Applications of digital signal processing - Continuous and Discrete Time Signals and Systems - page 777

Fig. 17.19. (a) Original
(512 512) pixel Lena image.
(b) Power spectrum of the Lena
image.

200

0

−200

p p

0.5 p

0.5 p

(a) (b)

17.6 Image filtering

Real images consist of a combination of smooth regions and active regions
with edges. In smooth regions, the intensity values of the pixels do not change
significantly. Therefore, the smooth regions represent lower-frequency com-
ponents in the 2D frequency space. On the other hand, the intensity values in
the active regions change significantly over edges. The active regions represent
higher-frequency components. Extracting the low- and high-frequency compo-
nents from a real image has important applications in image processing. In this
section, we introduce frequency-selective filtering in two dimensions.
The mathematical model for filtering a 2D image g [ m , n ] by a filter with an
impulse response h [ m , n ]isgivenby

∞ ∞

y [ m , n ] = g [ m , n ] ∗ h [ m , n ] = g [ m − q , n − r ] h [ q , r ] , (17.25)

q =−∞ r =−∞

where y [ m , n ] is the output response of the filter and ∗ denotes the convolution
operation. Alternatively, the filtering can be performed in the frequency domain
using the following equation:

Y ( Ω x , Ω y ) = G ( Ω x , Ω y ) H ( Ω x , Ω y ) , (17.26)

where G ( Ω x , Ω y ) is the Fourier transform of the input image, H ( Ω x , Ω y )isthe
2D transfer function of the filter, and Y ( Ω x , Ω y ) is the Fourier transform of the
resulting output. Like 1D filters, 2D filters can be broadly classified into four
categories: lowpass, bandpass, highpass, and bandstop filters. Some examples
of these filters are given in the following.