representing an image projection. Because the acquisition data are discrete,
the maximum number of peaks possible in a projection would be in a situ-
ation in which peaks and valleys occur in every alternate pixel (i.e., one
cycle per two pixels or 0.5 cycle/pixel), which is the Nyquist frequency. If
the pixel size is known for a given matrix, then the Nyquist frequency can
be determined. For example, if the pixel size in a 64 ×64 matrix is 4.5 mm
for a given detector, then the Nyquist frequency will be
Nyquist frequency =0.5 cycle/pixel
=0.5 cycle/0.45 cm
=1.11 cycles/cm
A common and well-known filter is the ramp filter (name derived from
its shape in the frequency domain) shown in Figure 12.8 in the frequency
domain. An undesirable characteristic of the ramp filter is that it amplifies
the noise associated with high frequencies in the image even though it
removes the blurring effect of simple backprojection. To eliminate the
high-frequency noise, a window is applied to the ramp filter. A window is a
function that is designed to eliminate high-frequency noises and retain the
low-frequency patient data. Typical filters that are commonly used in recon-
struction are basically the products of a ramp filter that has a sharp cut-off
at the Nyquist frequency (0.5 cycle/pixel) and a window with amplitude 1.0
at low frequencies but gradually decreasing at higher frequencies. A few of
these windows (named after those who introduced them) are illustrated in
Figure 12.9, and the corresponding filters (more correctly, filter–window
combinations) are shown in Figure 12.10.
The effect of a decreasing window at higher frequencies is to eliminate
the noise associated with them. The frequency above which the noise is
Single Photon Emission Computed Tomography 163Fig. 12.8. The ramp filter in the frequency domain.