Physics and Engineering of Radiation Detection

428 Chapter 7. Position Sensitive Detection and Imaging

d

x

C

ontrast

Reconstructed Image

x

C

ontrast

Sampled Points

Fitted Function

4d/5 Detector Array

Figure 7.1.3: Sampling of the

image having the Nyquist fre-

quency of 2/dby a detector ar-

ray having spatial frequency of

5 / 4 d. Since the sampling con-

dition is not met, the recon-

structed image is aliased.

expect that the reconstructed image would not get aliased. For such a system, the
detector array elements or pixels must be a distanced/2 apart from each other as
shown in Fig.7.1.4. It is clear from the figure that sampling at this frequency helps
in reconstructing the image fairly well. A point to note here is that although the
reconstruction of the image at Nyquist frequency is shown to be perfect, but in reality
since the images are not always so regular therefore sampling at a higher frequency
is generally preferred. This process is sometimes referred to asoversamplingand is
a common practice.
Up until now we have talked about sampling in spatial domain. However aliasing
is best understood if we look at the sampling in the frequency domain. It should be
noted that sampling in spatial domain is equivalent to multiplying by a spike function
at each spatial point (see Fig.7.1.5(a)), a process generally known asmultiplication.
Now, if we wanted to perform the same operation in frequency domain, we will have
toconvolutethe frequency spectrum of the image by a spike function. This process
of convolution is mathematically represented by

f(ω)⊗g(ω)=

∫

f(u)g(ω−u)du. (7.1.6)

wherefandgare the two functions (for example frequency spectrum and spike
function) in frequency domain. This implies that convolving in frequency domain
is equivalent to multiplying in spatial domain. The opposite is also true, that is,
convolution in spatial domain is like multiplication in frequency domain. Mathe-
matically, these equivalences for two functionsfandgin frequencyωand spatialx