.., S4} whose
positions are (a1, b1)...(a4, b4) respectively and a shadow edge represented by
a line L: y = ?®x+??, and sensor S1 is under the shadow as shown in Figure 9.
Using the dual-space transform, each sensor defines a line and a shadow edge
defines a point in the dual space, as shown in Figure 10.
According to the first two properties of dual-space transform, since the
shadow edge L in the primal space is below sensor S2, S3 and is above S1, its
corresponding point l in the dual space will also be below line s2, s3 and above
line s1. In other words, in the dual space, point l is bounded by line s2, s3 and
s1. Furthermore, if the half plane shadow moves toward the direction of P4
in the primal space, based on the third property, point l will also move in the
dual space and it has to cross one of the boundaries (i.e., s2 or s3) before
entering another cell, which means the shadow edge must come across either
P2 or P3 before covering P4.
This observation is significant to sensor management, because non-frontier
sensor nodes like P4 are unlikely to detect a change in their readings before
any of the frontier nodes like P2 or P3 does. Thus, they can be turned o?® or
put into deep sleep mode temporarily in order to save power.
For the purpose of tracking the movement of a shadow edge and sensor
management, it is essential to solve the problem of finding the frontier nodes.
Thanks to dual-space transform, this problem can be also ???transformed??? to
finding the cell where point l resides, and this can be solved easily by linear
programming in the dual space:
As illustrated in Figure 9, if the shadow is below its boundary, then any
sensor Pi with a 0 reading is above the shadow edge.
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