# Method of Multi-Angle Transmission Radiowave Tomography of Dielectric Objects
## 1 Introduction
In most cases, the entire tomographic system can be considered as linear. Therefore, it can be described in terms of systems of linear algebraic equations, where the parameters of the medium are unknown quantities and the known values are the set of measured signals. To solve such systems, regularization methods are used [31; 32; 40], since, as a rule, the matrix describing the system has a high conditionality. There are iterative approaches to solving the inverse tomography problems, which are based on minimizing the difference between the numerical solution of the direct problem and the actual measured data [13]. In [13], a variant of the method of gradient descent is proposed, which makes it possible to reconstruct flat distributions of electrical conductivity. The disadvantages of this method are the computationally complex iterative process and the probability of finding not a global but a local minimum. To avoid falling into the local minimum stochastic, Monte Carlo methods are used [25]; however, there remains the computationally costly necessity of the iterative search of solutions. Nevertheless, in view of the constant development of computer technology, the problem of the large amount of computation becomes less significant. An effective description of the residual diffraction effects is given by the phase approximation of the Huygens-Kirchhoff method [41], in which the main interaction of radiation with translucent objects is described by a phase screen. The phase distribution in the plane of the phase screen at each point is given in the geometric optics approximation, as if the corresponding wave were flat and propagated strictly along the line through the probing object. The best results can be achieved by methods based on taking into account the wave nature of radiation [42; 43; 44; 45; 46; 47] and considering phase information. The use of broadband signals or measurement at multiple frequencies makes it possible to minimize the artifacts of reconstructed tomographic images [42; 43] even with unfilled spatial measurements. The radiowave tomographic problem can be solved even in the case of the absence of phase measurements [48] by the application of frequency scanning and the synthetic aperture technique. In [45], the solution of the inverse problem reduces to the method of least squares, but such an approach is quite computationally consuming; however, it potentially allows the achievement of the most accurate solution. For restoring the properties of the medium, the most accurate results allow the obtaining of iterative methods. The optimized search schemes [47] allow the obtaining of a solution within an acceptable time frame. However, there is a need to speed up the solution by searching for the most accurate first approximation for iterative methods. In this paper, we propose a method of the inverse problem numerical solution for reconstructing the distribution of the refractive index in a medium from the data of multi-angle transmission broadband radio sounding. The proposed approach is new and has not been previously considered. In contrast to the classical method of back-projections with a ray approximation, our method takes into account the propagation of waves of finite length, and the shadow delay of the propagation of waves caused by the presence of dielectric inhomogeneities is considered as shadow projections. The existing methods of diffraction tomography consider cases of slightly contrasting inhomogeneities with an insignificant phase incursion in the inhomogeneities relative to the unperturbed case [49; 50; 51; 52; 53; 54]. In our case, the contrast of the refractive index can be significant; the ambiguity of the phase change is eliminated by applying measurements in a wide frequency band. In contrast to holographic tomography, in proposed article, the field diffraction at a variety of frequencies is considered [55]. In this case, the synchronization of the time counting for all frequencies is necessary. We considered multi-frequency sounding.