Physics and Engineering of Radiation Detection

7.1. Some Important Terminologies and Quantities 429

d

x

C

ontrast

Reconstructed Image

x

C

ontrast

Sampled

Points

Function

Fitted

d/2 Detector Array

Figure 7.1.4: Sampling of the

image having the Nyquist fre-

quency of 2/dby a detector ar-

ray having spatial frequency of

2 /d. Since the sampling con-

dition is satisfied, the recon-

structed image is not aliased.

domains can be represented by

f(ω)⊗g(ω) ⇔ f(x)×g(x) and (7.1.7)

f(ω)×g(ω) ⇔ f(x)⊗g(x), (7.1.8)

where⊗and×represent convolution and multiplication respectively.
Going back to our discussion, we first examine the shape of the convoluted spec-
trum as shown in Fig.7.1.5(b). It is apparent that convolution has introduced copies
of the replicated image. This is an unwanted byproduct of the process and must
somehow be filtered out. This is most conveniently done bymultiplyingthe function
by a box function (see Fig.7.1.5(c)). If the spectra were well separated, as they
should, the result is good reconstruction of the original spectrum as shown in the
figure. The original image can then be obtained by taking the inverse Fourier trans-
form of this spectrum. However, if the copies of the convoluted power spectrum
were overlapped due to sampling at lower than Nyquist frequency, then the high
and low frequencies will get overlapped at end positions as shown in Fig.7.1.5(d).
This would result in an effective frequency that is different from the actual one by
an amount determined by the overlap. The next step in the reconstruction process,
that is, multiplication by a box function is shown in Fig.7.1.5(e). Clearly the result
would be a spectrum different from the actual one. The inverse Fourier transform of
thisdistortedfrequency spectrum will not be able to faithfully reproduce the original
image. The resultant image would then be aliased, that is, it would contain spatial
frequencies that were not present in the actual image.