Image Processing Reference
In-Depth Information
from the space of valid solutions []. It is not known whether
D
will lead to a stable solution. As a
result, the well-posedness of Problem . when
D
is used is not yet established.
∗
9.5 Distributed Algorithms for Calculating Generalized
Projection
As we mentioned before, a very interesting aspect of the generalized projections formulation is that
the solution
P
∗
∈Q
i
,
k
.Inthis
section, we first calculate the generalized projection of a given power spectrum onto the sets
can be found using a series of projections onto the intermediate sets
Q
i
,
k
for
thesampledistancefunctionsintroducedinExample..hen,weproposeadistributedalgorithm
for calculating the final solution
P
∗
from these intermediate projections.
Let
P
[
Q
denote the power spectrum resulting from projecting a given power spectrum
P
P
↦Q
i
,
k
;
D
j
]
onto the set
Q
i
,
k
usingagivendistancefunctions
D
j
.hatis,
△
=
P
arg min
P
D
j
(
P
,
P
)
.
(.)
[
P
↦Q
i
,
k
;
D
j
]
∈Q
i
,
k
Using standard techniques from calculus of variations, we can show that the generalized distances
D
,
D
,and
D
introduced in Example . result in projections of the form
e
j
ω
e
j
ω
P
[
]
=
P
(
)−
α
G
i
(
)
cos
(
Mk
ω
)
,
P
↦Q
i
,
k
;
D
e
j
ω
e
j
ω
P
]
=
P
(
)
exp
(−
β
G
i
(
)
cos
(
Mk
ω
))
,
[
P
↦Q
i
,
k
;
D
−
,
e
j
ω
)
−
e
j
ω
P
]
=(
P
(
+
γ
G
i
(
)
cos
(
Mk
ω
))
[
P
↦Q
i
,
k
;
D
where α, β, and γ are parameters (Lagrange multipliers). hese parameter should be chosen such that
in each case
P
[
]
∈Q
i
,
k
.hatis,
P
↦Q
i
,
k
;
D
j
π
π
e
j
ω
e
jMk
ω
dω
P
[
G
i
(
)
=
R
v
i
(
k
)
.
(.)
∫
P
↦Q
i
,
k
;
D
j
]
−
π
The reader may observe that the above equation leads to a closed-form formula for α but in gen-
eral finding β and γ requires numerical methods. he projection formulae developed above can be
employed in a variety of iterative algorithms to fined a solution in the intersection of
Q
i
,
k
.Wediscuss
two example algorithms below.
9.5.1 Ring Algorithm
Ring Algorithm is a very simple algorithm: it starts with an initial guess
P
(
)
for
P
x
e
j
ω
(
)
and then
calculates a series of successive projections onto the constraint sets
Q
i
,
k
. .Then, it takes the last pro-
jection, now called
P
(
)
, and projects it back onto the first constraint set. Continuing this process
will generate a sequence of solutions
P
(
)
,
P
(
)
,
P
(
)
,...whichwilleventuallyconvergetoasolution
P
∗
∈
⋂
i
,
k
[]. Steps of the Ring Algorithm are summarized in the text box below. A graphical
representation of this algorithm is shown in Figure ..
Q
i
,
k
∗
Well-posedness of the minimization problem (Equation .) when
D
is the Kullback-Leibler divergence
D
has been
established in several works including [-]. Well-posedness results exist for certain classes of generalized distance
functionsaswell[,].Unfortunately,theBurgcrossentropy
D
does not belong to any of these classes. While Burg cross
entropy lacks theoretical support as a regularizing functional, it has been used successfully to resolve ill-posed problems
in several applications including spectral estimation and image restoration (see, for example, [] and references therein).
The desirable feature of Burg cross entropy in the context of spectrum estimation is that its minimization (subject to linear
constraints
P
x
e
j
ω
(
)∈Q
) leads to rational power spectra.