Publication 2008 Issue No. 8 - August Abstract - A Geometric Transversal Approach to Analyzing Track Coverage in Sensor Networks
A Geometric Transversal Approach to Analyzing Track Coverage in Sensor Networks
August 2008 (vol. 57 no. 8)
pp. 1113-1128
 ASCII Text x Kelli Baumgartner, Silvia Ferrari, "A Geometric Transversal Approach to Analyzing Track Coverage in Sensor Networks," IEEE Transactions on Computers, vol. 57, no. 8, pp. 1113-1128, August, 2008.
 BibTex x @article{ 10.1109/TC.2008.56,author = {Kelli Baumgartner and Silvia Ferrari},title = {A Geometric Transversal Approach to Analyzing Track Coverage in Sensor Networks},journal ={IEEE Transactions on Computers},volume = {57},number = {8},issn = {0018-9340},year = {2008},pages = {1113-1128},doi = {http://doi.ieeecomputersociety.org/10.1109/TC.2008.56},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on ComputersTI - A Geometric Transversal Approach to Analyzing Track Coverage in Sensor NetworksIS - 8SN - 0018-9340SP1113EP1128EPD - 1113-1128A1 - Kelli Baumgartner, A1 - Silvia Ferrari, PY - 2008KW - SensorsKW - Remote sensingKW - Geometrical problems and computationsKW - Nonlinear programmingKW - Wireless sensor networksVL - 57JA - IEEE Transactions on ComputersER -
Kelli Baumgartner, Duke University, Durham
Silvia Ferrari, Duke University Duke University, Durham Durham
This paper presents a new coverage formulation addressing the quality of service of sensor networks that cooperatively detect targets traversing a region of interest. The problem of track coverage consists of finding the positions of n sensors such that a Lebesgue measure on the set of tracks detected by at least k sensors is optimized. This paper studies the geometric properties of the network, addressing a deterministic track-coverage formulation and binary sensor models. It is shown that the tracks detected by a network of heterogeneous omnidirectional sensors are the geometric transversals of non-translates families of circles. A novel methodology based on cone theory is presented for representing and measuring sets of transversals in closed-form. Then, the solution of the track-coverage problem can be formulated as a nonlinear program (NLP). The numerical results show that this approach can improve track coverage by up to two orders of magnitude compared to grid and random deployments. Also, it can be used to reduce the number of sensors required to achieve a desired detection performance by up to 50%, and to optimally replenish or reposition existing sensor networks.

[1] Y. Zou and K. Chakrabarty, “A Distributed Coverage- and Connectivity-Centric Technique for Selecting Active Nodes in Wireless Sensor Networks,” IEEE Trans. Computers, vol. 54, no. 8, pp. 978-991, Aug. 2005.
[2] L.M. Kaplan, “Global Node Selection for Localization in a Distributed Sensor Network,” IEEE Trans. Aerospace and Electronic Systems, vol. 42, no. 1, pp. 113-135, 2006.
[3] S. Iyengar and R. Brooks, “Special Issue Introduction—The Road Map for Distributed Sensor Networks in the Context of Computing and Communication,” J. Parallel and Distributed Computing, vol. 64, no. 7, pp. 785-787, 2004.
[4] K. Chakrabarty, S.S. Iyengar, H. Qi, and E. Cho, “Grid Coverage for Surveillance and Target Location in Distributed Sensor Networks,” IEEE Trans. Computers, vol. 51, no. 12, pp. 1448-1453, Dec. 2002.
[5] X.-Y. Li, P.-J. Wan, and O. Frieder, “Coverage in Wireless Ad-Hoc Sensor Networks,” IEEE Trans. Computers, vol. 52, no. 6, pp. 753-763, June 2003.
[6] S. Megerian, F. Koushanfar, M. Potkonjak, and M.B. Srivastava, “Worst and Best-Case Coverage in Sensor Networks,” IEEE Trans. Mobile Computing, vol. 4, no. 2, pp. 84-92, Jan./Feb. 2005.
