The use of resonance effect in X-ray tomography

1. When photons pass through materials, one of the manifestations of the resonance effect is that some mathematical characteristics of the radiation considered as a function of energy have jump discontinuities at certain values of the argument [1, 2]. This property can be used in X-ray tomography, where information about the internal structure of an unknown medium is obtained by analyzing the passage of sensing radiation through it. The problem is studied in the framework of the photon migration mathematical model; the main part of this model is the stationary integro-differential transport equation. As in the classical tomography problem, which is based on the inversion of the ray transform, we want to determine the radiation attenuation coefficient. In this paper, we use the approach that proved to be effective in inverse problems. This approach is based on the separation and analysis of relatively simple components of a complex and cumbersome mathematical expression (e.g., see [3, 4]). In such cases, different properties of the constituent parts of the known data are used. In this paper, we propose a novel implementation of this approach based on using external resonance-type sources. It is important that, although all the coefficients of the transport equation are assumed to be unknown, only one of them is to be determined. This feature is rarely encountered in inverse problem theory. It can be interpreted as the absence of substantial a priori information about the object under examination, which helps the practical application of theoretical results. Although the mathematical model is interpreted in terms of X-rays in this paper, it is fairly general, and I hope that the results can be helpful in other kinds of radiation tomography. 2. We assume that the migration of photons in a medium G is described by the transport equation