Digital Signal Processing Reference
In-Depth Information
used in Equation 12.48. On the other hand, if the vectors under considera-
tion have the same direction in the vector space (collinear vectors), the first
part of
S
(directional information) equals to one and the similarity
measure of Equation 12.48 is based only on the magnitude of the difference
part.
Utilizing this similarity measure, an adaptive vector processing filter based
on the general framework of Equation 12.41 and the weighting formula of
Equation 12.48 was devised in Reference 111. The
adaptive nearest neighbor
multichannel filter
(ANNMF) belongs to the adaptive vector processing fil-
ter family defined through Equation 12.41. However, ANNMF combines
the weighting formula of Equation 12.47 with the new distance measure of
Equation 12.48 to evaluate its weights.
(
F
i
,
F
j
)
12.4.3
Nonparametric Adaptive Multichannel Filter
Consider the following model for the color image degradation process:
=
+
F
j
X
j
G
j
,
(12.49)
where
X
j
is a
three-dimensional
uncorrupted image vector,
F
j
is the corre-
sponding noisy vector to be filtered, and
G
j
is an additive noise vector. In
our analysis, it is assumed that the color image vectors are unknown and that
the noise vectors are uncorrelated at the different image locations and are
signal independent.
Let us denote with
the minimum variance estimator of the color
vector
X
, given the noisy measurement vector
F
. The expected square error
of the filter, when the image vectors are corrupted by additive noise as in
Equation 12.49, can be written as
(
F
)
[
X
]
T
V
=
−
(
F
)
][
X
−
(
F
)
f
(
X
|
F
)
f
(
F
)
d
X
d
F
,
(12.50)
∞
d
X
f
∞
]
T
V
=
[
X
−
(
F
)
][
X
−
(
F
)
f
(
X
|
F
)
(
F
)
d
F
,
(12.51)
−∞
−∞
where
z
T
denotes the transpose of
z
. Because
(
F
)
does not enter into
the outer integral and
is always positive, it is sufficient for the optimal
minimum variance estimator to minimize the expected value of the estimation
cost (conditional Bayesian risk), given the observation
F
. Thus, it is sufficient
to minimize the quantity
f
(
F
)
∞
]
T
=
−
(
)
−
(
)
(
|
)
.
V
BR
[
X
F
][
X
F
f
X
F
d
X
(12.52)
−∞
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