The throughout knowledge of a X-ray beam spectrum is mandatory to assess the quality of its source device. Since the techniques to directly measurement such spectra are expensive and laborious, the X-ray spectrum reconstruction using attenuation data has been a promising alternative. However, such reconstruction corresponds mathematically to an inverse, nonlinear and ill-posed problem. Therefore, to solve it the use of powerful optimization algorithms and good regularization functions is required. Here, we present a generalized simulated annealing algorithm combined with a suitable smoothing regularization function to solve the X-ray spectrum reconstruction inverse problem. We also propose an approach to set the initial acceptance and visitation temperatures and a standardization of the objective function terms to automatize the algorithm to address with different spectra range. Numerical tests considering three different reference spectra with its attenuation curve are presented. Results show that the algorithm provides good accuracy to retrieve the reference spectra shapes corroborating the central importance of our regularization function and the performance improvement of the generalized simulated annealing compared to its classical version.

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The image reconstruction using the EIT (Electrical Impedance Tomography) technique is a nonlinear and ill-posed inverse problem which demands a powerful direct or iterative method. A typical approach for solving the problem is to minimize an error functional using an iterative method. In this case, an initial solution close enough to the global minimum is mandatory to ensure the convergence to the correct minimum in an appropriate time interval. The aim of this paper is to present a new, simple and low cost technique (quadrant-searching) to reduce the search space and consequently to obtain an initial solution of the inverse problem of EIT. This technique calculates the error functional for four different contrast distributions placing a large prospective inclusion in the four quadrants of the domain. Comparing the four values of the error functional it is possible to get conclusions about the internal electric contrast. For this purpose, initially we performed tests to assess the accuracy of the BEM (Boundary Element Method) when applied to the direct problem of the EIT and to verify the behavior of error functional surface in the search space. Finally, numerical tests have been performed to verify the new technique.