Number of pixels
Number of pixels
Figure 8. Spatial domain median filter 3*3
50000
40000
30000
20000
10000
0
0 10 20 30 40
Gray level
Figure 9. Histogram spatial domain
3*3 median filter for image at 8.
Figure 10. Spatial domian mode filter 3*3
50000
40000
30000
20000
10000
0 10 20 30 40
Gray level.
Figure 11. Histogram spatial domain
3*3 mode filter for image at 10.
50
Frequency domain
Prior to applying filters in the frequency domain, the
power spectrum must be computed and displayed. Fast
Fourier Transform (FFT) is used to transfer the image from
spatial to frequency domain. The power spectrum is a
graphical display of the horizontal and vertical frequency
components of the image. Mathematically, this can be
expressed as follows:
F(u,v) T T f Gc, y)expI- j2n(ux 4 vy)ldxdy seii 2
f(x,y) = | | F(u,v)explj2n(ux +vy)ldudy ............. 3
where:
F(u,v) is the Fourier transformation of f(x,y).
Equation 3 would transfer the image (f(x,y)) from the
spatial to frequency domain and 4 inverse it back.
Equations 3 and 4 are referred to as Fourier transformation
pairs. Fourier transformation of real function is generally
contain real and imaginary terms. R(u,v)is the real and
I(u,v) is the imaginary term respectively. The |F(u,v)| and
|F (uv)? are called Fourier spectrum, and power spectrum
respectively.
1
|F(u,v)| = [R*(u,v) + Pure 4
P(u,v) 2 FQ v? 2 RAu,v) - Qv) s 5
The discrete form of equation 2 and 3 are:
] N-1N-1
F(u,»») 2— X Y f(x,y)expl—j2n(ux+vy)/ N] ..6
N x=0 y=0
] N-1N-1
jeu 5 5 YF(u,v)exp[j27(ux + vy)/ NT 7
u=0 v=0
where: all parameters are as previously defined.
Figure 12 shows the Log power spectrum of the image in
figure 2. In general, high spatial frequencies correspond to
areas of changing intensity on the image, such as areas
containing noise, while low frequency components
correspond to areas of less varying, or uniform, intensity.
Ideal low-pass filter with cut -off frequency radii ranged
from 0.25 to 2.75 inches was applied. Histograms and
their corresponding images were collected at all cut-off
frequencies (figure 13, 14, 15, and 16,)
Figure 12. Log power spectrum of the original image