[7] J. Ai and A.A. Abouzeid, “Coverage by Directional Sensors in Randomly Deployed Wireless Sensor Networks,” J. Combinatorial Optimization, vol. 11, pp. 21-41, 2006.
[8] V. Isler, S. Khanna, and K. Daniilidis, “Sampling Based Sensor-Network Deployment,” Proc. IEEE/RSJ Int'l Conf. Intelligent Robots and Systems), vol. 100, pp. 1780-1785, 2004.
[9] L. Benyuan and D. Towsley, “A Study of the Coverage of Large-Scale Sensor Networks,” Proc. First IEEE Int'l Conf. Mobile Ad-Hoc and Sensor Systems, pp. 475-483, 2004.
[10] N. Heo and P.K. Varshney, “Energy-Efficient Deployment of Intelligent Mobile Sensor Networks,” IEEE Trans. Systems, Man, and Cybernetics—Part A: Systems and Humans, vol. 35, no. 1, pp. 78-92, Jan. 2005.
[11] W. Huang, Y. Li, and C. Li, “Greedy Algorithms for Packing Unequal Circles into a Rectangular Container,” J. Operational Research Soc., vol. 56, pp. 539-548, 2005.
[12] J.-A. George, J.-M. George, and B. Lamer, “Packing Different-Sized Circles into a Rectangular Container,” European J. Operational Research, vol. 84, pp. 693-712, 1995.
[13] S. Martinez and F. Bullo, “Optimal Sensor Placement and Motion Coordination for Target Tracking,” Automatica, vol. 42, pp. 661-668, 2006.
[14] M. Marengoni, B.A. Draper, A. Hanson, and R.A. Sitaraman, “System to Place Observers on a Polyhedral Terrain in Polynomial Time,” Image and Vision Computing, vol. 18, pp. 773-780, 1996.
[15] J. O'Rourke, Art Gallery Theorems and Algorithms. Oxford Univ. Press, 1987.
[16] T.A. Wettergren, “Statistical Analysis of Detection Performance for Large Distributed Sensor Systems,” Technical Report ADA417136, Naval Undersea Warfare Center, http:/stinet.dtic. mil, 2008.
[17] T.A. Wettergren, R.L. Streit, and J.R. Short, “Tracking with Distributed Sets of Proximity Sensors Using Geometric Invariants,” IEEE Trans. Aerospace and Electronic Systems, vol. 40, no. 4, pp. 1366-1374, Oct. 2004.
[18] J.-P. LeCadre and G. Souris, “Searching Tracks,” IEEE Trans. Aerospace and Electronic Systems, vol. 36, no. 4, pp. 1149-1166, 2000.
[19] B. Koopman, Search and Screening: General Principles with Historical Applications. Pergamon Press, 1980.
[20] H. Cox, “Cumulative Detection Probabilities for a Randomly Moving Source in a Sparse Field of Sensors,” Proc. 23rd Asilomar Conf. Signals, Systems and Computers, pp. 384-389, 1989.
[21] J. Goodman, R. Pollack, and R. Wenger, “Geometric Transversal Theory,” New Trends in Discrete and Computational Geometry, J.Pach, ed., pp. 163-198, Springer Verlag, 1991.
[22] S. Meguerdichian, F. Koushanfar, G. Qu, and M. Potkonjak, “Exposure in Wireless Ad-Hoc Sensor Networks,” Mobile Computing and Networking, pp. 139-150, 2001.
[23] Q. Huang, “Solving an Open Sensor-Exposure Problem Using Variational Calculus,” Technical Report WUCS-03-1, Washington Univ., http://www.cs.wustl.edu/qingfeng/papersExposureTR Short.pdf , 2003.
[24] T. Clouqueur, V. Phipatanasuphorn, P. Ramanathan, and K. Saluja, “Sensor Deployment for Detection of Targets Traversing a Region,” Mobile Networks and Applications, vol. 8, pp. 453-461, Aug. 2003.
[25] E. Helly, “Über Mengen Konvexer Körper mit Gemeinschaftlichen Punkten,” Jahresbericht der Deutschen MathematikerVereiningung, vol. 32, pp. 175-176, 1923.
