Image Processing Reference
In-Depth Information
Other desirable features of the convex feasibility approach include the following:
. Globalsolutionisuniqueandstableinthesensethatsmallperturbationsintheobserved
datawillcauseasmallchangeinthesolution.
. Functional form of the solution will depend on the choice of the generalized distance used
in the projections. herefore, a functional form which is easy to manipulate or interpret
for a specific application (for instance, a rational function) can be obtained using a proper
generalized distance.
. Formulation can be applied to a variety of network topologies. Some topologies allow for
the most efficient computation, some allow for the most robust setup, and others lead to
various degrees of compromise between these desirable properties.
. Formulation has a very rich mathematical structure relying on recent results in sev-
eral fields of applied mathematics including convex analysis, parallel optimization, and
regularization theory.
To maintain clarity and simplicity, we will focus on solving a concrete distributed estimation prob-
lem. However, the fusion algorithms that result from our formulations are very general and can be
used to solve other sensor network signal-processing problems as well.
9.1.1 Notation
Vectors are denoted by capital letters. Boldface capital letters are used for matrices. Elements of a
matrix
A
are referred to as
M
andusethenotation
[
A
]
ij
. We denote the set of real
M
-tuples by
R
R
+
for positive real numbers. The expected value of a random variable
x
is denoted by
E
{
x
}
.helinear
convolution operator is denoted by
⋆
. ⋆.The spaces of Lebesgue-measurable functions are represented
by
L
,
L
(
a
,
b
)
(
a
,
b
)
,etc.heendofanexampleisindicatedusingthesymbol
♢
.
9.2 Case Study: Spectrum Analysis Using Sensor Networks
9.2.1 Background
A spectrum analyzer or spectral analyzer is a device used to examine the spectral composition
of some electrical, acoustic, or optical waveform. Questions such as “Does most of the power of
the signal reside at low or high frequencies?” or “Are there resonance peaks in the spectrum?”
are often answered as a result of a spectral analysis. Spectral analysis finds frequent and extensive
use in many areas of physical sciences. Examples abound in oceanography, electrical engineer-
ing,geophysics,astronomy,andhydrology.hroughoutthischapter,wewillusespectrumanaly-
sis as a benchmark signal-processing problem to demonstrate our distributed information fusion
algorithms.
Consider the scenario in Figure ., where a sound source (a speaker) is monitored by a col-
lection of Motes put at various known locations in a room . Because of reverberation, noise, and
other artifacts, the signal arriving at each Mote location is diferent. he Motes (which constitute the
sensor nodes in our network) are equipped with microphones, sampling devices, sufficient signal-
processing hardware and some communication means. Each Mote can process its observed data,
come up with some statistical inference about it, and share the result with other nodes in the net-
work. However, to save energy and communication bandwidth, the “Motes are not allowed to share
their raw observed data with each other.” Now, how should the network operate so that an estimate
of the frequency spectrum of the sound source consistent with the observations made by all Motes
is obtained?
