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 [16] 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.

Was this article helpful?

0 0

Post a comment