Image Processing Reference
In-Depth Information
In order to select the best template configuration for each pixel, a quality
measure must be defined. Ahn [3] proposed the use of the local standard deviation
of the pixel values on the templates. For each template T
, the local standard
j
deviation
σ
(m, n) of the pixel values is given by:
j
1
∑
σ
j
(,)
mn
=
[(,)
xi j
−
x mn
(, ]
2
N
−
1
j
ij T
,
∈
j
(6.8)
1
N
,
∑
mn
,
x
(
)
=
xi j
(, )
j
ij T
j
and x(m, n) is the input pixel value at the (m, n) coordinates.
For an implementation of the adaptive filtering procedure, first, local standard
deviations for each pixel in the image and for each template must be evaluated.
Next, templates are classified into two categories based on standard deviation
value: templates with local standard deviation less than the threshold value (cor-
responding to the random noise standard deviation) and templates with local
standard deviation larger than the threshold value. When more elements reflect
the former condition, then the optimal template is the one having the maximum
filter size. As far as the latter condition is concerned, the optimal template is the
one having the minimum standard deviation
σ
(,)(,)
mnxmn
σ
x mn
(,)
2
2
+
+
y
n
ymn
(,)
=
σ
2
(,)
mn
σ
2
(6.9)
y
n
{
}
σ
(,) max , (,)
mn
0
σ
mn
σ
2
2
2
=
−
y
x
n
where y(m, n) is the filtered output, the local variance evaluated at the (m, n)
pixel, and the noise variance. Local variance close to the noise variance implies
almost constant regions. The relevant filtered output is a smoothed version of the
input image on the template. Otherwise, i.e., in edge regions, the filtering con-
tribution is negligible.
A key issue in the optimization of the adaptive filtering algorithm based on
templates is the choice of the threshold value. A popular method is to define the
threshold as
σ
2
σ
2
τσ
=
a
(6.10)
n
where
σ
is the estimated noise standard deviation on the image and
a
is a scale
n
factor. Values from 1.2
σ
to 1.6
σ
may be chosen to optimize filter performance
n
n
[3]. Estimation of
from MR images can be done by multiplying the standard
deviation of the image background
σ
n
by the scale factor 1.526 as in Equation 6.5.
The computational complexity of the algorithm for a brute-force search of
templates is M
σ
b
NT, where M is the image dimension and NT the number of
possible templates. In order to reduce the algorithm complexity, templates can
2
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