Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 64

TABLE 2.2
Ideal Impulse Response of Symmetric and Nonsymmetric Filters

Support of
H d ( j v 1 , j v 2 ) Filter Type h d (n 1 , n 2 )

sin (v c x p n 1 )
p n 1

sin (v c y p n 2 )
p n 2

Low pass

d( n 1 , n 2 ) sin (v c x p n 1 )
p n 1

sin (v c y p n 2 )
p n 2

High pass

sin (v c x 2 p n 1 )
p n 1

sin (v c y 2 p n 2 )
p n 2

Band pass

sin (v c x 1 p n 1 )
p n 1

sin (v c y 1 p n 2 )
p n 2

p

n 1 þ n 2 J 1 v c

v c
2 p

ω c

p

n 1 þ n 2

Low pass
(circularly symmetric)

p

n 1 þ n 2

J 1 v c

d( n 1 , n 2 ) v c
2

High pass
(circularly symmetric)

p

n 1 þ n 2

p

v c 2

p

v c 1

p

n 1 þ n 2 n 1 þ n 2

J 1 J 1

v c 2
2 p

v c 1
2 p

Band pass
(circularly symmetric)

p

n 1 þ n 2

p

n 1 þ n 2

They are tabulated in Table 2.2. The support for the impulse response h d ( n 1 , n 2 ) is

the entire ( n 1 , n 2 ) plane. Therefore, it has to be truncated for a finite support. The

truncation is done using a 2-D window function, which is

h ( n 1 , n 2 ) ¼ h d ( n 1 , n 2 ) w ( n 1 , n 2 ) ( 2 : 99 )

The 2-D window w ( n 1 , n 2 ) is generally designed using 1-D windows. For example,

for symmetrical filters the 2-D window is

q

n 1 þ n 2 w ( n 1 , n 2 ) ¼ w 1 ( 2 : 100 )