[26] N. Megiddo, “Linear Programming in Linear Time When the Dimension Is Fixed,” J. ACM, vol. 31, no. 1, pp. 114-127, 1984.
[27] E. Edelsbrunner, H. Maurer, F. Preparata, A. Rosenberg, E. Welzl, and D. Wood, “Stabbing Line Segments,” BIT Numerical Math., vol. 22, pp. 274-281, 1982.
[28] M. Atallah and C. Bajaj, “Efficient Algorithms for Common Transversals,” Information Processing Letters, vol. 25, pp. 87-91, 1987.
[29] E. Edelsbrunner, “Finding Transversals for Sets of Simple Geometric Figures,” Theoretical Computer Science, vol. 35, pp. 55-69, 1985.
[30] N. Ansari, J.-G. Chen, and Y.-Z. Zhang, “Adaptive Decision Fusion for Unequiprobable Sources,” IEEE Proc. Radar, Sonar, and Navigation, vol. 144, no. 3, pp. 105-111, June 1997.
[31] D. Avis, J. Robert, and R. Wenger, “Lower Bounds for Line Stabbing,” Information Processing Letters, vol. 33, pp. 59-62, 1989.
[32] D.P. Bertsekas, Convex Analysis and Optimization. Athena Scientific, 2003.
[33] H.F. Davis and A.D. Snider, Vector Analysis. William C. Brown, 1987.
[34] S. Skiena, “Generating $k\hbox{-}{\rm Subsets}$ ,” Implementing Discrete Math.: Combinatorics and Graph Theory with Mathematica, pp. 44-46, Addison-Wesley, 1990.
[35] S. Ferrari, “Track Coverage in Sensor Networks,” Proc. Am. Control Conf., pp. 2053-2059, 2006.
[36] S. Ross, Introduction to Stochastic Dynamic Programming. Academic Press, 1983.
[37] H.T. Croft, K.J. Falconer, and R.K. Guy, Unsolved Problems in Geometry. Springer-Verlag, 1991.
[38] D.P. Bertsekas, Nonlinear Programming. Athena Scientific, 2007.
[39] M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinear Programming: Theory and Algorithms. Wiley Interscience, 2006.
[40] R.J. Vanderbei, “LOQO: An Interior Point Code for Quadratic Programming,” Optimization Methods and Software, vol. 11, nos. 1-4, pp. 451-484, 1999.
[41] Mathworks, Matlab Optimization Toolbox, http:/www. mathworks.com, function: fmincon, 2004.
[42] D. Hammond, “Sonobuoy Field Drift Prediction,” “Naval Air Warfare Center Aircraft Division,” Technical Report A115034, Patuxent River, Md., http://www.stormingmedia.us/11/1150A115034.html , 2008.
[43] G.J. Juselis, “Station Keeping Buoy System,” Secretary of the Navy Patent number: 5577942, http://www.google.compatents?id= tMwhAAAAEBAJ , 1996.
[44] G. Petryk and M. Buehler, “Dynamic Object Localization via a Proximity Sensor Network,” Proc. IEEE/SICE/RSJ Int'l Conf. Multisensor Fusion and Integration for Intelligent Systems, pp. 337-341, 1996.
[45] W. Ridely, “Automation of Buoy Positioning,” OCEANS, vol. 17, pp. 179-183, 1985.
[46] K.A.C. Baumgartner, “Control and Optimization of Track Coverage in Underwater Sensor Networks,” PhD thesis, Duke Univ., Dec. 2007.
[47] R.V. Hogg, J.W. McKean, and A.T. Craig, Introduction to Mathematical Statistics. Prentice Hall, 2005.

Index Terms:
Sensors, Remote sensing, Geometrical problems and computations, Nonlinear programming, Wireless sensor networks
Citation:
Kelli Baumgartner, Silvia Ferrari, "A Geometric Transversal Approach to Analyzing Track Coverage in Sensor Networks," IEEE Transactions on Computers, vol. 57, no. 8, pp. 1113-1128, Aug. 2008, doi:10.1109/TC.2008.56