In this case the 2D Radon transform is sampled for
, ,
and , .
Here is the radius of the reconstruction region, i.e. we assume *f(x) = 0* for , . This means that the measured rays come in *p* parallel
bundles with directions evenly distributed over , each bundle consisting of *2q+1* equispaced lines. This was the scanning geometry of the first commercial
scanner for which Hounsfield received the Nobel prize in 1979. This geometry has been replaced by more efficient ones in present day's scanners (see below), but it is still
used in scientific and technical imaging.

We evaluate the integral in (2.2) by the trapezoidal rule:

The accuracy of this approximation can be assessed by sampling theory, according to which the trapezoidal rule for an inner product is exact provided the stepsize *h* satisfies
the Nyquist criterion, i.e. where is the bandwidth of the
factors in the inner product. In our case the first factor is (as a function of *s*) which has bandwidth . The second factor is (again as a function of *s*). This is our data and does not, in general, have finite bandwidth. At this point we have to make an assumption.

We assume *f* to be essentially band-limited with
essential bandwidth . The *n*D Fourier transform of *f* and the 1D Fourier transform *Rf* (with
respect to the second variable) are interrelated by

This is the famous (and easy to prove) ``projection'' or ``central slice'' theorem of computerized tomography. In the present context we need it only to deduce that *f*
and *g = Rf* have the same (essential) bandwidth. Thus the *s*-integral in (2.2)
is accurately represented by the -sum in (2.4) provided that the stepsize
in that sum satisfies the Nyquist criterion . In other
words,

The condition for the number *p* of directions which makes the *j*-sum in (2.4) a good approximation for the -integral in (2.2) is less obvious.
Based on Debye's asymptotic relation for the Bessel functions one can show that the
essential bandwidth of *Rf* as a function of , , is , see Natterer (1986). The stepsize *h* for
the -integral being the Nyquist criterion requires , i.e.

(2.6), (2.7) are the conditions for a good accuracy in (2.4), assuming *f* to be zero outside the ball of radius and essentially band-limited with bandwidth .

The double sum in (2.4) has to be evaluated for each reconstruction point *x*.
This leads to an unbearable complexity. This complexity can be reduced by
introducing the function

Then, (2.4) reads

This requires only a simple sum for each reconstruction point *x*, at the expense
of an additional interpolation in the second argument of *h*. In most cases linear interpolation suffices (but nearest neighbour does not!). This leads us to the filtered
backprojection algorithm.

**Algorithm 1** (Filtered backprojection algorithm for standard parallel
geometry.)

**Data:**- The values ,
, ,
*g*is the 2D Radon transform of*f*. **Step 1:**- For carry out the discrete convolution
**Step 2:**- For each reconstruction point
*x*, compute the discretebackprojection

where

*k = k(j,x)*and are determined by **Result:**- is an approximation to
*f(x)*.

**1.**- The condition (2.6) has to be strictly satisfied.
Otherwise the
*s*integral in (2.2) is not even approximately equal to the sum in (2.4), leading to unacceptable errors. **2.**- If (2.7) is not satisfied, the reconstruction is still accurate for .
**3.**- Filter functions whose ``kernel sum''
does not vanish should not be used, see Natterer and Faridani (1990).

**4.**- Usually linear interpolation in step 2 is sufficient. However, for difficult
functions
*f*- e.g. functions containing large objects at the boundary of the reconstruction region - linear interpolation generates visible artefacts. In that case an oversampling procedure similar to the one of Algorithm 2 below is advisable. Alternatively one may use the circular harmonic algorithm from section 5. **5.**- The algorithm needs O
*(p)*operations for each reconstruction point. Algorithms with lower complexity (such as O ) can be obtained either by Fourier reconstruction, see section 6, or by the fast backprojection algorithm in section 2.5. **6.**- The conditions (2.6), (2.7) suggest to take .
This much debated condition is usually not complied with in radiological
applications, where
*p*is chosen considerably smaller. This is due to the special requirements in radiological imaging.

Thu Sep 10 10:51:17 MET DST 1998