Eigenvector Synchronization, Graph Rigidity and the Molecule Problem

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend on previous work and propose the 3D-ASAP algorithm, for the graph realization problem in $\mathbb{R}^3$, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in $\mathbb{R}^3$, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms.Comment: 49 pages, 8 figure

Synchronization Problems in Computer Vision

The goal of \u201csynchronization\u201d is to infer the unknown states of a network of nodes, where only the ratio (or difference) between pairs of states can be measured. Typically, states are represented by elements of a group, such as the Symmetric Group or the Special Euclidean Group. The former can represent local labels of a set of features, which refer to the multi-view matching application, whereas the latter can represent camera reference frames, in which case we are in the context of structure from motion, or local coordinates where 3D points are represented, in which case we are dealing with multiple point-set registration. A related problem is that of \u201cbearing-based network localization\u201d where each node is located at a fixed (unknown) position in 3-space and pairs of nodes can measure the direction of the line joining their locations. In this thesis we are interested in global techniques where all the measures are considered at once, as opposed to incremental approaches that grow a solution by adding pieces iteratively

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Distributed Finite-Time Cooperative Localization for Three-Dimensional Sensor Networks

This paper addresses the distributed localization problem for a network of sensors placed in a three-dimensional space, in which sensors are able to perform range measurements, i.e., measure the relative distance between them, and exchange information on a network structure. First, we derive a necessary and sufficient condition for node localizability using barycentric coordinates. Then, building on this theoretical result, we design a distributed localizability verification algorithm, in which we propose and employ a novel distributed finite-time algorithm for sum consensus. Finally, we develop a distributed localization algorithm based on conjugate gradient method, and we derive a theoretical guarantee on its performance, which ensures finite-time convergence to the exact position for all localizable nodes. The efficiency of our algorithm compared to the existing ones from the state-of-the-art literature is further demonstrated through numerical simulations.Comment: 39 pages, 7 figures, under revie

Bearing-based Network Localizability: A Unifying View

This paper provides a unifying view and offers new insights on bearing-based network localizability, that is the problem of establishing whether a set of directions between pairs of nodes uniquely determines (up to translation and scale) the position of the nodes in d-space. If nodes represent cameras then we are in the context of global structure from motion. The contribution of the paper is theoretical: first, we rewrite and link in a coherent structure several results that have been presented in different communities using disparate formalisms; second, we derive some new localizability results within the edge-based formulation

Distance-based sensor node localization by using ultrasound, RSSI and ultra-wideband - A comparision between the techniques

Wireless sensor networks (WSNs) have become one of the most important topics in wireless communication during the last decade. In a wireless sensor system, sensors are spread over a region to build a sensor network and the sensors in a region co-operate to each other to sense, process, filter and routing. Sensor Positioning is a fundamental and crucial issue for sensor network operation and management. WSNs have so many applications in different areas such as health-care, monitoring and control, rescuing and military; they all depend on nodes being able to accurately determine their locations. This master’s thesis is focused on distance-based sensor node localization techniques; Received signal strength indicator, ultrasound and ultra-wideband. Characteristics and factors which affect these distance estimation techniques are analyzed theoretically and through simulation the quality of these techniques are compared in different scenarios. MDS, a centralized algorithm is used for solving the coordinates. It is a set of data analysis techniques that display the structure of distance-like data as a geometrical picture. Centralized and distributed implementations of MDS are also discussed. All simulations and computations in this thesis are done in Matlab. Virtual WSN is simulated on Sensorviz. Sensorviz is a simulation and visualization tool written by Andreas Savvides.fi=Opinnäytetyö kokotekstinä PDF-muodossa.|en=Thesis fulltext in PDF format.|sv=Lärdomsprov tillgängligt som fulltext i PDF-format

Distance-based sensor node localization by using ultrasound, RSSI and ultra-wideband - A comparision between the techniques

Wireless sensor networks (WSNs) have become one of the most important topics in wireless communication during the last decade. In a wireless sensor system, sensors are spread over a region to build a sensor network and the sensors in a region co-operate to each other to sense, process, filter and routing. Sensor Positioning is a fundamental and crucial issue for sensor network operation and management. WSNs have so many applications in different areas such as health-care, monitoring and control, rescuing and military; they all depend on nodes being able to accurately determine their locations. This master’s thesis is focused on distance-based sensor node localization techniques; Received signal strength indicator, ultrasound and ultra-wideband. Characteristics and factors which affect these distance estimation techniques are analyzed theoretically and through simulation the quality of these techniques are compared in different scenarios. MDS, a centralized algorithm is used for solving the coordinates. It is a set of data analysis techniques that display the structure of distance-like data as a geometrical picture. Centralized and distributed implementations of MDS are also discussed. All simulations and computations in this thesis are done in Matlab. Virtual WSN is simulated on Sensorviz. Sensorviz is a simulation and visualization tool written by Andreas Savvides.fi=Opinnäytetyö kokotekstinä PDF-muodossa.|en=Thesis fulltext in PDF format.|sv=Lärdomsprov tillgängligt som fulltext i PDF-format

Localization and security algorithms for wireless sensor networks and the usage of signals of opportunity

In this dissertation we consider the problem of localization of wireless devices in environments and applications where GPS (Global Positioning System) is not a viable option. The _x000C_rst part of the dissertation studies a novel positioning system based on narrowband radio frequency (RF) signals of opportunity, and develops near optimum estimation algorithms for localization of a mobile receiver. It is assumed that a reference receiver (RR) with known position is available to aid with the positioning of the mobile receiver (MR). The new positioning system is reminiscent of GPS and involves two similar estimation problems. The _x000C_rst is localization using estimates of time-di_x000B_erence of arrival (TDOA). The second is TDOA estimation based on the received narrowband signals at the RR and the MR. In both cases near optimum estimation algorithms are developed in the sense of maximum likelihood estimation (MLE) under some mild assumptions, and both algorithms compute approximate MLEs in the form of a weighted least-squares (WLS) solution. The proposed positioning system is illustrated with simulation studies based on FM radio signals. The numerical results show that the position errors are comparable to those of other positioning systems, including GPS. Next, we present a novel algorithm for localization of wireless sensor networks (WSNs) called distributed randomized gradient descent (DRGD), and prove that in the case of noise-free distance measurements, the algorithm converges and provides the true location of the nodes. For noisy distance measurements, the convergence properties of DRGD are discussed and an error bound on the location estimation error is obtained. In contrast to several recently proposed methods, DRGD does not require that blind nodes be contained in the convex hull of the anchor nodes, and can accurately localize the network with only a few anchors. Performance of DRGD is evaluated through extensive simulations and compared with three other algorithms, namely the relaxation-based second order cone programming (SOCP), the simulated annealing (SA), and the semi-de_x000C_nite programing (SDP) procedures. Similar to DRGD, SOCP and SA are distributed algorithms, whereas SDP is centralized. The results show that DRGD successfully localizes the nodes in all the cases, whereas in many cases SOCP and SA fail. We also present a modi_x000C_cation of DRGD for mobile WSNs and demonstrate the e_x000E_cacy of DRGD for localization of mobile networks with several simulation results. We then extend this method for secure localization in the presence of outlier distance measurements or distance spoo_x000C_ng attacks. In this case we present a centralized algorithm to estimate the position of the nodes in WSNs, where outlier distance measurements may be present