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

Partial Relaxation Approach: An Eigenvalue-Based DOA Estimator Framework

In this paper, the partial relaxation approach is introduced and applied to DOA estimation using spectral search. Unlike existing methods like Capon or MUSIC which can be considered as single source approximations of multi-source estimation criteria, the proposed approach accounts for the existence of multiple sources. At each considered direction, the manifold structure of the remaining interfering signals impinging on the sensor array is relaxed, which results in closed form estimates for the interference parameters. The conventional multidimensional optimization problem reduces, thanks to this relaxation, to a simple spectral search. Following this principle, we propose estimators based on the Deterministic Maximum Likelihood, Weighted Subspace Fitting and covariance fitting methods. To calculate the pseudo-spectra efficiently, an iterative rooting scheme based on the rational function approximation is applied to the partial relaxation methods. Simulation results show that the performance of the proposed estimators is superior to the conventional methods especially in the case of low Signal-to-Noise-Ratio and low number of snapshots, irrespectively of any specific structure of the sensor array while maintaining a comparable computational cost as MUSIC.Comment: This work has been submitted to IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

A Compact Formulation for the ℓ2,1\ell_{2,1} Mixed-Norm Minimization Problem

Parameter estimation from multiple measurement vectors (MMVs) is a fundamental problem in many signal processing applications, e.g., spectral analysis and direction-of- arrival estimation. Recently, this problem has been address using prior information in form of a jointly sparse signal structure. A prominent approach for exploiting joint sparsity considers mixed-norm minimization in which, however, the problem size grows with the number of measurements and the desired resolution, respectively. In this work we derive an equivalent, compact reformulation of the $\ell_{2,1}$ mixed-norm minimization problem which provides new insights on the relation between different existing approaches for jointly sparse signal reconstruction. The reformulation builds upon a compact parameterization, which models the row-norms of the sparse signal representation as parameters of interest, resulting in a significant reduction of the MMV problem size. Given the sparse vector of row-norms, the jointly sparse signal can be computed from the MMVs in closed form. For the special case of uniform linear sampling, we present an extension of the compact formulation for gridless parameter estimation by means of semidefinite programming. Furthermore, we derive in this case from our compact problem formulation the exact equivalence between the $\ell_{2,1}$ mixed-norm minimization and the atomic-norm minimization. Additionally, for the case of irregular sampling or a large number of samples, we present a low complexity, grid-based implementation based on the coordinate descent method

3-D Hybrid Localization with RSS/AoA in Wireless Sensor Networks: Centralized Approach

This dissertation addresses one of the most important issues present in Wireless Sensor Networks (WSNs), which is the sensor’s localization problem in non-cooperative and cooperative 3-D WSNs, for both cases of known and unknown source transmit power PT . The localization of sensor nodes in a network is essential data. There exists a large number of applications for WSNs and the fact that sensors are robust, low cost and do not require maintenance, makes these types of networks an optimal asset to study or manage harsh and remote environments. The main objective of these networks is to collect different types of data such as temperature, humidity, or any other data type, depending on the intended application. The knowledge of the sensors’ locations is a key feature for many applications; knowing where the data originates from, allows to take particular type of actions that are suitable for each case. To face this localization problem a hybrid system fusing distance and angle measurements is employed. The measurements are assumed to be collected through received signal strength indicator and from antennas, extracting the received signal strength (RSS) and angle of arrival (AoA) information. For non-cooperativeWSN, it resorts to these measurements models and, following the least squares (LS) criteria, a non-convex estimator is developed. Next, it is shown that by following the square range (SR) approach, the estimator can be transformed into a general trust region subproblem (GTRS) framework. For cooperative WSN it resorts also to the measurement models mentioned above and it is shown that the estimator can be converted into a convex problem using semidefinite programming (SDP) relaxation techniques.It is also shown that the proposed estimators have a straightforward generalization from the known PT case to the unknown PT case. This generalization is done by making use of the maximum likelihood (ML) estimator to compute the value of the PT . The results obtained from simulations demonstrate a good estimation accuracy, thus validating the exceptional performance of the considered approaches for this hybrid localization system