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Yang Yang, Miao Jin, Hongyi Wu Presenter: Buri Ban The Center for Advanced Computer Studies (CACS) University of Louisiana at Lafayette 3D Surface Localization with Terrain Model

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Applications of Wireless Sensor Networks

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Localization in Sensor Networks Location information is important Devices need to know where they are. We want to know where the data is from. It helps infrastructure establishment. For examples, geographical routing, sensor coverage, et al.

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Localization in Sensor Networks Wireless Sensors Deployed on 2D plane (ground) Deployed on 3D space (air, underwater) Deployed on 3D surface (terrain)

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3D Surface Localization Applications Volcano Monitoring ZebraNet

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Challenges in 3D Surface Localization Connectivity and surface distance information only are not enough for 3D surface localization. (b) Deformation to cylinder. (c) Deformation to wave shape. (a) A 2D Surface

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Previous Methods Y. Zhao, H. Wu, M. Jin, S. Xia, "Localization in 3D Surface Sensor Networks: Challenges and Solutions”, INFOCOM'12, pp. 55-63,2012. Y. Zhao, H. Wu, M. Jin, Y. Yang, H. Zhou, and S. Xia, “Cut-and- sew:A distributed autonomous localization algorithm for 3d surface wireless sensor networks,” MobiHoc'13, pp. 69-78, 2013.

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Previous Methods Assume each sensor node knows the distance between its neighboring nodes. Assume each sensor node can measure its own height information.

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Our Approach To reduce the cost of hardware, is it possible not using height information? If possible, we still need some extra information, since a surface network is non-localizable with pure connectivity and surface distance information.

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Digital Terrain Model (DTM) A 3D representation of a terrain’s surface. Commonly built using remote sensing technology. Available to public with a variable resolution up to one meter.

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Outline Theoretical background and motivation of our approach Our approach Discussions Simulations Conclusion and future works

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Theoretical Background Conformal structure is an intrinsic geometric structure of surfaces: Tolerate a small local deformation of a surface; Surfaces sharing the same conformal structure exist conformal mapping between them. A conformal mapping is a one-to-one and continuous mapping/function that preserves angles and local shape.

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Motivation of Our Approach The triangular mesh of a DTM and the triangular mesh extracted from the connectivity graph of a network deployed over the terrain surface approximate the geometric structure of the same terrain surface. Theoretically, the two triangular meshes share the same conformal structure. There exists a conformal mapping between them. DTM mesh M1 Sensor Network mesh M2

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Motivation of Our Approach It is extremely difficult to directly construct a conformal mapping between two 3D surfaces. We can conformally map the two surfaces to 2D plane, and then construct a conformal mapping between the mapped two planar domains. The three conformal mappings induces a conformal mapping between the two 3D surfaces. Based on this mapping, each sensor node of the network can easily locate reference grid points from the DTM to calculate its own geographic location.

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Outline Theoretical background and motivation of our approach Our approach Discussions Simulations Conclusion and future works

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Overview of Our Approach 2D triangular mesh D1 DTM mesh M1 2D triangular mesh D2 Sensor Network mesh M2 Alignment Localization

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Step 1: Conformal Mapping to Plane Construct Triangular Mesh for both DTM and Sensor Network. A DTM is represented by a grid of squares. It is straightforward to convert the grid into a triangulation, e.g., by simply connecting a diagonal of each square. For Sensor Network with one-hop distance information available, a simple distributed algorithm can extract a refined triangular mesh from the network connectivity graph.

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Step 1: Conformal Mapping to Plane Given a triangular mesh M embedded in 3D. We apply Discrete Surface Ricci Flow to conformally map M to a planar region, denoted as D in 2D. The mapping result is stored at each vertex V as a complex number, which serves as the planar coordinates of V when M is mapped to D.

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Overview of Our Approach 2D triangular mesh D1 DTM mesh M1 2D triangular mesh D2 Sensor Network mesh M2 Alignment Localization

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Step 2: Alignment Randomly deployed three Anchor Nodes (Sensors with GPS information) Sensor Network mesh M2 2D triangulation mesh D2

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Step 2: Alignment A Mobius Transformation is a conformal mapping between complex plane to itself, with a set of three points mapped to another set of three points. D2D1 Property: aligns to

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Overview of Our Approach 2D triangular mesh D1 DTM mesh M1 2D triangular mesh D2 Sensor Network mesh M2 Alignment Localization

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Step 3: Localization If DTM has a high density, a sensor node simply determines its own 3D coordinates according to the nearest triangle vertex of DTM. If DTM has a low density, with the aligned planar coordinates, each sensor node locates three nearest grid points on D1, and use the Barycentric Coordinates and grid points’3D coordinates to get sensors’ approximate 3D location.

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Step 3: Barycentric Coordinates Barycentric Coordinates provides a convenient way to interpolate a function on triangles as long as the function’s value is known at all vertices. bb b

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Outline Theoretical background and motivation of our approach Our approach Discussions Simulations Conclusion and future works

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Discussions: Performance with # of Anchor Nodes The size of Anchor Nodes

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Discussions: Anchor Node Free

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Outline Theoretical background and motivation of our approach Our approach Discussions Simulations Conclusion and future works

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Simulations: Different Digital Terrain Models Different Digital Terrain Models

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Simulations: The Distribution of Localization Errors under Different Sets of Anchor Nodes The distribution of localization errors under different sets of anchor nodes

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Simulations: Terrain models with Various Resolutions

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Simulations: Range Distance Measurement Error

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Simulations: Connectivity Only Networks with Connectivity Information Only

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Outline Theoretical background and motivation of our approach Our approach Discussions Simulations Conclusion and future works

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Conclusion and Future Works A fully distributed algorithm to localize a wireless sensor network deployed on the surface of complex 3D terrains with range distance measurement only. Both the 3D terrain surface and the network can be any complicated shape, not necessarily convex. Future works: Incorporate those useful contour features of surfaces like the peaks of valleys into the alignment algorithm.

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Q & A

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Thank You !

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