MART

When the correction in the iterative algorithms are multiplicative rather than additive, the algorithms are grouped under the family of MART (Verhoeven, [311]). Gordon et al. [97] and Gordon and Herman [100] have suggested different forms of MART. The MART algorithms presented below are similar to those considered by Verhoeven [31].

The major difference between ART and MART algorithms is in the method of computing the corrections. While ART uses the difference between the calculated projections and measured projections, MART uses the ratio between the two. Hence the corrections applied to each cell during calculations are via the multiplication operation. The structure otherwise is similar to Gordon’s ART (ART2).

The individual steps of three versions of MART (1,2, and3) are summarized below.
start: 1start iterations :
start: 2 For each   projection angle
start: 3 For each ray
Compute the numerical projection
Calculate the correction as:

Start: 4   For each cell

If is non-zero then:
MART1:

MART2:

MART3:

where is a relaxation factor.

close: 4
close: 3
close: 2
Check for convergence as:
If

where is a suitable stopping criterion.
 STOP:
Else: Continue
close: 1

Steps 3 and 4 form the essence of the reconstruction algorithm. All three versions include the relaxation factor . Typical values of the relaxation factor reported are in the range 0.1 - 1.0, larger values leading to divergence It is to be noted that the correction calculated in step 3 is the ratio of the recorded projection data () and that calculated from the guessed field, namely which is being iterated.  The three versions of MART differ in the manner in which the corrections are implemented. In MART 1. The  weight function is prescribed in binary form, being unity if a particular ray passes through a pixel and zero otherwise . In MART 2 and MART 3, the weight function is precisely calculated as the ratio of the length of the ray intercepted by the pixel and the maximum dimension of a segment enclosed by it.