1. 1 Shape Segmentation and Applications in Sensor Networks Xianjin Xhu, Rik Sarkar, Jie Gao Department of CS, Stony Brook University INFOCOM 2007

2. 2 Outline Introduction Notations and Definitions Segmentation Algorithm Simulation Applications Conclusion

3. 3 Introduction (1) Many sensor networks algorithms implicitly assume nodes are deployed inside a simple geometric region. Approaches to deal with irregularly shaped field.  Re-develop algorithms on virtual coordinates  non-trivial  Partition field into nicely shaped pieces. Algorithm is applied inside each piece and data is collected across segments.

4. 4 Introduction (2) Challenges in shape segmentation for a discrete sensor field.  Sensor nodes start with no idea of the global picture.  Sensor nodes may not know their geographical location.  Distance between two nodes is often approximated by their hop count, when sensor locations are not available.

5. 5 Introduction (3) The paper proposes to adapt a shape segmentation algorithm by using flow complex [11] to sensor networks. The algorithm uses only the connectivity information and does not assume sensors know their location. [11] T.K. Dey et al, “Shape segmentation and matching with flow discretization”

6. 6 Notation and Definitions  The medial axis is the set of points {p1, s1, s2, s3} with at least two closest points on the boundaries.  A point x is called sink if x is inside the convex hull of its closest points on boundaries. [11] T.K. Dey et al, “Shape segmentation and matching with flow discretization”

7. 7 Segmentation Algorithm (1) Implement the flow and the segmentation in a discrete sensor field.  Approximate the distance to the boundary by the minimum hop count to boundary nodes.  Closest node of the boundary for a interior node becomes the closest intervals.  A flow pointer of a nodes now points to one of its neighbor. Minimum hop count = 2 Closest Interval contains 3 boundary nodes

8. 8 Segmentation Algorithm (2) Algorithm steps 1) Boundary detection 2) Construct the distance field 3) Compute the flow 4) Merge nearby sinks 5) Segmentation 6) Final clean-up

9. 9 Segmentation Algorithm (3) 1) Boundary detection [16] Y.Wang et al, “Boundary recognition in sensor networks by topological methods.”

10. 10 Segmentation Algorithm (4) 2) Construct the distance field  A interval I is (j, start I, end I, len), each node p keeps track of the set S p of closest intervals information.  Let the boundary nodes flood the message (I, h) at roughly the same time.  On receiving message (I, h), p compares h to h p : if h > h p  discard the message. if h < h p  discard existing intervals, h p = h, S p = { I }, and sends {I, h+1} to all neighbors. If h = h p  merge I with adjacent and overlapping intervals of S p if there is any, or add I to S p. Sends {I, h+1} to all neighbors.

11. 11 Segmentation Algorithm (5)  Definition 3.1 Nodes on medial axis: A node p is a medial axis node if | S p | > 1 (i) Nodes on medial axis

12. 12 Segmentation Algorithm (6) 3) Compute the flow mid  Definition 3.2 Mid point: The mid point mid(I) for an interval I is the -th element of I, if | I | is odd, else mid(I) is the mean of the -th and the -th elements.  Definition 3.3. Angular distance: The angular distance between neighboring nodes p and q is defined as, where h p < h q, I and I’ must be on the same boundary.

13. 13 Segmentation Algorithm (7)  Definition 3.4. Flow pointer: Let H p be p’s neighbors with higher hop count from the boundary, i.e., h p < h q, for. Then the parent of p, v(p) is defined as the neighbor in H p with minimum angular distance,

14. 14 Segmentation Algorithm (8)  Definition 3.5. Sink: A node c is a sink, if c is a medial axis node, and has locally maximum hop count from the boundary. (i) Medial axis (green) and sinks (red)(ii) Directed trees rooted at sinks

15. 15 Segmentation Algorithm (9) 4) Merge nearby sinks  Taking the trees rooted at the sinks as the segments may result in a heavily fragmentation. (i) Fish(i) Corridor

16. 16 Segmentation Algorithm (10)  Merge nearby sinks with similar distance from boundary to sink clusters. A sink cluster K is (id, h max, h min ), Initially each sink is by itself a sink cluster and its cluster leader. A user-defined threshold t is used to guarantee |h max – h min | < t. A sink cluster leader c sends search message (id, h max, h min ) to all neighboring nodes on the medial axis.

17. 17 Segmentation Algorithm (11) Medial axis node p of cluster (id’, h max ’, h min ’) on receiving a search message executes:  H max = max(h max, h max ’), H min = min(h min, h min ’)  If |H max – H min | > t  Discard message  If |H max – H min | <= t  Forward the message to all neighboring medial axis nodes. And if p is cluster leader, p sends merging request message with (id’, h max ’, h min ’) to c. A cluster leader c makes merging decision by current situation on receiving merging message and sends merging response message to p.

18. 18 Segmentation Algorithm (12)  The threshold t determines the granularity of segmentation. t is smaller: merge fewer sinks  more segments. t is larger: merge more sinks  fewer and larger segments. (i) t=2(ii) t=4

19. 19 Segmentation Algorithm (13) 5) Segmentation  Each sink cluster defines a segment.  Each sink node c propagates the ID of the cluster to all the nodes in the tree rooted at sink c.

20. 20 Segmentation Algorithm (14) 6) Final clean-up  Some nodes may have locally max hop count to the boundary, but not medial axis nodes.  Nodes in the tree rooted at that will becomes orphans.  Final clean-up merges orphans to a nearby segment. Clean Up Orphans

21. 21 Simulation

22. 22 Applications - Random Sampling Basic random sampling Benefit of shape segmentation No. of trialscrosscorridorfish Without shape segmentation168149136 With shape segmentation112115123 Cost per samplingcrosscorridorfish Without shape segmentation477.84511.95361.80 With shape segmentation102.49182.32238.47 ． random a geographic location ． route to the closest node x ． p is picked with a acceptable probability: min(A T /A x,1)

23. 23 Applications - Distributed Index Benefit of shape segmentation  Storage load  Communication cost Cost per event insertioncrosscorridorfish Without shape segmentation293.69359.21254.37 With shape segmentation84.15151.05204.58 (i) Basic DIM peaks = 248(ii) DIM with segmentation peaks = 65

24. 24 Conclusion The paper introduced a simple distributed algorithm the partitions an irregular field into nicely shaped segments. More application of shape segmentation  Load-balanced routing  Data aggregation Future work is to classify applications into several categories so that more precise segmentation definitions can be found for each category.