Physics and Engineering of Radiation Detection

7.1. Some Important Terminologies and Quantities 427

as shown in Fig.7.1.1(b). The Fourier transform of the function as sketched in
Fig.7.1.1(c) shows a single peak at a frequency that corresponds to thewavelength
of the sine function.
Let us see what happens if we use an array detector having a spatial frequency
equal to the frequency of the sine wave or half the Nyquist frequency. Such a detector
would consist of pixels or individual detection elements arranged at a distancedfrom
each other (see Fig.7.1.2). Of course the detector can record information only at the
pixel locations. This means that, for a periodic function, the recorded contrast will
be the same at all points. The reconstructed image, as shown in Fig.7.1.2 will then
simply be a single shade. The wrong sampling frequency has thus added information
in the reconstructed image that was not there in the actual object. In other words
the image has been aliased due to sampling at a frequency less than the Nyquist
frequency.

d

x

C

ontrast

Fitted Function

Detector Array

Reconstructed Image

d

Sampled Points

x

C

ontrast

Figure 7.1.2: Sampling of the

image having the Nyquist fre-

quency of 2/dby a detector ar-

ray having spatial frequency of

1 /d. Since the acquired data

points lie on a single straight

line, the reconstructed image is

severely aliased. That is, the fi-

nal image contains information

that was not there in the actual

image. The contrast in the final

image depends on the position

of detector pixels relative to the

image being sampled.

Another example of aliasing is shown in Fig.7.1.3, where the detector has a regu-
lar array structure with a sampling frequency offs=5/ 4 d. Note that here also the
detector does not meet the condition set by the sampling theorem (that is,fs≥ 2 /d).
As shown in the figure, the image is sampled only at the pixel locations. The recon-
structed image how has more information as compared to the case of Fig.7.1.2 where
the sampling frequency was even lower. Hence increasing the sampling frequency
has increased the image quality. However still the reconstructed image contains a
lot of information not contained in the actual image. Or in other words, it is still
aliased.
Let us now increase the sampling frequency further such that it becomes equal
to the Nyquist frequency. Since now it satisfies the sampling condition, we would