Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 65

TABLE 2.3
Spectral Property of Different Windows

Distance between the Peak
of the Mainlobe and the Peak
of the First Sidelobe (dB) Window Type w(n), n ¼M, Mþ1, ..., M

Uniform 1 13

n M

Hamming 0 : 54 þ 0 : 46 cos 43

n M

Hanning 0 : 5 þ 0 : 5 cos 46

þ

n M 2n M

Blackman 0 : 42 þ 0 : 5 cos 0 : 08 cos 50

For separable filters,

w ( n 1 , n 2 ) ¼ w 1 ( n 1 ) w 2 ( n 2 ) ( 2 : 101 )

The functions w 1 ( n 1 ) and w 2 ( n 2 ) are 1-D windows. The frequency response of the

designed filter can be obtained by taking the Fourier transform from both sides of

Equation 2.99. Using the convolution property of the Fourier transform, we get

H (v 1 , v 2 ) ¼ H d (v 1 , v 2 ) * W (v 1 , v 2 ) ( 2 : 102 )

This means that the frequency response of the designed filter is the convolution

of the desired frequency response and the frequency response of the truncated
window. If the truncated window is a uniform window, the designed

filter will
have ripples in its passband. This is due to the side lobes in the spectrum of the
uniform window. To minimize the effect of these side lobes and reduce the ripples in
the passband, we can use other windows such as Hamming, Hanning, or Blackman
windows. The purpose of a window is to smooth the frequency response. The
window must have a small mainlobe width so that the transition width of

H (v 1 ,

v 2 )

is small. We also need to have a window with the smallest sidelobe
amplitude to make the ripple in the passband and stop-band regions as small as
possible. The most frequently used 1-D windows and their spectral properties are
listed in Table 2.3.
The 1-D uniform, Hamming, Hanning, and Blackman windows of size 32 in.
spatial and frequency domains are shown in Figures 2.28 and 2.29, respectively.

Example 2.14

Design a circularly symmetric 2-D low-pass FIR filter for an image of size 57 57

v c ¼ 4 . with a cutoff frequency of