Image Processing Reference
In-Depth Information
e
j
ω
e
j
ω
can be used to measure the generalized distance between P
(
)
and P
(
)
.hesefunctionsare
nonnegative and become zero if and only if P
P
.NotethatD
is simply the Euclidean distance
between P
and P
.hefunctionsD
and D
have roots in information theory and statistics. They
are known as the Kullback-Leibler divergence and Burg cross entropy, respectively.
=
♢
By using a suitable generalized distance, we can convert our original sensor network spectrum
estimation problem (Problem .) into the following minimization problem:
be defined as in Problem .. Find P
x
e
j
ω
PROBLEM .
Let
Q
(
)
in
Q
such that
P
∗
=
arg min
P
D
(
P
,
P
)
,
(.)
∈Q
e
j
ω
e
j
ω
where P
(
)
is an arbitrary power spectrum, say P
(
)
=
,
−
π
≤
ω
<
π
.
When a unique
P
∗
exists, it is called the generalized projection of
P
[]. In gen-
eral, a projection of a given point onto a convex set is defined as another point, which has two
properties: First, it belongs to the set onto which the projection operation is performed and, sec-
ond, it renders a minimal value to the distance between the given point and any point of the set
(Figure .).
If the Euclidean distance,
onto
Q
is used in this context then the projection is called a met-
ric projection. In some cases, such as the spectrum estimation problem considered here, it turns
outtobeveryusefultointroducemoregeneralmeanstomeasurethedistancebetweentwovec-
tors.hemainreasonisthatthefunctionalformofthesolutionwilldependonthechoiceofthe
distance measure used in the projection. Often, a functional form which is easy to manipulate or
interpret (for instance, a rational function) cannot be obtained using the conventional Euclidean
metric.
It can be shown that the distances
D
and
D
in Example . lead to well-posed solutions for
P
∗
.
The choice
D
will lead to a unique solution given that certain singular power spectra are excluded
∣∣
X
−
Y
∣∣
D
(
X
,
Y
)
||
X
-
Y
||
Y
Y
*
X
*
X
Y
X
Q
Q
(a)
(b)
FIGURE .
(a) Symbolic depiction of metric projection and (b) generalized projection of a vector
Y
into a closed
. In (a) the projection
X
∗
is selected by minimizing the metric
while in (b)
X
∗
is
convex set
Q
∣∣
X
−
Y
∣∣
over all
X
∈Q
found by minimizing the generalized distance
D
(
X
,
Y
)
overthesameset.
