7.1. Some Important Terminologies and Quantities 431
that is not aliased. Aliasing can also creep into the image at the reconstruction
stage, which is the topic of our next discussion.
Aliasing due to Reconstruction
Aliasing can also occur due to poor reconstruction. We saw earlier that recon-
struction in the frequency domain can be done by multiplying the convoluted fre-
quency spectrum by a box function. Now, suppose instead of this standard practice,
one resorts to using a sinc function instead. The sinc function actually represents
the Fourier transform of the box function. In other words, convolution of a function
in the spatial domain by a box function is equivalent to multiplying it by a sinc
function in spatial domain. For any variablex, the sinc function is defined by
sinc(x)=sin(πx)
πx. (7.1.9)
Fig.7.1.6 shows the process of multiplication of the convoluted frequency spectrum
by a sinc function. It is apparent that even when there is no overlap between the
neighboring copies of the spectrum, the final spectrum has unwanted frequencies.
The inverse Fourier transform of this spectrum will contain a high degree of aliasing.
Note that this kind of aliasing has nothing to do with the sampling frequency and
is solely due to poor reconstruction.
A.3 Point Spread Function (PSF)
PSFdetermines how well a detector is able to distinguish between two perfect points
separated in space. Let us suppose there is a perfect point whose image is recorded
by an imaging system consisting of a large number of pixels. These pixels can be
regarded as individual detectors separated by some small distance or pitch. If the
radiation being used to create the image of the spot on these pixels is monochromatic
and perpendicular to both surfaces, the image will consist of a round shape having
highest intensity in the middle and rapidly decreasing intensity with distance away
from the center (see Fig.7.1.7(a)). If one now plots this intensity with respect to
any one coordinate (the image will be radially symmetrical for a perfect spot), a
Gaussian-like distribution can be obtained. This one-dimensional distribution is our
point spread function. The distribution, of course, is never perfectly Gaussian and
can even be skewed, but, for most practical purposes, it does not really matter how
it looks like since we can still define a quantity calledFull Width at Half Maximum
orFWHMas shown in Fig.7.1.7(b). The reason for defining this quantity can
be understood from Fig.7.1.8, which shows the distributions of two closely spaced
spots. It is quite clear that if the distance between the two spots is equal to or larger
thanFWHM, the detector will be able to distinguish between them. The spatial
resolution of the system therefore depends on the value ofFWHM.
ThePSFfor most imaging systems can be characterized by a Gaussian distri-
bution, given by
f(x)=1
√
2 πσ^2exp[
−(x−x 0 )^2
2 σ^2