7.1. Some Important Terminologies and Quantities 425
image reconstruction. Let us discuss these two causes of aliasing in some detail.
Aliasing due to Sampling Frequency
Most of the modern imaging systems generate the so calleddigital imageof the
object bysamplingthe analog information. The term sampling is used to character-
ize the process of gathering spatial information at regular time intervals and passing
it on to the analog to digital converters. Of course there is no reason to exclude
the possibility of sampling at irregular time intervals but the general practice is to
do it at well defined regular intervals since it simplifies the further processing of the
information. There are also imaging systems that perform only thespatial sampling,
that is, the image is sampled at only one point in time. An obvious example would
be the digital cameras that sample and store information through charged coupled
photon detection devices. In most medical imaging systems both spatial and tem-
poral sampling is done at the same time to preserve and reconstruct as many details
of the object as possible.
Let us concentrate now on the process of spatial sampling since it can cause
aliasing of the final image. To understand this we first note that the object, whose
image is to be captured, has some spatial frequencies. In other words there are
shades in the image occurring at different spatial intervals. In general, the spatial
variation of shades in an image is fairly irregular and therefore contains a number of
frequencies. The spatial frequency content of the object can be obtained by taking
the Fourier transform of variation of its shades. In general the Fourier spectrum
shows a number of frequencies, the maximum of which corresponds to the band-
width of the system (this is true only for baseband systems but most images can be
characterized to be baseband). Now, if we want to recreate the image of this object
through an array detector then the natural question to ask is how many pixels (or
individual detection elements) per unit distance should be enough to create an ac-
ceptable image. In other words, what should the sampling frequency of the detector
array be? Intuitively we would think that the sampling frequency must be higher
than the maximum frequency of the shades in the actual image. In fact, in reality
the sampling frequency must be greater than twice the maximum frequency. This
is formally stated by the followingsampling theorem.
An image can be faithfully reconstructed if and only if the sampling
frequency is greater than or equal to twice the bandwidth of the input
signal.
Mathematically, this can be written as
fs≥ 2 B, (7.1.1)whereFsis the sampling frequency andBis the bandwidth of the detector. In the
specific case of a baseband system, for which the highest frequencyfmaxcoincides
with the bandwidth, the sampling condition can be written as
fs≥ 2 fmax, (7.1.2)where 2fmaxis sometimes also called theNyquist frequency, a term that was origi-
nally introduced for sampling in time domain. Hence we can say that an image will