Sensor Networks TDOA Self-Calibration: 2D Complexity Analysis and Solutions

Given a network of receivers and transmitters, the process of determining their positions from measured pseudo-ranges is known as network self-calibration. In this paper we consider 2D networks with synchronized receivers but unsynchronized transmitters and the corresponding calibration techniques,known as TDOA techniques. Despite previous work, TDOA self-calibration is computationally challenging. Iterative algorithms are very sensitive to the initialization, causing convergence issues.In this paper, we present a novel approach, which gives an algebraic solution to three previously unsolved scenarios. Our solvers can lead to a position error <1.2% and are robust to noise

Localization using Distance Geometry : Minimal Solvers and Robust Methods for Sensor Network Self-Calibration

In this thesis, we focus on the problem of estimating receiver and sender node positions given some form of distance measurements between them. This kind of localization problem has several applications, e.g., global and indoor positioning, sensor network calibration, molecular conformations, data visualization, graph embedding, and robot kinematics. More concretely, this thesis makes contributions in three different areas.First, we present a method for simultaneously registering and merging maps. The merging problem occurs when multiple maps of an area have been constructed and need to be combined into a single representation. If there are no absolute references and the maps are in different coordinate systems, they also need to be registered. In the second part, we construct robust methods for sensor network self-calibration using both Time of Arrival (TOA) and Time Difference of Arrival (TDOA) measurements. One of the difficulties is that corrupt measurements, so-called outliers, are present and should be excluded from the model fitting. To achieve this, we use hypothesis-and-test frameworks together with minimal solvers, resulting in methods that are robust to noise, outliers, and missing data. Several new minimal solvers are introduced to accommodate a range of receiver and sender configurations in 2D and 3D space. These solvers are formulated as polynomial equation systems which are solvedusing methods from algebraic geometry.In the third part, we focus specifically on the problems of trilateration and multilateration, and we present a method that approximates the Maximum Likelihood (ML) estimator for different noise distributions. The proposed approach reduces to an eigendecomposition problem for which there are good solvers. This results in a method that is faster and more numerically stable than the state-of-the-art, while still being easy to implement. Furthermore, we present a robust trilateration method that incorporates a motion model. This enables the removal of outliers in the distance measurements at the same time as drift in the motion model is canceled

Mapping and Merging Using Sound and Vision : Automatic Calibration and Map Fusion with Statistical Deformations

Over the last couple of years both cameras, audio and radio sensors have become cheaper and more common in our everyday lives. Such sensors can be used to create maps of where the sensors are positioned and the appearance of the surroundings. For sound and radio, the process of estimating the sender and receiver positions from time of arrival (TOA) or time-difference of arrival (TDOA) measurements is referred to as automatic calibration. The corresponding process for images is to estimate the camera positions as well as the positions of the objects captured in the images. This is called structure from motion (SfM) or visual simultaneous localisation and mapping (SLAM). In this thesis we present studies on how to create such maps, divided into three parts: to find accurate measurements; robust mapping; and merging of maps.The first part is treated in Paper I and involves finding precise – on a subsample level – TDOA measurements. These types of subsample refinements give a high precision, but are sensitive to noise. We present an explicit expression for the variance of the TDOA estimate and study the impact that noise in the signals has. Exact measurements is an important foundation for creating accurate maps. The second part of this thesis includes Papers II–V and covers the topic of robust self-calibration using one-dimensional signals, such as sound or radio. We estimate both sender and receiver positions using TOA and TDOA measurements. The estimation process is divided in two parts, where the first is specific for TOA or TDOA and involves solving a relaxed version of the problem. The second step is common for different types of problems and involves an upgrade from the relaxed solution to the sought parameters. In this thesis we present numerically stable minimal solvers for both these steps for some different setups with senders and receivers. We also suggest frameworks for how to use these solvers together with RANSAC to achieve systems that are robust to outliers, noise and missing data. Additionally, in the last paper we focus on extending self-calibration results, especially for the sound source path, which often cannot be fully reconstructed immediately. The third part of the thesis, Papers VI–VIII, is concerned with the merging of already estimated maps. We mainly focus on maps created from image data, but the methods are applicable to sparse 3D maps coming from different sensor modalities. Merging of maps can be advantageous if there are several map representations of the same environment, or if there is a need for adding new information to an already existing map. We suggest a compact map representation with a small memory footprint, which we then use to fuse maps efficiently. We suggest one method for fusion of maps that are pre-aligned, and one where we additionally estimate the coordinate system. The merging utilises a compact approximation of the residuals and allows for deformations in the original maps. Furthermore, we present minimal solvers for 3D point matching with statistical deformations – which increases the number of inliers when the original maps contain errors

A Unifying Approach to Minimal Problems in Collinear and Planar TDOA Sensor Network Self-Calibration

