Distributed Information Storage and Retrieval in 3D Sensor Networks with General Topologies

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

1Applications of Wireless Sensor Networks

Wireless sensors are used in a wide range of applications nowadays, They can be used underwater for detection, they can be deployed on the mountain surface to study the activity of the volcanoes. The sensors form a network and cooperate with each other to perform assigned tasks. 2Localization 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.

For some applications,.3Localization in Sensor Networks Wireless Sensors

Deployed on 2D plane (ground)

Deployed on 3D space (air, underwater)

Deployed on 3D surface (terrain)

The majority of current research on sensor network localization focuses on wireless sensor networks deployed on plane or in three dimensional space, but very few on 3D surface.43D Surface Localization Applications

Volcano Monitoring ZebraNet

However, in real-world applications, many large-scale sensor networks are deployed over complex terrains, such as the volcano monitoring project and ZebraNet.

5Challenges 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 SurfaceSo there are needs, but still, there are also challenges. This figure is an illustration of the challenges in 3D surface localization. A surface can be deformed to another one without changing the surface distance between any pair of points, so pure surface distance information is not enough to localize a surface.

6Previous 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.

Here are previous methods targeting 3D surface network localization problem. 7Previous Methods Assume each sensor node knows the distance between its neighboring nodes.

Assume each sensor node can measure its own height information.

They take the following assumptions:8Our 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.

Since they need another device to measure height, we had this question in our mind.9Digital Terrain Model (DTM)

A 3D representation of a terrains surface. Commonly built using remote sensing technology. Available to public with a variable resolution up to one meter.

A DTM is commonly represented by a grid of squares. And the 3D coordinates of all grid points are known. 10Outline Theoretical background and motivation of our approach

Our approach

Discussions

Simulations

Conclusion and future works

Next is Theoretical background and motivation of our approach

11Theoretical 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.

12Motivation 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

What is the relation between conformal structure and our problems here? Well..13Motivation 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.

14Outline Theoretical background and motivation of our approach

Our approach

Discussions

Simulations

Conclusion and future works

Next is our approach steps.15Overview of Our Approach

2D triangular mesh D1

DTM mesh M1

2D triangular mesh D2Sensor Network mesh M2AlignmentLocalization16Step 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.

17Step 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.

Then, for mapping, we used Discrete Surface Ricci Flow to conformally map 3D surface to a planar region and the mapping result is stored at each vertex as a complex number, which serves as the planar coordinates of the vertex18Overview of Our Approach

2D triangular mesh D1

DTM mesh M1

2D triangular mesh D2Sensor Network mesh M2AlignmentLocalizationThe second step is alignment.

19Step 2: Alignment

Randomly deployed three Anchor Nodes (Sensors with GPS information)

Sensor Network mesh M22D triangulation mesh D2In this step, we need 3 anchor nodes with GPS information. They are randomly deployed on the sensor network.20Step 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 toFor alignment, our goal is to find the mapping from D2 to D1.After the first step, now we have coordinates of 3 anchor nodes in D2, and after we get the mapped coordinates in D1, we used Mobius Transformation to get the mapping function, and all other nodes then can get their mapped points in D1 by the function.So after 2nd step, each sensor nodes in D2 will get the aligned planar coordinates in D1

how to align the two 2D planes?1. Find the feature points of two planes, then use Mobius Transformation explain which three points from D2 mapped which three points of D1 3 anchor nodes in D2, denoted as z1,z2,z3 are mapped to w1,w2,w3 correspondingly in D1, then the map function f(z) is set by this function.All other nodes then can get their mapped points in D1 by the function.21Overview of Our Approach

2D triangular mesh D1

DTM mesh M1

2D triangular mesh D2Sensor Network mesh M2AlignmentLocalizationThe last step is localization22Step 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 points3D coordinates to get sensors approximate 3D location.23Step 3: Barycentric Coordinates

Barycentric Coordinates provides a convenient way to interpolate a function on triangles as long as the functions value is known at all vertices.

bbbas this figure shows, the Barycentric Coordinates of one node in a triangle is determined by the area ratio.24Outline Theoretical background and motivation of our approach

Our approach

Discussions

Simulations

Conclusion and future works

25Discussions: Performance with # of Anchor Nodes

The size of Anchor NodesTheoretically, the proposed localization algorithm requires only three anchor nodes to align two triangular meshes on plane. With increased anchor nodes, we can improve the localization accuracy with a different conformal alignment method.26Discussions: Anchor Node Free

Also, it is possible to do alignment with Anchor Node Free.1. Locally conformal mapping introduces no distortion, only scaling. Such scaling is called conformal factor.2. Sensor nodes, with the highest conformal factors are around the peaks of a terrain surface and we can apply them as anchor nodes (feature points) for alignment.So we compute conformal factors of the triangular meshes of the DTM and the network, then pick 3 vertices with highest conformal factors for each model and pair them for alignment.compute conformal factors of the triangular meshes of the DTM and the network, then pick n vertices with highest conformal factors; then if a vertex in D1 shares a similar conformal factor with a vertex in D2, then they are a map pair. In total we will have n pairs.

27Outline Theoretical background and motivation of our approach

Our approach

Discussions

Simulations

Conclusion and future works

28Simulations: Different Digital Terrain ModelsDifferent Digital Terrain Models

In this figure, the first row shows a set of DTMs of representative terrain surfaces. The second row shows wireless sensor networks marked with black points deployed on these terrain surfaces. The third row shows the localized sensor networks with randomly deployed anchor nodes marked with red. 29Simulations: The Distribution of Localization Errors under Different Sets of Anchor NodesThe distribution of localization errors under different sets of anchor nodes

We assume sensor nodes with accurate range distance measurement and DTMs with high resolutions. For each network, we randomly deploy three anchor nodes and calculate the localization errors of the network. We repeat eight times for each network.u -- arithmetic mean of errorssigma--standard deviationx bar--the median30Simulations: Terrain models with Various Resolutions

The results show that the resolution of a DTM has a small impact on the performance of the localization algorithm unless it is too low.31Simulations: Range Distance Measurement Error

We have also evaluated our algorithm when the one-hop distance information exists measurement error.32Simulations: Connectivity Only

Networks with Connectivity Information OnlyWhen range distance measurement is not available, we can still extract a sparse triangular mesh from a network connectivity graph based on a simple landmark-based algorithm and then do localization. Here show two extracted sparse triangular meshes.33Outline Theoretical background and motivation of our approach

Our approach

Discussions

Simulations

Conclusion and future works

34Conclusion 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 valleysinto the alignment algorithm.

35Q & A

36Thank You !37