AVMART 

The reconstruction of a function from a finite number of projections recorded at different view angles leads to an ill-posed matrix inversion problem. The problem is accentuated when the projection data is limited. The resulting matrix is rectangular with the number of unknowns being greater than the number of equations. In view of the ill-conditioning of the matrix, the convergence of the matrix, the convergence of the iterations to any particular solution is dependent on the initial on the initial guess, the noise level in the projection data and the under-relaxation parameter employed. In the present study, the MART algorithm has been extended so as to

• enlarge the range of the usable relaxation factor.
• diminish the influence of noise in the projection data, and
• guarantee a meaningful solution when the initial guess is simply a constant.

The original MART algorithm described above has been modified in the present work to form a new approach to applying the corrections. In the proposed algorithm the corrections are calculated by considering all the rays from all the angles passing through a given pixel. Instead of a single correction obtained from individual rays, a correction that is determined as determined as the average of all the rays is used. The difference between the conventional MART and the present implementation is the following. The correction at each pixel is updated on the basis of the N-th root of the product of all the corrections from all the N rays of all view angles passing through a pixel. This idea is based on the fact that average corrections are expected to behave better in the presence of noisy projection data. Since an average correction is introduction, the algorithm is desensitized to noise. There is however a potential drawback. Since an average pixel correction based on a set of rays is computed, the reconstructed field is not required to satisfy exactly the recorded projection data. This was seen to be no cause for concern for the three application considered. The projection data was exactly satisfied by the reconstructed field in each case.
The modified algorithms have been identified below as AVMART, The prefix AV referring to average. The important step, namely step 4 alone is presented here, with the understanding that all other steps remain unchanged.

Start: 4 For each cell
Identify all the rays passing through a given cell Let be the total number of rays passing through the j- th cell.
Apply correction as:

AVMART1:    

AVMART2:    

AVMART3:

Close:4
The symbol in the three algorithms above represents a product over the variable . The -th root of this product is evaluated in each approach. The relaxation factor has been retained in the statements above for completeness. In all calculations, it was set equal to unity to bring out a mixture of the “smooth “ and “sharp” features of the temperature field. The proposed algorithms require a smaller CPU time per iteration, when compared to the existing ones, Section 6.7 evaluates the benefits derived by modifying step 4 for problems of practical importance.