This work presents a study of sensor network calibration from time-difference-of-arrival (TDOA) measurements for cases when the dimensions spanned by the receivers and the transmitters differ. This could for example be if receivers are restricted to a line or plane or if the transmitting objects are moving linearly in space. Such calibration arises in several applications such as calibration of (acoustic or ultrasound) microphone arrays, and radio antenna networks. We propose a non-iterative algorithm based on recent stratified approaches: (i) rank constraints on modified measurement matrix, (ii) factorization techniques that determine transmitters and receivers up to unknown affine transformation and (iii) determining the affine stratification using remaining non-linear constraints. This results in a unified approach to solve almost all minimal problems. Such algorithms are important components for systems for self-localization. Experiments are shown both for simulated and real data with promising results

Multiple Offsets Multilateration : A New Paradigm for Sensor Network Calibration with Unsynchronized Reference Nodes

Positioning using wave signal measurements is used in several applications, such as GPS systems, structure from sound and Wifi based positioning. Mathematically, such problems require the computation of the positions of receivers and/or transmitters as well as time offsets if the devices are unsynchronized. In this paper, we expand the previous state-of-the-art on positioning formulations by introducing Multiple Offsets Multilateration (MOM), a new mathematical framework to compute the receivers positions with pseudoranges from unsynchronized reference transmitters at known positions. This could be applied in several scenarios, for example structure from sound and positioning with LEO satellites. We mathematically describe MOM, determining how many receivers and transmitters are needed for the network to be solvable, a study on the number of possible distinct solutions is presented and stable solvers based on homotopy continuation are derived. The solvers are shown to be efficient and robust to noise both for synthetic and real audio data.©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.fi=vertaisarvioitu|en=peerReviewed

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

Beyond Gr\"obner Bases: Basis Selection for Minimal Solvers

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Gr\"obner basis for the polynomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Gr\"obner bases leads to more efficient solvers in many cases. We present a novel basis sampling scheme that we evaluate on a number of problems

Mathematical modelling ano optimization strategies for acoustic source localization in reverberant environments

La presente Tesis se centra en el uso de técnicas modernas de optimización y de procesamiento de audio para la localización precisa y robusta de personas dentro de un entorno reverberante dotado con agrupaciones (arrays) de micrófonos. En esta tesis se han estudiado diversos aspectos de la localización sonora, incluyendo el modelado, la algoritmia, así como el calibrado previo que permite usar los algoritmos de localización incluso cuando la geometría de los sensores (micrófonos) es desconocida a priori. Las técnicas existentes hasta ahora requerían de un número elevado de micrófonos para obtener una alta precisión en la localización. Sin embargo, durante esta tesis se ha desarrollado un nuevo método que permite una mejora de más del 30\% en la precisión de la localización con un número reducido de micrófonos. La reducción en el número de micrófonos es importante ya que se traduce directamente en una disminución drástica del coste y en un aumento de la versatilidad del sistema final. Adicionalmente, se ha realizado un estudio exhaustivo de los fenómenos que afectan al sistema de adquisición y procesado de la señal, con el objetivo de mejorar el modelo propuesto anteriormente. Dicho estudio profundiza en el conocimiento y modelado del filtrado PHAT (ampliamente utilizado en localización acústica) y de los aspectos que lo hacen especialmente adecuado para localización. Fruto del anterior estudio, y en colaboración con investigadores del instituto IDIAP (Suiza), se ha desarrollado un sistema de auto-calibración de las posiciones de los micrófonos a partir del ruido difuso presente en una sala en silencio. Esta aportación relacionada con los métodos previos basados en la coherencia. Sin embargo es capaz de reducir el ruido atendiendo a parámetros físicos previamente conocidos (distancia máxima entre los micrófonos). Gracias a ello se consigue una mejor precisión utilizando un menor tiempo de cómputo. El conocimiento de los efectos del filtro PHAT ha permitido crear un nuevo modelo que permite la representación 'sparse' del típico escenario de localización. Este tipo de representación se ha demostrado ser muy conveniente para localización, permitiendo un enfoque sencillo del caso en el que existen múltiples fuentes simultáneas. La última aportación de esta tesis, es el de la caracterización de las Matrices TDOA (Time difference of arrival -Diferencia de tiempos de llegada, en castellano-). Este tipo de matrices son especialmente útiles en audio pero no están limitadas a él. Además, este estudio transciende a la localización con sonido ya que propone métodos de reducción de ruido de las medias TDOA basados en una representación matricial 'low-rank', siendo útil, además de en localización, en técnicas tales como el beamforming o el autocalibrado

Methods for Optimal Model Fitting and Sensor Calibration

The problem of fitting models to measured data has been studied extensively, not least in the field of computer vision. A central problem in this field is the difficulty in reliably find corresponding structures and points in different images, resulting in outlier data. This thesis presents theoretical results improving the understanding of the connection between model parameter estimation and possible outlier-inlier partitions of data point sets. Using these results a multitude of applications can be analyzed in respects to optimal outlier inlier partitions, optimal norm fitting, and not least in truncated norm sense. Practical polynomial time optimal solvers are derived for several applications, including but not limited to multi-view triangulation and image registration. In this thesis the problem of sensor network self calibration is investigated. Sensor networks play an increasingly important role with the increased availability of mobile, antenna equipped, devices. The application areas can be extended with knowledge of the different sensors relative or absolute positions. We study this problem in the context of bipartite sensor networks. We identify requirements of solvability for several configurations, and present a framework for how such problems can be approached. Further we utilize this framework to derive several solvers, which we show in both synthetic and real examples functions as desired. In both these types of model estimation, as well as in the classical random samples based approaches minimal cases of polynomial systems play a central role. A majority of the problems tackled in this thesis will have solvers based on recent techniques pertaining to action matrix solvers. New application specific polynomial equation sets are constructed and elimination templates designed for them. In addition a general improvement to the method is suggested for a large class of polynomial systems. The method is shown to improve the computational speed by significant reductions in the size of elimination templates as well as in the size of the action matrices. In addition the methodology on average improves the numerical stability of the solvers

