Relax and unfold: Microphone localization with Euclidean distance matrices

Abstract

Recent methods for microphone position calibration work with sound sources at a priori unknown locations. This is convenient for ad hoc arrays, as it requires little additional infrastructure. We propose a flexible localization algorithm by first recognizing the problem as an instance of multidimensional unfolding (MDU)-a classical problem in Euclidean geometry and psychometrics-and then solving the MDU as a special case of Euclidean distance matrix (EDM) completion. We solve the EDM completion using a semidefinite relaxation. In contrast to existing methods, the semidefinite formulation allows us to elegantly handle missing pairwise distance information, but also to incorporate various prior information about the distances between the pairs of microphones or sources, bounds on these distances, or ordinal information such as "microphones 1 and 2 are more apart than microphones 1 and 15". The intuition that this should improve the localization performance is justified by numerical experiments.

Loudspeaker and Listening Position Estimation Using Smart Speakers

“…The reverberation was added using a RIR-generator [21]. The proposed method was compared to two different reference methods which is a variation of [13] and the multidimensional unfolding method (MDU) in [7,8]. The former consists in that we use the source localisation method of the proposed method to compute the local maps and classical multidimensional scaling for combining these maps.…”

“…If the knowledge of the local geometry is taken into account, more accurate estimates can be obtained with only a few subarrays. Exactly this was recently demonstrated in [7,8], but the proposed multidimentional unfolding (MDU) method requires many sources and sensors to work. When this is the case, however, MDU outperforms existing methods.…”

“…Acoustic source and sensor geometry calibration has been a research topic for several decades. A lot of work has focused on creating a map for individual sensors and sources which were not necessarily synchronised (see, e.g., [3][4][5][6][7][8]). However, at least four sources, four sensors, and a total of at least ten transducers (sources + sensors) are required to solve the geometry calibration in the synchronised case [4], and many current audio reproduction systems consist of fewer sources and sensors than that.…”

From acoustic room reconstruction to slam

“…The relaxation of (8) works well for multidimensional unfolding (that is, it returns a point set with the correct embedding dimension) when the number of points is large enough (empirically at least 10 or 15 microphones and equally many acoustic events [22]). This number grows larger as the quality of measurements deteriorates.…”

Kinetic Euclidean Distance Matrices

“…T He famous distance geometry problem (DGP) [1] asks to reconstruct the geometry of a point set from a subset of interpoint distances. It models a wide gamut of practical problems, from sensor network localization [2], [3], [4] and microphone positioning [5], [6], [7], [8] to clock synchronization [9], [10], to molecular geometry reconstruction from NMR data [11], [12]. Among the most successful vehicles for the design of DGP algorithms are the Euclidean distance matrices (EDM) [13].…”