Signal Processing - March 2016 - 101
X (t+τ )
X (t+τ )
X (t )
(a)
X (t )
(b)
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.5
0
(c)
-0.8
0.5
-0.5
0
(d)
0.5
figure 9. The wheeze-detection process. In (a), the top row corresponds to normal signals, while the bottom two rows correspond to wheeze signals.
(a) Delay embeddings. (b) Various samplings of a wheeze. (c) A triangulation of a wheeze. (d) A triangulation with large edges removed.
nonwheeze signal appears as chaotic, and its topology is also
nicely captured at a minimal computational cost.
Sensor networks
The application of algebraic topology in sensor networks is
very illustrative of its utility. Owing to the difficulties of
obtaining location information of the sensors in the field and
to the need for additional hardware to compute precise distances between pairs of sensors, it may be prohibitively
expensive to obtain geometric information. Interestingly,
(a)
problems like verifying coverage are purely topological in
nature, and, as discussed previously, computational topology
provides a coordinate-free solution to quantifying the coverage status as topological information.
Consider a set of sensors randomly deployed in a region to be
monitored. Two problems are often of interest:
■ verifying if the region being monitored is actually fully
covered and accounted for
■ discovering uncovered regions and identifying their surrounding nodes.
(b)
(c)
(d)
figure 10. A way to approximate the coverage area, shown in (b), in sensor networks using a complex constructed from the communication graph. If
the communication radius is twice the individual sensor coverage radius, the Rips shadow is a good topological approximation to the coverage area, as
shown in (d). (a) Sensors in a plane. (b) Sensor coverage. (c) The Cˇ ech complex. (d) The Rips shadow.
IEEE Signal Processing Magazine
|
March 2016
|
101
Table of Contents for the Digital Edition of Signal Processing - March 2016
Signal Processing - March 2016 - Cover1
Signal Processing - March 2016 - Cover2
Signal Processing - March 2016 - 1
Signal Processing - March 2016 - 2
Signal Processing - March 2016 - 3
Signal Processing - March 2016 - 4
Signal Processing - March 2016 - 5
Signal Processing - March 2016 - 6
Signal Processing - March 2016 - 7
Signal Processing - March 2016 - 8
Signal Processing - March 2016 - 9
Signal Processing - March 2016 - 10
Signal Processing - March 2016 - 11
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Signal Processing - March 2016 - 15
Signal Processing - March 2016 - 16
Signal Processing - March 2016 - 17
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Signal Processing - March 2016 - 19
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Signal Processing - March 2016 - 21
Signal Processing - March 2016 - 22
Signal Processing - March 2016 - 23
Signal Processing - March 2016 - 24
Signal Processing - March 2016 - 25
Signal Processing - March 2016 - 26
Signal Processing - March 2016 - 27
Signal Processing - March 2016 - 28
Signal Processing - March 2016 - 29
Signal Processing - March 2016 - 30
Signal Processing - March 2016 - 31
Signal Processing - March 2016 - 32
Signal Processing - March 2016 - 33
Signal Processing - March 2016 - 34
Signal Processing - March 2016 - 35
Signal Processing - March 2016 - 36
Signal Processing - March 2016 - 37
Signal Processing - March 2016 - 38
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Signal Processing - March 2016 - 40
Signal Processing - March 2016 - 41
Signal Processing - March 2016 - 42
Signal Processing - March 2016 - 43
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Signal Processing - March 2016 - 45
Signal Processing - March 2016 - 46
Signal Processing - March 2016 - 47
Signal Processing - March 2016 - 48
Signal Processing - March 2016 - 49
Signal Processing - March 2016 - 50
Signal Processing - March 2016 - 51
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Signal Processing - March 2016 - 53
Signal Processing - March 2016 - 54
Signal Processing - March 2016 - 55
Signal Processing - March 2016 - 56
Signal Processing - March 2016 - 57
Signal Processing - March 2016 - 58
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Signal Processing - March 2016 - 60
Signal Processing - March 2016 - 61
Signal Processing - March 2016 - 62
Signal Processing - March 2016 - 63
Signal Processing - March 2016 - 64
Signal Processing - March 2016 - 65
Signal Processing - March 2016 - 66
Signal Processing - March 2016 - 67
Signal Processing - March 2016 - 68
Signal Processing - March 2016 - 69
Signal Processing - March 2016 - 70
Signal Processing - March 2016 - 71
Signal Processing - March 2016 - 72
Signal Processing - March 2016 - 73
Signal Processing - March 2016 - 74
Signal Processing - March 2016 - 75
Signal Processing - March 2016 - 76
Signal Processing - March 2016 - 77
Signal Processing - March 2016 - 78
Signal Processing - March 2016 - 79
Signal Processing - March 2016 - 80
Signal Processing - March 2016 - 81
Signal Processing - March 2016 - 82
Signal Processing - March 2016 - 83
Signal Processing - March 2016 - 84
Signal Processing - March 2016 - 85
Signal Processing - March 2016 - 86
Signal Processing - March 2016 - 87
Signal Processing - March 2016 - 88
Signal Processing - March 2016 - 89
Signal Processing - March 2016 - 90
Signal Processing - March 2016 - 91
Signal Processing - March 2016 - 92
Signal Processing - March 2016 - 93
Signal Processing - March 2016 - 94
Signal Processing - March 2016 - 95
Signal Processing - March 2016 - 96
Signal Processing - March 2016 - 97
Signal Processing - March 2016 - 98
Signal Processing - March 2016 - 99
Signal Processing - March 2016 - 100
Signal Processing - March 2016 - 101
Signal Processing - March 2016 - 102
Signal Processing - March 2016 - 103
Signal Processing - March 2016 - 104
Signal Processing - March 2016 - 105
Signal Processing - March 2016 - 106
Signal Processing - March 2016 - 107
Signal Processing - March 2016 - 108
Signal Processing - March 2016 - 109
Signal Processing - March 2016 - 110
Signal Processing - March 2016 - 111
Signal Processing - March 2016 - 112
Signal Processing - March 2016 - 113
Signal Processing - March 2016 - 114
Signal Processing - March 2016 - 115
Signal Processing - March 2016 - 116
Signal Processing - March 2016 - 117
Signal Processing - March 2016 - 118
Signal Processing - March 2016 - 119
Signal Processing - March 2016 - 120
Signal Processing - March 2016 - 121
Signal Processing - March 2016 - 122
Signal Processing - March 2016 - 123
Signal Processing - March 2016 - 124
Signal Processing - March 2016 - 125
Signal Processing - March 2016 - 126
Signal Processing - March 2016 - 127
Signal Processing - March 2016 - 128
Signal Processing - March 2016 - Cover3
Signal Processing - March 2016 - Cover4
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