Acoustic sensor network geometry calibration and applications

In the modern world, we are increasingly surrounded by computation devices with communication links and one or more microphones. Such devices are, for example, smartphones, tablets, laptops or hearing aids. These devices can work together as nodes in an acoustic sensor network (ASN). Such networks are a growing platform that opens the possibility for many practical applications. ASN based speech enhancement, source localization, and event detection can be applied for teleconferencing, camera control, automation, or assisted living. For this kind of applications, the awareness of auditory objects and their spatial positioning are key properties. In order to provide these two kinds of information, novel methods have been developed in this thesis. Information on the type of auditory objects is provided by a novel real-time sound classification method. Information on the position of human speakers is provided by a novel localization and tracking method. In order to localize with respect to the ASN, the relative arrangement of the sensor nodes has to be known. Therefore, different novel geometry calibration methods were developed. Sound classification The first method addresses the task of identification of auditory objects. A novel application of the bag-of-features (BoF) paradigm on acoustic event classification and detection was introduced. It can be used for event and speech detection as well as for speaker identification. The use of both mel frequency cepstral coefficient (MFCC) and Gammatone frequency cepstral coefficient (GFCC) features improves the classification accuracy. By using soft quantization and introducing supervised training for the BoF model, superior accuracy is achieved. The method generalizes well from limited training data. It is working online and can be computed in a fraction of real-time. By a dedicated training strategy based on a hierarchy of stationarity, the detection of speech in mixtures with noise was realized. This makes the method robust against severe noises levels corrupting the speech signal. Thus it is possible to provide control information to a beamformer in order to realize blind speech enhancement. A reliable improvement is achieved in the presence of one or more stationary noise sources. Speaker localization The localization method enables each node to determine the direction of arrival (DoA) of concurrent sound sources. The author's neuro-biologically inspired speaker localization method for microphone arrays was refined for the use in ASNs. By implementing a dedicated cochlear and midbrain model, it is robust against the reverberation found in indoor rooms. In order to better model the unknown number of concurrent speakers, an application of the EM algorithm that realizes probabilistic clustering according to auditory scene analysis (ASA) principles was introduced. Based on this approach, a system for Euclidean tracking in ASNs was designed. Each node applies the node wise localization method and shares probabilistic DoA estimates together with an estimate of the spectral distribution with the network. As this information is relatively sparse, it can be transmitted with low bandwidth. The system is robust against jitter and transmission errors. The information from all nodes is integrated according to spectral similarity to correctly associate concurrent speakers. By incorporating the intersection angle in the triangulation, the precision of the Euclidean localization is improved. Tracks of concurrent speakers are computed over time, as is shown with recordings in a reverberant room. Geometry calibration The central task of geometry calibration has been solved with special focus on sensor nodes equipped with multiple microphones. Novel methods were developed for different scenarios. An audio-visual method was introduced for the calibration of ASNs in video conferencing scenarios. The DoAs estimates are fused with visual speaker tracking in order to provide sensor positions in a common coordinate system. A novel acoustic calibration method determines the relative positioning of the nodes from ambient sounds alone. Unlike previous methods that only infer the positioning of distributed microphones, the DoA is incorporated and thus it becomes possible to calibrate the orientation of the nodes with a high accuracy. This is very important for all applications using the spatial information, as the triangulation error increases dramatically with bad orientation estimates. As speech events can be used, the calibration becomes possible without the requirement of playing dedicated calibration sounds. Based on this, an online method employing a genetic algorithm with incremental measurements was introduced. By using the robust speech localization method, the calibration is computed in parallel to the tracking. The online method is be able to calibrate ASNs in real time, as is shown with recordings of natural speakers in a reverberant room. The informed acoustic sensor network All new methods are important building blocks for the use of ASNs. The online methods for localization and calibration both make use of the neuro-biologically inspired processing in the nodes which leads to state-of-the-art results, even in reverberant enclosures. The high robustness and reliability can be improved even more by including the event detection method in order to exclude non-speech events. When all methods are combined, both semantic information on what is happening in the acoustic scene as well as spatial information on the positioning of the speakers and sensor nodes is automatically acquired in real time. This realizes truly informed audio processing in ASNs. Practical applicability is shown by application to recordings in reverberant rooms. The contribution of this thesis is thus not only to advance the state-of-the-art in automatically acquiring information on the acoustic scene, but also pushing the practical applicability of such methods