## Discrete Implementation of the FBP

The discrete implementation of Eqs. (12) and (13) using the measured samples of p(s, <j) described in the section on sinogram data and sampling, above (Eq. (5)), involves four approximations:

(i) The approximation of the kernel h(s) by an apodized kernel hw(s) = J dv\v\ w(v) exp(2nisv)

where w(v) is a low-pass filter which suppresses the high spatial frequencies, and will be discussed later in the section on the ill-posedness of the inverse X-ray transform. (ii) The approximation of the convolution integral by a discrete quadrature. Usually standard trapezoidal quadrature is used: Ns pF (kAs,<)—As X p(k'As,< )hw((k - k')As)

The calculation of this discrete convolution can be accelerated using the discrete Fourier transform (FFT) (see  section 13.1). In this case, some care is needed when defining the discrete filter: to avoid bias, this filter must be calculated as the FFT of the sampled convolution kernel hw(kAs), k = 0, ±1,±2,...,and not by simply sampling the continuous filter function \v\w(v).

(iii) The approximation of the backprojection by a discrete quadrature

f (x,y) — A<j X pF (s = x cos <j + y sin <,<) (17)

for a set of image points (x, y) (usually a square pixel grid)(6).

(iv) The estimation of pF (s = x cos jj + y sin j, jj) in Eq. (17) from the available samples pF (kAs, jj). This is usually done using linear interpolation:

where k is the integer index such that kAs < s < (k + 1)As. Instead of linear interpolation some implementations apply a faster nearest-neighbor interpolation to filtered projections which have first been linearly interpolated on a finer grid (typically sampled at a rate As/4).

Remarkably, most FBP implementations only use simple tools of numerical analysis, such as linear interpolation and trapezoidal quadrature, despite many attempts to demonstrate the benefits of more sophisticated techniques.