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

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.

💡 Research Summary

The paper tackles the problem of estimating plane‑wave coefficients for sound‑field reconstruction when microphone arrays suffer from calibration errors that manifest as phase perturbations. Traditional sparse recovery methods such as LASSO, LAD‑LASSO, or Tikhonov regularization assume that the measurement model is only corrupted by additive Gaussian noise. When the sensor positions or responses are uncertain, the measured complex pressures acquire an extra multiplicative phase term, leading to biased estimates and non‑convex optimization problems.

To address this, the authors introduce a novel framework based on optimal transport (OT) barycenters. The key idea is to “lift’’ each complex coefficient αℓ into a non‑negative measure μℓ defined on the unit circle T =