Another property of the Fourier transform is the projection-slice theorem, which is the mathematical basis of computed tomography (CT). Computed tomography has a number of applications, including the medical application of reconstructing cross sections of a human body from x-ray images. The impact of computed tomography on medicine requires no elaboration.

Consider a 2-D analog function f_{c}(t_{1},t_{2})
where t_{1} and t_{2} are continuous variables. The subscript
c denotes that the signal is a function of a continuous variable or variables.
The analog Fourier transform F_{c}(W_{1},W_{2}) is related to f_{c}(t_{1},t_{2})
by

_{ }(1.49 a)

_{} (1.49 b)

_{}

Let us
integrate _{ }along
the parallel rays shown in Figure 1.40. The angle that the rays make

with the t_{2}-axis is denoted by q. The result of
the integration at a given q is a 1-D function, and we denote it by p_{q}(t). In
this figure, p_{q}(t) is the result of integrating f_{c}(t_{1},t_{2})
along the ray passing through the origin. The function p_{6}(t), which
is called the **projection** of f_{c}(t_{1},t_{2})
at angle q or **Radon transform** of f_{c}(t_{1},t_{2}),
can be expressed in terms of f_{c}(t_{1},t_{2}) by

_{} (1.50)

Equation (1.50) arises naturally from the analysis of an
x-ray image. Consider a 2-D object (a slice of a 3-D object, for example)
through which we radiate a monoenergetic x-ray beam, as shown in Figure 1.40.
On the basis of the LambertBeer law, which describes the attenuation of the
x-ray beam as it passes through an object, and of a model of a typical film used
to record the output x-ray beam, the image recorded on film can be modeled by p_{q}(t) in
(1.50). where _{} is
the attenuation coefficient of the 2-D object as a function of two spatial
variables t_{1 }and t_{2}. The function _{ }depends on the material that
composes the 2-D object at the spatial position _{}. To the extent that the attenuation
coefficients of different types of material such as human tissue and bone
differ, _{}can
be used to determine the types of material. Reconstructing _{ }from the recorded p_{q}(t) is,
therefore, of considerable interest.

Consider the 1-D analog Fourier
transform of p_{q}(t) with respect to the variable and denote it by _{} so
that

_{} (1.51)

It can be shown that there is a simple relationship
between _{} and
_{}, given by

_{} (1.52)

Expressed graphically, (1.52) states that the 1-D
Fourier transform of the projection p_{q}(t) is F_{c}(W_{1},W_{2}) evaluated along the slice that
passes through the origin and makes an angle of q with the W_{1 }axis, as shown in Figure 1.41. The relationship in (1.52) is called
the **projection-slice theorem**.

The projection-slice theorem of
(1.52) can be used in developing methods to reconstruct the 2-D function f_{c}(t_{1},t_{2})
from its projections p_{q}(t). One method is to compute the inverse Fourier transform of F_{c}(W_{1},W_{2}) obtained from p_{q}(t). Specifically,
if we compute the 1-D Fourier transform of p_{q}(t) with respect to t
for all 0 £ q < p, we will have complete
information on F_{c}(W_{1},W_{2}). In practice. of course, p_{q}(t) cannot be
measured for all possible angles 0 £
q < p, so F_{c}(W_{1},W_{2}) must be estimated by interpolating
known slices of F_{c}(W_{1},W_{2}).

Another reconstruction method, known
as the **filtered back-projection method,** is more popular in practice and
can be derived from (1 .49b) and (1.52). It can be shown that

_{} (1.53)

where _{}is related to p_{q}(t) by

_{} (1.54)

The function _{}, which can be viewed as
the impulse response of a filter, is given by

_{} (1.55)

where _{} is the frequency above
which the energy in any projection p_{q}(t) can he assumed
to be zero. From (1.53) and (1.54), we can see that one method of
reconstructing f_{c}(t_{1},t_{2}) from p_{q}(t) is to
first compute q_{q}(t) by filtering (convolving) p_{q}(t) with h(t) and
then to determine f_{c}(t_{1},t_{2}) from q_{q}(t) by
using (1.53). The process of determining f_{c}(t_{1},t_{2})
from q_{q}(t) using (1.53) can be viewed as a back-projection. Consider a
particular q and t, say q’ and t’. From (1.53), the values of (t_{1},t_{2})
for which f_{c}(t_{1},t_{2}) is affected by q_{q}(t’) are
given by t’ = t_{1}.cosq’+t_{2}.sinq’. These values are shown by the straight line in Figure 1.42.
Furthermore, the contribution that q_{q}_{’}(t’) makes to f_{c}(t_{1},t_{2}) is equal
at all points along this line. in essence, q_{q}_{’}(t’) is back-projected in the (t_{1},t_{2}) domain.
This back-projection takes place for all values of t’ and is integrated over
all values of q’. Since q_{q}(t) is a filtered version of p_{q}(t), this technique
is called the **filtered back-projection method**. In practice, p_{q}(t) is not
available for all values of q. As a result, q_{q}(t) must be interpolated from the known
slices of q_{q}(t).

In addition to the interpolation involved in both
the direct Fourier transform method and the filtered back-projection method, a
number of practical issues arise in reconstructing f_{c}(t_{1},t_{2})
from p_{q}(t). For example, the Fourier transform, inverse Fourier transform,
filtering, and integration require a discretization of the problem, which
raises a variety of important issues, including sampling and aliasing. In
practice, the measured function p_{q}(t) may be only an approximate
projection of f_{c}(t_{1},t_{2}). In addition, the
measured data may not have been obtained from parallel-beam projection, but
instead from fan-beam projection, in which case a different set of equations
governs.