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.
The first sensor localization algorithm we consider is @cite_10 . This algorithm has two phases. First, the Euclidean distance of far off sensors (i.e., the ones that are not in each other's communication range) are approximated by the shortest path between them. It was recently shown that having local connectivity, the shortest path is a reliable estimate of far off sensors @cite_12 . Second, to estimate the relative positions of sensors, multidimensional scaling is applied to the approximated distance matrix. However, one can easily see that given faraway sensor's distances, the shortest path is a very coarse estimate of the distance between the close-by sensors. This makes perform very poorly in our setting.
@cite_10 @cite_12
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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. This paper focuses on the principled study of distance reconstruction for distance-based node localization. We address an important issue in node localization by showing that a highly incomplete set of inter-node distance measurements obtained in ad-hoc node deployments carries sufficient information for the accurate reconstruction of the missing distances, even in the presence of noise and sensor node failures. We provide an efficient and provably accurate algorithm for this reconstruction, and we show that the resulting error is bounded, decreasing at a rate that is inversely proportional to radicn, the square root of the number of nodes in the region of deployment. Although this result is applicable to many localization schemes, in this paper we illustrate its use in conjunction with the popular multidimensional scaling algorithm. Our analysis reveals valuable insights and key factors to consider during the sensor network setup phase, to improve the quality of the position estimates This paper deals with the determination of the position and orientation of a mobile robot from distance measurements provided by a belt of onboard ultrasonic sensors. The environment is assumed to be two-dimensional, and a map of its landmarks is available to the robot. In this context, classical localization methods have three main limitations. First, each data point provided by a sensor must be associated with a given landmark. This data-association step turns out to be extremely complex and time-consuming, and its results can usually not be guaranteed. The second limitation is that these methods are based on linearization, which makes them inherently local. The third limitation is their lack of robustness to outliers due, e.g., to sensor malfunctions or outdated maps. By contrast, the method proposed here, based on interval analysis, bypasses the data-association step, handles the problem as nonlinear and in a global way and is (extraordinarily) robust to outliers. Sensor localization from only connectivity information is a highly challenging problem. To this end, our result for the first time establishes an analytic bound on the performance of the popular MDS-MAP algorithm based on multidimensional scaling. For a network consisting of n sensors positioned randomly on a unit square and a given radio range r = o(1), we show that resulting error is bounded, decreasing at a rate that is inversely proportional to r, when only connectivity information is given. The same bound holds for the range-based model, when we have an approximate measurements for the distances, and the same algorithm can be applied without any modification.