We study the application of matrix completion in the process of calibrating physical devices. In particular we propose an algorithm together with reconstruction bounds for calibrating circular ultrasound tomography devices. We use the time-of-flight (ToF) measurements between sensor pairs in a homogeneous medium to calibrate the system. The calibration process consists of a low-rank matrix completion algorithm to de-noise and estimate random and structured missing ToFs, and the classic multi-dimensional scaling method to estimate the sensor positions from the ToF measurements. We provide theoretical bounds on the calibration error. Several simulations are conducted to evaluate the theoretical results presented in this paper.
Calibration for circular tomography devises is a variant of sensor localization, a problem that has been extensively studied for the past decade @cite_15 @cite_20 . In sensor localization, given the (i.e., which sensors are in the communication range of which others), the objective is to devise an algorithm that can infer the global position of the sensors. In practice, several methods are deployed as a means of obtaining this local information: the Signal Strength @cite_1 , the Angle of Arrival (AOA) @cite_21 , and the Time Difference of Arrival (TDOA) @cite_7 . Our problem is naturally related to sensor localization when estimated TDOAs are used to measure the pairwise distances between nearby nodes. One should note that due to energy constraints, each node has a small communication range compared to the field size they are installed. As a result, only nodes within the communication range of each other can communicate and hence estimate their pairwise TDOA's. This situation is depicted in Fig. .
@cite_7 @cite_21 @cite_1 @cite_15 @cite_20
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Calibration of ultrasound tomography devices is a challenging problem and of highly practical interest in medical and seismic imaging. This work addresses the position calibration problem in circular apertures where sensors are arranged on a circular ring and act both as transmitters and receivers . We introduce a new method of calibration based on the timeof-flight (ToF) measurements between sensors when the enclo sed medium is homogeneous. Knowing all the pairwise ToFs, one can find the positions of the sensors using multi-di mensional scaling (MDS) method. In practice, however, we are facing two major sources of loss. One is due to the transitional behaviour of the sensors, which makes the ToF measurements for close-by sensors unavailable. The other is due to the random malfunctioning of the sensors, that leads to random missing ToF measurements. On top of the missing entries, since in practice the impulse response of the piezoelectric and the time origin in the measurement procedure are not present, a time mismatch is also added to the measurements. In this work, we first show that a matrix defined from all the ToF measurements is of rank at most four. In order to estimate the structured and random missing entries, utilizing the fact that the matrix in question is shown to be low-rank, we apply a state-of-the-art low-rank matrix completion algorithm. Then we use MDS in order to find the correct positions of the sensors. To confirm the function ality of our method in practice, simulations mimicking the measurements of an ultrasound tomography device are performed. The localization of sensor nodes is a fundamental problem in sensor networks and can be implemented using powerful and expensive beacons. Beacons, the fewer the better, can acquire their position knowledge either from GPS devices or by virtue of being manually placed. In this paper, we propose a distributed method to localization of sensor nodes using a single moving beacon, where sensor nodes compute their position estimate based on the range-free technique. Two parameters are critical to the location accuracy of sensor nodes: the radio transmission range of the beacon and how often the beacon broadcasts its position. Theoretical analysis shows that these two parameters determine the upper bound of the estimation error when the traverse route of the beacon is a straight line. We extend the position estimate when the traverse route of the beacon is randomly chosen in a real-world situation, where the radio irregularity might cause a node to miss some crucial coordinate information from the beacon. We further point out that the movement pattern of the beacon plays a pivotal role in the localization task for sensors. To minimize estimation errors, sensor nodes can carry out a variety of algorithms in accordance with the movement of the beacon. Simulation results compare variants of the distributed method in a variety of testing environments. Real experiments show that the proposed method is feasible and can estimate the location of sensor nodes accurately, given a single moving beacon. This paper describes a distributed, linear-time algorithm for localizing sensor network nodes in the presence of range measurement noise and demonstrates the algorithm on a physical network. We introduce the probabilistic notion of robust quadrilaterals as a way to avoid flip ambiguities that otherwise corrupt localization computations. We formulate the localization problem as a two-dimensional graph realization problem: given a planar graph with approximately known edge lengths, recover the Euclidean position of each vertex up to a global rotation and translation. This formulation is applicable to the localization of sensor networks in which each node can estimate the distance to each of its neighbors, but no absolute position reference such as GPS or fixed anchor nodes is available. We implemented the algorithm on a physical sensor network and empirically assessed its accuracy and performance. Also, in simulation, we demonstrate that the algorithm scales to large networks and handles real-world deployment geometries. Finally, we show how the algorithm supports localization of mobile nodes. The problem of position estimation from time difference of arrival (TDOA) measurements occurs in a range of applications from wireless communication networks to electronic warfare positioning. Correlation analysis of the transmitted signal to two receivers gives rise to one hyperbolic function. With more than two receivers, we can compute more hyperbolic functions, which ideally intersect in one unique point. With TDOA measurement uncertainty, we face a non-linear estimation problem. We suggest and compare a Monte Carlo based method for positioning and a gradient search algorithm using a nonlinear least squares framework. The former has the feature of being easily extended to a dynamic framework where a motion model of the transmitter is included. A small simulation study is presented.