the ill-posed nature of the problem, due to the large number of plane waves, by adding a constant to the ground cost, ensuring the sparsity of the transport matrix. Convex consistency of the solution is maintained. Simulation results confirm that our proposed method provides more accurate coefficient estimations compared to baseline approaches in scenarios with both additive noise and phase perturbations.
Paper Structure (8 sections, 21 equations, 2 figures)

This paper contains 8 sections, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Signal model
  3. Proposed method
  4. Lifting
  5. Optimal Transport Barycenter
  6. Resulting estimator
  7. Numerical results
  8. Conclusions

Figures (2)

  • Figure 1: Top row: Magnitude of the sound field generated by three plane waves. Bottom row: Magnitude of the plane-wave coefficients, where ${\mathbf n}=(n_x,n_y)$ represents the unit-length direction vector. Non-zero positions indicate the incident wave direction. Reconstructions using LAD-Lasso, Lasso, Tikhonov, and the proposed method, based on nine measurement points, are compared to the ground truth. Measurement sensor locations are marked with red crosses in the ground truth panel, and cross-validation points are shown in blue.
  • Figure 2: NMSE as a function of system parameters based on 30 Monte Carlo experiments. Left: NMSE versus the number of microphones; Middle: NMSE versus frequency; Right: NMSE versus the standard deviation of the phase error.