Euclidean Distance Matrices: Essential Theory, Algorithms and Applications

TL;DR

This paper provides a comprehensive tutorial on Euclidean distance matrices (EDMs), covering their fundamental properties, algorithms for data completion and denoising, and various practical applications across multiple fields.

Contribution

It offers a concise overview of EDM theory, algorithms, and applications, including new insights into their properties and practical implementation guidance for signal processing tasks.

Findings

Algorithms for EDM completion and denoising demonstrated

Applications in microphone calibration and ultrasound tomography shown

Matlab code provided for reproducibility

Abstract

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.