Array element localization using multidimensional scaling (MDS) and matrix completion, with comparisons to more conventional non-linear least squares optimization

Abstract

Multidimensional scaling (MDS) is an approach from the social sciences which was invented to project qualitative notions of distance onto 2D or 3D coordinate axes to provide visual displays for clustering purposes. As it turns out, this provides a viable algorithm for array element localization. When we calibrate an array using acoustic or RF ranging signals we measure inter-element distances and form a Euclidean Distance Matrix (EDM), and then convert them to 3D orthogonal coordinates. MDS converts the EDM to AEL coordinates. This process has some drawbacks. We may not get all the inter-element distances, because our ranging signals may have limited detection ranges, or because of obstructions to the line of sight. Also, it is not clear how to incorporate knowledge of unequal measurement errors among the EDM entries. Matrix completion is an approach from compressed sensing that allows us to perform the MDS paradigm, despite missing values in the EDM. We will present simulations for typical AEL scenarios and compare AEL results from MDS and matrix completion with results obtained using non-linear least squares optimization.