2.3.3. Mach Band Effect and Spatial
Frequency Response
Consider an image whose
intensity is constant along the vertical dimension hut increases in a staircase
manner along the horizontal dimension, as shown in Figure 2.21(a). The
intensities along the horizontal direction are sketched in Figure 2.21(h). Even
though the intensity within each rectangular region is constant, each region
looks brighter towards the left and darker towards the right. This is known as
the Mach band effect. This phenomenon
is consistent with the presence of spatial filtering in the peripheral-level
model of the visual system in Figure 2.15. When a filter is applied to a signal
with sharp discontinuities, an overshoot and undershoot occur. This is partly
responsible for uneven brightness perception within the region of uniform
intensity. This suggests that precise preservation of the edge shape is not
necessary in image processing.
The presence in the
visual system of a spatial bandpass filter can
he seen by looking at the image in Figure 2.22. The image I(x,y) in Figure 2.22 is given by
I(x,y)
= Io(y).cos(w (x)x) + constant (2.11)
where the constant is chosen such
that I(x,y)
is positive for all (x,y). As we move in the
horizontal direction from left to right, the spatial frequency w(x) increases. As we move in the vertical direction from top to bottom,
the amplitude Io(y) increases, If
the spatial frequency response were constant across the frequency range,
sensitivity to intensity would be constant along the horizontal direction. In
Figure 2.22, we are more sensitive to the contrast in mid-frequency regions
than in low- and high-frequency regions, indicating the bandpass
character of the visual system. A spatial filter frequency response H(Ωx,Ωy)
which is more accurately measured by assuming the model in Figure 2.15 is
correct, is shown in Figure 2.23. The horizontal axis is the spatial
frequency/angle of vision. The perceived spatial frequency of an image changes
as a function of the distance between the eye and the image. As the distance
increases, the perceived spatial frequency increases. To take
this effect into account. the spatial
frequency/angle of vision (spatial frequency relative to the spatial domain in
the retina) is often used in determining H(Ωx,Ωy). The frequency response H(Ωx,Ωy) is maximum at the spatial frequency in
the range of approximately 5 — 10
cycles/degree and decreases as the spatial frequency increases or decreases
from 5 — 10 cycles/degree.