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Shape Segmentation and Applications in Sensor Networks

Xianjin Xhu, Rik Sarkar, Jie Gao

Department of CS, Stony Brook University

INFOCOM 2007

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Outline

Introduction Notations and Definitions Segmentation Algorithm Simulation Applications Conclusion

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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.

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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.

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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”

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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”

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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

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Segmentation Algorithm (2)

Algorithm steps1) Boundary detection

2) Construct the distance field

3) Compute the flow

4) Merge nearby sinks

5) Segmentation

6) Final clean-up

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Segmentation Algorithm (3)

1) Boundary detection

[16] Y.Wang et al, “Boundary recognition in sensor networks by topological methods.”

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Segmentation Algorithm (4)

2) Construct the distance field A interval I is (j, startI, endI, len), each node p keeps

track of the set Sp 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 hp:

if h > hp discard the message. if h < hp discard existing intervals, hp = h, Sp = { I },

and sends {I, h+1} to all neighbors. If h = hp merge I with adjacent and overlapping

intervals of Sp if there is any, or add I to Sp. Sends {I, h+1} to all neighbors.

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Segmentation Algorithm (5)

Definition 3.1 Nodes on medial axis: A node p is a medial axis node if |Sp| > 1

(i) Nodes on medial axis

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Segmentation Algorithm (6)

3) Compute the flow Definition 3.2 Mid point: The mid point midmid(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 hp < hq , I and I’ must be on the same boundary.

),( qp

SpI SpI ImidImidqp )'()(min),( ''

2

1|| I

)2

||(I

)12

||( I

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Segmentation Algorithm (7)

Definition 3.4. Flow pointer: Let Hp be p’s neighbors with higher hop count from the boundary , i.e., hp < hq, for . Then the parent of p, v(p) is defined as the neighbor in Hp with minimum angular distance,

pHq

),(minarg)( qppvpHq

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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

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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

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Segmentation Algorithm (10)

Merge nearby sinks with similar distance from boundary to sink clusters.

A sink cluster K is (id, hmax, hmin), Initially each sink is by itself a sink cluster and its cluster leader.

A user-defined threshold t is used to guarantee |hmax – hmin| < t.

A sink cluster leader c sends search message (id, hmax, hmin) to all neighboring nodes on the medial axis.

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Segmentation Algorithm (11)

Medial axis node p of cluster (id’, hmax’, hmin’) on receiving a search message executes: Hmax = max(hmax, hmax’), Hmin = min(hmin, hmin’) If |Hmax – Hmin| > t Discard message If |Hmax – Hmin| <= t Forward the message to all

neighboring medial axis nodes. And if p is cluster leader, p sends merging request message with (id’, hmax’, hmin’) to c.

A cluster leader c makes merging decision by current situation on receiving merging message and sends merging response message to p.

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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

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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.

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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

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Simulation

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Applications - Random Sampling Basic random sampling

Benefit of shape segmentationNo. of trials cross corridor fish

Without shape segmentation 168 149 136

With shape segmentation 112 115 123

Cost per sampling cross corridor fish

Without shape segmentation 477.84 511.95 361.80

With shape segmentation 102.49 182.32 238.47

． random a geographic location

． route to the closest node x

． p is picked with a acceptable probability: min(AT/Ax,1)

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Applications - Distributed Index Benefit of shape segmentation

Storage load

Communication costCost per event insertion cross corridor fish

Without shape segmentation 293.69 359.21 254.37

With shape segmentation 84.15 151.05 204.58

(i) Basic DIM peaks = 248 (ii) DIM with segmentation peaks = 65

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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.