Assuming sensor nodes are deployed on a plane,
a collection of their values of event predicates can be regarded as a twodimensional
matrix. Edges can be easily detected with the image processing
approach and the classifier approach discussed in previous sections.
However, other schemes do exist. In this section, a simple but elegant
algorithm based on dual-space transform[6] for centralized edge determination
is presented.
3.1 Dual-space approach
The basic idea behind the the dual-space approach is to transform lines and
points in the primal space to points and lines in the dual space. A dual space
transform has the following useful properties that can be used to assist boundary
detection in wireless sensor networks:
??? In the primal space, if a point (a, b) is on a line y = ?®x + ??, then, the
corresponding line ' = a?µ+b goes through the corresponding point (??’?®, ??)
in the dual space.
??? In the primal space, if a point (a, b) is above a line y = ?®x + ??, then, the
corresponding line ' = a?µ +b is above the corresponding point (??’?®, ??) in
the dual space.
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Chapter 6 Boundary Detection for Sensor Networks
Fig. 8. The mapping between the primal space and the dual space.
??? In the primal space, if a line y = ?®x + ?? performs a continuous motion,
the corresponding point (??’?®, ??) also performs a continuous motion in the
dual space.
Now, consider, in the primal space, a set of sensors {S1, .
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