**Author details**

Faycal Kharfi

*Iteration*(*n*) 1 2 3 4 5 6 7 8 9

104 Imaging and Radioanalytical Techniques in Interdisciplinary Research - Fundamentals and Cutting Edge Applications

*<sup>n</sup>* 3 1.78 2.99 2.73 2.11 2.08 2.01 2.01 2.00

*<sup>n</sup>* 3.66 2.66 0.81 2.68 2.75 2.96 2.97 2.99 2.99

(E.6)

*1*

(E.7)

*<sup>n</sup>* 3.16 2 2.44 2.71 2.27 2.30 2.25 2.25 /

*<sup>n</sup>* 4.49 2.44 1.35 2.69 2.59 2.74 2.73 2.74 /

*This method is similar to the Jacobi method and use the same updating function (E.5) but just for the first iteration which*

*Iteration*(*n*) 1 2 3 4

*<sup>n</sup>* 3 1.18 1.90 1.98

*<sup>n</sup>* 2.66 3.27 3.03 3.00

*We remark that this algorithm and method gives good results just after 4 iterations; so it is the very fast one in term of*

*End of practical example 1*

The iterative process for image reconstruction must be stopped at the end of the conver‐ gence phase before it starts diverging. This can be done using some regularization meth‐ ods. Indeed, well selected regularization parameter can controls the amount of stabilization imposed on the solution. In iterative methods one can use the stopping index as regulariza‐ tion parameter. When an iterative method is employed for image reconstruction, the user can also study on-line adequate visualizations of the iterates as soon as they are computed, and simply halt the iteration when the approximations reach the desired quality. This may actually be the most appropriate stopping rule in many practical applications, but it requires a good intuitive imagination of what to expect. If this is not the case, a computer can give us some aid to determine the optimal approximation. The stopping rules are divided into two categories: rules which are based on knowledge of the norm of the errors, and rules which do notrequire such information. If the error norm is known within reasonable accuracy, the perhaps most well known stopping rule is the discrepancy principle Morozov [8]. Examples of the second category of methods are the L-curve criterion [9], and the general‐

After a substantial effort, major breakthroughs have been achieved in the last fourteen years in the mathematical modeling of CT. The aim of this chapter is to survey this progress and to describe the relevant models, mathematical problems and reconstruction procedures used in CT. We give a summary on the mathematics of computed tomography. We start with a short

*1 is estimated from p2 equation considering the obtained value f1*

*2 from p1 equation and so on (by matching alternatively between p1 and p2)*

*f* 1

*f* 2

*p*1

*p*2

*3. SIRT algorithm with Gauss-Seidel method*

*which itself will be used to determinate f1*

**2.3. Iteration stopping rules**

ized cross-validation criterion [10].

**3. Conclusions**

*allows the determination of f1*

*convergence.*

*With this algorithm and method the best solution is obtained after 9 iterations.*

*1 value. After that f2*

*until obtaining the best results. The results obtained are shown on the following matrix:*

*f* 1

*f* 2

Department of Physics, Faculty of Science, University of Ferhat Abbas-Sétif, Algeria
