Ad Hoc Microphone Array Calibration: Euclidean Distance Matrix Completion Algorithm and Theoretical Guarantees

This paper addresses the problem of ad hoc microphone array calibration where only partial information about the distances between microphones is available. We construct a matrix consisting of the pairwise distances and propose to estimate the missing entries based on a novel Euclidean distance matrix completion algorithm by alternative low-rank matrix completion and projection onto the Euclidean distance space. This approach confines the recovered matrix to the EDM cone at each iteration of the matrix completion algorithm. The theoretical guarantees of the calibration performance are obtained considering the random and locally structured missing entries as well as the measurement noise on the known distances. This study elucidates the links between the calibration error and the number of microphones along with the noise level and the ratio of missing distances. Thorough experiments on real data recordings and simulated setups are conducted to demonstrate these theoretical insights. A significant improvement is achieved by the proposed Euclidean distance matrix completion algorithm over the state-of-the-art techniques for ad hoc microphone array calibration.

💡 Research Summary

The paper tackles the problem of calibrating ad‑hoc microphone arrays when only a subset of the pairwise distances between microphones is available. The authors first estimate short‑range distances using the coherence of microphone signals in a diffuse noise field, which follows a sinc‑based model. Because this model breaks down for larger separations, distances beyond a maximum range d_max are missing, creating a structured pattern of missing entries. In addition, random failures or model mismatches cause sporadic missing entries.

The core idea is to treat the squared distance matrix M as a low‑rank matrix. By expressing M = 1_NΛ^T + Λ1_N^T – 2XX^T, where X contains the unknown microphone coordinates, the authors show that rank(M) ≤ ζ + 2 (ζ is the spatial dimension). For planar arrays ζ = 2, giving rank = 4. This low‑rank property enables the use of matrix completion techniques to recover the full matrix from partially observed entries.

The proposed algorithm alternates between (i) a standard low‑rank matrix completion step (implemented via OptSpace) that enforces consistency with the observed entries and maintains the target rank η = ζ + 2, and (ii) a projection onto the Euclidean Distance Matrix (EDM) cone. The projection forces the intermediate estimate to satisfy the geometric constraints of an EDM (positive semidefinite after double centering, non‑negative distances, etc.), thereby eliminating non‑physical artifacts such as negative squared distances. Repeating these two steps yields a refined estimate ˆM that converges to a matrix close to the true distance matrix.

Theoretical analysis assumes that each observed entry is corrupted by sub‑Gaussian noise whose variance scales with the true distance (reflecting the fact that longer distances are measured less accurately). Random sampling is modeled by an independent Bernoulli process with probability p. Under these conditions the authors prove an error bound of the form ‖ˆM – M‖_F ≤ C·(σ/√p)·√N, where σ characterizes the noise level and C is a constant. This bound shows that the reconstruction error decays as N⁻¹ᐟ², matching classic low‑rank matrix completion results, and that the algorithm remains robust even when a large fraction of entries are missing. For the structured missing pattern (distances > d_max) they demonstrate that as long as the “local connectivity” graph induced by the short‑range distances is sufficiently dense, the same error scaling holds.

Empirical validation is performed on two fronts. First, real recordings are collected in a reverberant meeting room with eight microphones placed around a circular table. Distances up to 73 cm are estimated via sinc‑fitting of the coherence; longer distances are unavailable. The proposed method is compared against classical multidimensional scaling (MDS), vanilla OptSpace, and a recent Cadzow‑based distance recovery technique. The new algorithm achieves a mean positional error reduction of roughly 30 %–45 % relative to the baselines, and it remains stable when the observed fraction drops below 50 %. Second, extensive simulations vary the number of microphones (N = 20 … 200), the sampling probability p (0.3–0.8), and the noise level σ. Results confirm the theoretical N⁻¹ᐟ² decay of the error and illustrate that the algorithm consistently outperforms the competitors across all regimes.

In summary, the paper makes three principal contributions: (1) a novel calibration framework that fuses low‑rank matrix completion with EDM cone projection, (2) rigorous error bounds that accommodate both random and structured missing entries under realistic sub‑Gaussian noise, and (3) thorough experimental evidence on both real‑world and synthetic data showing superior performance over state‑of‑the‑art methods. The approach eliminates the need for known acoustic sources, making it attractive for a wide range of applications such as smart‑room audio capture, distributed sensor networks, and robotic audition where microphone positions are unknown or only partially observable.