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.Comment: In Press, available online, August 1, 2014. http://www.sciencedirect.com/science/article/pii/S0165168414003508, Signal Processing, 201

Ad-Hoc Microphone Array Calibration from Partial Distance Measurements

We address the problem of ad~hoc microphone array calibration where some of the distances between the microphones can not be measured. The conventional techniques require information about all the distances for accurate reconstruction of the array geometry. To alleviate this condition, we propose to exploit the properties of Euclidean distance matrices within the framework of low-rank matrix completion to recover the missing entries. We provide rigorous analysis to bound the calibration error using noisy measurements. This study elucidates the links between the performance and the structure of the missing distances, along with the size of the network. The experiments carried out on real data recordings 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

Euclidean Distance Matrix Completion for Ad-hoc Microphone Array Calibration

This paper addresses the application of missing data recovery via matrix completion for audio sensor networks. We propose a method based on Euclidean distance matrix completion for ad-hoc microphone array location calibration. This method can calibrate a full network from partial connectivity informa- tion. The pairwise distances of microphones in close proximity are estimated using the coherence model of the diffuse noise field. The distance matrix of the ad-hoc network is constructed where the distances of the microphones above a threshold are missing. We exploit the low-rank property of the squared distance matrix and apply a matrix completion method to recover the missing entries. In order to constrain the Euclidean space geometry, we propose the additional use of the Cadzow algorithm for matrix completion. The applicability of the proposed method is evaluated on real data recordings where a significant improvement over the state-of-the-art is achieved

Theoretical Analysis of Euclidean Distance Matrix Completion for Ad hoc Microphone Array Calibration

We consider the problem of ad~hoc microphone array calibration where the distance matrix consisted of all microphones pairwise distances have entries missing corresponding to distances greater than $d_{\text{max}}$. Furthermore, the known entries are noisy modeled through additive independent random variables with strictly sub-Gaussian distribution, \textsc{S}\textsc{ub}(c^2(d)) with a bounded constant dependent on the distance $d$ between the microphone pairs. In this report, we exploit matrix completion approach to recover the full distance matrix. We derive the theoretical guarantees of microphone calibration performance which demonstrates that the error of calibrating a network of $N$ microphones using matrix completion decreases as $\mathcal{O}(N^{-1/2})$

Euclidean Distance Matrices: Essential Theory, Algorithms and Applications

Euclidean distance matrices (EDM) are matrices of squared distances between points. The definition is deceivingly simple: thanks to their many useful properties they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness. We show how various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine - change of title in the last revisio

Self-Localization of Ad-Hoc Arrays Using Time Difference of Arrivals

This work was supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/K007491/1

ON APPLICATION OF NON-NEGATIVE MATRIX FACTORIZATION FOR AD HOC MICROPHONE ARRAY CALIBRATION FROM INCOMPLETE NOISY DISTANCES

ABSTRACT We propose to use non-negative matrix factorization (NMF) to estimate the unknown pairwise distances and reconstruct a distance matrix for microphone array position calibration. We develop new multiplicative update rules for NMF with incomplete input matrix that take into account the symmetry of the distance matrix. Additionally, we develop a convex matrix completion method which is related to an l2-regularized symmetric NMF. Thorough experiments demonstrate that the proposed methods lead to substantial improvement over the state-of-the-art techniques in a wide range of signalto-noise and unknown-distance ratios. The convex symmetric matrix completion method was found to be the most robust method with less computational cost

On Application Of Non-Negative Matrix Factorization for Ad Hoc Microphone Array Calibration from Incomplete Noisy Distances

We propose to use non-negative matrix factorization (NMF) to estimate the unknown pairwise distances and reconstruct a distance matrix for microphone array position calibration. We develop new multiplicative update rules for NMF with incomplete input matrix that take into account the symmetry of the distance matrix. Additionally, we develop a convex matrix completion method which is related to an $l_2$-regularized symmetric NMF. Thorough experiments demonstrate that the proposed methods lead to substantial improvement over the state-of-the-art techniques in a wide range of signal-to-noise and unknown-distance ratios. The convex symmetric matrix completion method was found to be the most robust method with less computational cost

Localization of sound sources : a systematic review

Sound localization is a vast field of research and advancement which is used in many useful applications to facilitate communication, radars, medical aid, and speech enhancement to but name a few. Many different methods are presented in recent times in this field to gain benefits. Various types of microphone arrays serve the purpose of sensing the incoming sound. This paper presents an overview of the importance of using sound localization in different applications along with the use and limitations of ad-hoc microphones over other microphones. In order to overcome these limitations certain approaches are also presented. Detailed explanation of some of the existing methods that are used for sound localization using microphone arrays in the recent literature is given. Existing methods are studied in a comparative fashion along with the factors that influence the choice of one method over the others. This review is done in order to form a basis for choosing the best fit method for our use

From Acoustic Room Reconstruction to SLAM

Recent works on reconstruction of room geometry from echoes assume that the geometry of the sensor array is known. In this paper, we show that such an assumption is not essential; echoes provide sufficient clues to reconstruct the room’s and the array’s geometries jointly, even from a single acoustic event. Rather than focusing on the combinatorial problem of matching the walls and the recorded echoes, we provide algorithms for solving the joint estimation problem in practical cases when this matching is known and the number of microphones is small. We then explore intriguing connections between this problem and simultaneous localization and mapping (SLAM), and show that SLAM can be solved by the same methods. Finally, we demonstrate how effective the proposed methods are by numerical simulations and experiments with real measured room impulse responses