Image Processing Reference
In-Depth Information
9.3.2.3 Quasi-Solution Method
The Quasi-Solution Method was developed by Ivanov [,]. In this method, one minimizes the
functional
△
=∥
R
I
(
x
)
Ax
−
y
δ
∥
E
subject to the constraint
S
(
x
)≤
σ,
where σ is a fixed constant. Again, the stabilizing functional
S
(
x
)
is defined as in Tikhonov's
method.
Note that the three regularization methods mentioned above contain one free parameter (ξ in
Tikhonov's method, µ for Phillips' method, and σ in Ivanov's method). It has been shown [] that
these methods are all equivalent in the sense that if one of the methods (say Phillips') for a given
value of its parameter (say µ
∗
)producesasolution
x
∗
, then there exist corresponding values of
parameters of the other two methods that produce the same solution. We remark in passing that
a smart choice of the free parameter is crucial in obtaining a good (fast converging) solution using
any of the regularization methods mentioned above. here exist several principles for choosing the
free parameter in an optimal fashion [, Section ., , Chapter ].
9.4 Spectrum Estimation Using Generalized Projections
The sensor network spectrum estimation problem (Problem .) posed in Section .. is essentially
finding a
P
e
j
ω
(
)
in the intersection of the feasible sets
Q
i
,
k
. It is easy to verify that the sets
Q
i
,
k
are closed and convex []. [].The problem of finding a point in the intersection of finitely many closed
convex sets is known as the convex feasibility problem and is an active area of research in applied
mathematics.
An elegant way to solve a convex feasibility problem is to employ a series of “generalized pro-
jections” []. A generalized projection is essentially a regularization method with a “generalized
distance” serving as the stabilizing functional. A great advantage of using the generalized projec-
tions formulation is that the solution
P
∗
∈Q
can be found using a series of projections onto
the intermediate sets
i
,
k
. These intermediate projections can be computed locally at each sen-
sor node thus allowing the computations to be done simultaneously and in a highly distributed
fashion.
A generalized distance is a real-valued nonnegative function of two vector variable
D
Q
)
defined in a specific way such that its value may represent the distance between
X
and
Y
in some
generalized sense. When defining generalized distances, it is customary not to require the symmetry
condition. hus,
D
(
X
,
Y
(
X
,
Y
)
may not be the same as
D
(
Y
,
X
)
. Moreover, we do not insist on the triangle
inequality that a traditional metric must obey either.
Example .
e
j
ω
e
j
ω
be two power spectra in
L
Let P
(
)>
and P
(
)>
(−
π, π
)
.hefunctions
π
dω
D
(
P
,
P
)=
∫
π
(
P
−
P
)
−
π
P
ln
P
D
(
P
,
P
)=
∫
π
(
P
+
P
−
P
)
dω
−
π
P
P
−
ln
P
D
(
P
,
P
)=
∫
π
(
P
−
)
dω
−
