Physics and Engineering of Radiation Detection

426 Chapter 7. Position Sensitive Detection and Imaging

be aliasedor will not be faithfully reconstructedif

fs<fNyq, (7.1.3)

wherefNyq=2fmax is the Nyquist frequency for a baseband system. An array
detector having pixels at a distance ofdfrom each other we havefmax=1/dand
the Nyquist frequency is given by

fNyq=

2

d

, (7.1.4)

and the condition that the image does not get aliased or gets faithfully recon-
structed is

fs≥

2

d

. (7.1.5)

In most instances the Nyquist frequency is reported in units of line pairs per
millimeter (lp/mm)orcycles/mm. Only the frequencies below this value faithfully
pass through the system and therefore the device acts as a spatial low pass filter.
The frequency spectrum beyond the Nyquist value is folded about that frequency
and added to the spectrum of lower frequencies causing aliasing of the reconstructed
image. It is obvious that aliasing results is an artificial increase in the spectral
content of the signal and therefore should be minimized for a faithful reconstruction
of the actual image.

(^1) d
d
Frequency
(b)
(a)
(c)
Amplitude

C

ontrast

d

x

Figure 7.1.1: (a) The shades

in an object’s image that can

be described by a sine func-

tion as shown in (b). Con-

trary to the figure shown, as-

sume that the shades are con-

tinuously distributed and do not

have defined boundaries. (c)

The Fourier transform of the

sine function is a single peak at

a frequency that is equal to the

inverse of the wavelength.

Let us now try to intuitively understand how sampling at a frequency lower than
the Nyquist frequency can cause aliasing. Fig.7.1.1(a) shows a one-dimensional
image having periodically occurring shades that can be described by a sine function