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 con≠stant, 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 pres≠ervation 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 fre≠quency 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(Ωxy) 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(Ωxy). The frequency response H(Ωxy) is maximum at the spatial frequency in the range of approximately 5 ó 10 cycles/degree and decreases as the spatial frequency increases or decreases from 5 ó 10 cycles/degree.