Nonlinear filtering

f(x,y) F(u,v)
g(x,y) G(u,v)
G(u,v) = F(u,v) • H(u,v)
spatial domain frequency domain
filter
•
f(x,y) F(u,v)
H(u,v)g(x,y)

u
v
H(u,v)
0 D0
1
D(u,v)
H(u,v)
H(u,v) =
1 D(u,v) D0
0 D(u,v) > D0
D(u,v) = u2 + v2
D0 = cut off frequency

H(u,v) - Ideal Filter
H(u,v) =
0 D(u,v) D0
1 D(u,v) > D0
D(u,v) = u2 + v2
0 D0
1
D(u,v)
H(u,v)
u
v
H(u,v)

H(u,v) =
1 D0- D(u,v) D0 +
0 D(u,v) > D0 +
D(u,v) = u2 + v2
u
v
H(u,v)
0
1
D(u,v)
H(u,v)
D0- w
2
D0+ w
2
D0
0 D(u,v) D0 -
w
2
w
2
w
2
w
2
w = band width

• Note: An OS Filter is Uniquely Defined by {wi}
• Example 1:
• Example 2:
)2()1()(1221 25.05.025.0},,,{ MMMM xxxxxxOS 
}0,,4/1,2/1,4/1,,0{}{ iw
(M+1)-thM-th (M+2)-th
then
12
1
)(1221
12
1
},,,{
M
i
iM x
M
xxxOS 
)12/(}1,,1,1{}{ Mwi 
}12,,1,,{}12,,1,{ )( MixMeanMixMean ii 
then

5
6
6
0
7
8
8
7
5
6
6
5
15
8
8
7
0 0 0 0
0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
5
6
6
0
7
8
8
7
5
6
6
5
15
8
8
7
0
6
5
5
6
7
0
7
5
5
6
0
7
0
7
6
•A 4x4 grayscale image is given by
1) Filter the image with a 3x3 median filter, after zero-padding
at the image borders
5
6
6
0
7
8
8
7
5
6
6
5
15
8
8
7
zero-padding
median
filtering
impulse?
impulse?

5
6
6
0
7
8
8
7
5
6
6
5
15
8
8
7
5 6 7 8
5 6 7 8
5
0
5
5
5
5
8
8
8
8
8
8
5
6
6
0
7
8
8
7
5
6
6
5
15
8
8
7
5
6
6
5
7
8
8
7
5
6
6
5
7
8
8
7
2) Filter the image with a 3x3 median filter, after replicate-
padding at the image borders
replicate
-padding
median
filtering
impulse cleaned!

5
6
6
5
7
8
8
7.25
5
6
6
5
7.25
8
8
7.25
3) Filter the image with a 3x3 OS filter, after replicate-
padding at the image borders. The weighting factors of
the OS filter are given by
{wi | i = 1, …, 9} = {0, 0, 0, ¼, ½, ¼, 0, 0, 0}
5
6
6
0
7
8
8
7
5
6
6
5
15
8
8
7
5 6 7 8
5 6 7 8
5
0
5
5
5
5
8
8
8
8
8
8
5
6
6
0
7
8
8
7
5
6
6
5
15
8
8
7
replicate
-padding
OS
filtering

Nonlinear filtering

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