Image Processing Reference
In-Depth Information
e
j
ω
The above formula shows that
P
x
(
)
uniquely specifies
R
v
i
(
k
)
for all values of
k
.However,the
reverseisnottrue.hatis,ingeneral,knowing
R
v
i
(
k
)
forsomeorallvaluesof
k
is not sufficient for
e
j
ω
characterizing
P
x
(
)
uniquely.
is a WSS signal so all the statistical information that can be gained about it is
confined in its autocorrelation coefficients. One might use the signal-processing hardware available
at each sensor node and estimate the autocorrelation coefficients
R
v
i
Recall that
v
i
(
n
)
(
k
)
for some
k
,say
≤
k
≤
L
−
.
This leads us to pose the sensor network spectrum estimation problem as follows:
PROBLEM .
i
,
k
denote the set of all power spectra which are consistent with the kth
autocorrelation coefficient R
v
i
Let
Q
e
j
ω
(
k
)
estimated at the ith sensor node. hat is, P
x
(
)∈Q
i
,
k
if
π
∫
π
e
j
ω
e
j
ω
e
jMk
ω
d
ω
P
x
(
)
G
i
(
)
=
R
v
i
(
k
)
,
−
π
e
j
ω
P
x
(
)≥
,
e
j
ω
e
−
j
ω
P
x
(
)=
P
x
(
)
,
e
j
ω
L
P
x
(
)∈
(−
π, π
)
.
△
=
⋂
N
i
L
−
Define
i
,
k
where N is the number of nodes in the network and L is the number of
autocorrelation coefficients estimated at each node. Find a P
x
Q
Q
⋂
=
k
=
e
j
ω
(
)
in
Q
.
If we ignore measurement imperfections and assume that the observed autocorrelation coefficients
R
v
i
(
k
)
are exact, then the sets
Q
i
,
k
are nonempty and admit a nonempty intersection
Q
as well. In
e
j
ω
this case,
Q
contains infinitely many
P
x
(
)
. When the measurements
v
i
(
n
)
are contaminated by
noise or
R
v
i
might be
empty due to the potential inconsistency of the autocorrelation coefficients estimated by different
sensors. hus, Problem . has either no solution or infinitely many solutions. Problems which have
such undesirable properties are called “ill-posed”. Ill-posed problems are studied in the next section.
(
k
)
are estimated based on finite-length data records, the intersection set
Q
9.3 Inverse and Ill-Posed Problems
The study of inverse problems has been one of the fastest-growing areas in applied mathematics in
the last two decades. This growth has largely been driven by the needs of applications in both nat-
ural sciences (e.g., inverse scattering theory, astronomical image restoration, and statistical learning
theory) and industry (e.g., computerized tomography and remote sensing). The reader is referred
to [-] for detailed treatments of the theory of ill-posed problems and to [,] for applications in
inverse scattering and statistical inference, respectively.
The definition, inverse problems are concerned with determining causes for a desired or an
observed effect. Most often, inverse problems are much more difficult to deal with (from a math-
ematical point of view) than their direct counterparts. his is because they might not have a solution
in the strict sense or solutions might not be unique or depend on data continuously. Mathemati-
cal problems having such undesirable properties are called “ill-posed problems” and cause severe
numerical difficulties (mostly because of the discontinuous dependence of solutions on the data).
Formally, a problem of mathematical physics is called “well-posed or well-posed in the sense of
Hadamard” if it fulfills the following conditions:
. For all admissible data, a solution exists.
. For all admissible data, the solution is unique.
. The solution depends continuously on the data.
