Series expansion methods being discussed in the present section can be classified into: ART (Algebraic Reconstruction Technique) and MART (multiplicative Algebraic Reconstruction Technique).The optimization of the entropy and minimization of the energy functions.

Figure 4.64: Discretization of a plane of fluid layer for art calculations

The ART and MART families of algorithms differ only in the method of updating the field parameters in each iteration In ART. The correction is additive while for MART, the correction is multiplicative. In both cases, the numerical procedure is based on the comparison of the estimated projection from an initial guess with the measured projection data obtained though experiments. This gives a correction term for the field variables. The values of the variables are then updated. Once an iteration is over, the field value differs from the previous guess. The extent of the difference in then calculated. If the difference is within acceptable limits, the field value is taken to represent the physical field. Otherwise the iterations continue until the convergence criterion is satisfied.

Since the original field in real experiments is unknown, an estimate of the number of iterations can be found by using test functions (called phantoms) that are similar in nature to the original field. The test functions are also perturbed with noise to gauge the sensitivity of the algorithms to issues such as initial guess and error in the projection data. This method can only be adopted where an exact estimate of noise in the projection data and a good knowledge of the   original field is known beforehand. Variations in the noise level and nature of the noise in the projection data can alter the convergence rates.

Tomographic algorithms used in the present work are iterative in nature and intermediate steps may also involve iterations in the form of FOR loops.  To identify   the beginning and the ending of each iterative loop, start and close labels with statement number have been indicated in the description of each  algorithm. These algorithms are briefly surveyed in the following sections.