Digital Signal Processing Reference
In-Depth Information
two-neighbor average is used to set the “
” pixel values. Last, the four origi-
nal samples neighboring each “
◦
” pixel are averaged to set the values of these
remaining samples.
A similar polyphase approach utilizing weighted median, or fuzzy
weighted median, filters can be developed as a straightforward extension of
the linear approach.
38
,
45
In this case, the algorithm first interpolates “
” pixels
as the weighted median of the original pixels at its four corners. As each of
these samples is equally distant from the sample to be estimated, they are
considered equally reliable and each is assigned a weight of 1. The remaining
pixels are determined by taking the weighted median of the four neighboring
samples for each case. More specifically, for each “
◦
” pixel the two original
pixels to its left and right are assigned weight 1, while the two interpolated
“
” pixels above and below are assigned weight 0.5. This lower weighting
reflects the fact that the interpolated samples are less reliable than the origi-
nal pixels. Similarly, the original pixels above and below each “
◦
” pixel are
assigned weight 1 and the interpolated pixels to the right and left are assigned
weight 0.5. The choice of 0.5 for the weights reflects the reliability of the esti-
mated samples. It should be noted that, due to the structure of the weighted
median and the number of samples utilized, any assignment of weight values
between zero and one for the estimated samples yields identical results.
A fuzzy weighted median filter can be implemented in the above-mentioned
algorithm simply by replacing the pixels used for interpolation by their fuzzy
counterparts. Thus, the interpolation process can be illustrated as follows.
Consider an image represented by an array of pixel values denoted as
.
Then the fuzzy weighted median polyphase interpolation performs the fol-
lowing mapping:
{
a
i, j
}
x
1
,
1
x
1
,
1
x
1
,
2
x
1
,
2
x
1
,
3
x
1
,
3
x
1
,
1
x
1
,
2
x
1
,
3
x
1
,
1
x
1
,
2
x
1
,
3
a
1
,
1
a
1
,
2
a
1
,
3
x
2
,
1
x
2
,
1
x
2
,
2
x
2
,
2
x
1
,
3
x
2
,
3
⇒
a
2
,
1
a
2
,
2
a
2
,
3
,
(3.19)
x
2
,
1
x
2
,
2
x
1
,
3
x
2
,
1
x
2
,
2
x
2
,
3
a
3
,
1
a
3
,
2
a
3
,
3
x
3
,
1
x
3
,
1
x
3
,
2
x
3
,
2
x
3
,
3
x
3
,
3
x
3
,
1
x
3
,
2
x
3
,
3
x
3
,
1
x
3
,
2
x
3
,
3
where
a
i, j
is the value of the pixel in the
i
th row and
j
th column of the original
image. The pixels in the interpolated image are determined as
x
i, j
=
a
i, j
,
(3.20)
MED
a
i, j
, a
i
+
1
,j
, a
i, j
+
1
, a
i
+
1
,j
+
1
,
x
i, j
=
(3.21)
MED
a
i, j
, a
i, j
+
1
,
0
x
i
+
1
,j
,
x
i, j
=
x
i
−
1
,j
,
0
.
5
.
5
(3.22)
MED
a
i, j
, a
i
+
1
,j
,
0
x
i, j
+
1
.
x
i, j
=
x
i, j
−
1
,
0
.
.
5
5
(3.23)
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