# Method of Multi-Angle Transmission Radiowave Tomography of Dielectric Objects
## 3 Method of Inverse Task Solution
Equations (7) and (8) are an approximate solution of the inverse problem of restoring the distribution of dielectric heterogeneities. In fact, radio waves of backward propagation are not direct rays, as in the case of X-ray radiation. In reality, there are effects of refraction, reflection, diffraction, and multiple scattering. However, in approximating that backward propagation waves are direct rays, Equations (7) and (8) should ensure the reconstruction of images of radio-permeable dielectric heterogeneities. Nevertheless, Equation (4) takes into account the wave nature of the probe radiation. In addition, numerical modeling was carried out for objects of various shapes. The results of image reconstruction using Equation (7) are shown in Figure 4:1 The dashed lines in Figure 8 show a simulated objects contour. Figure 4:2 Restored distribution of dielectric heterogeneities \(P(x,0,z)\). Figure 10a shows the result of the image reconstruction of two cylinders with refractive indices of 2 and 1.5. Because of the diffraction of waves on one object before falling onto another, the shape of the objects being reconstructed is distorted. As the radius of dielectric cylinders decreases, the diffraction effect no longer substantially distorts the images of objects (Figure 10b). The numerical simulation of a hollow cylinder was performed (Figure 10c), which showed that the circular shape of the object is restored correctly but that the internal volume is restored with distortions, which is explained by the refraction effects at the cylinder boundary. An object of a complex step-shaped form was also considered (Figure 10d). In the area of shading itself, a distortion of the propagation paths of the waves arises, because of which the steps are not clearly drawn. Figure 4:3 Result of the numerical modeling of the image reconstruction of various objects. (**a**) Two cylinders with refractive indices of 2 and 1.5; (**b**) Two cylinders of small diameter with refractive indices of 2 and 1.5; (**c**) Hollow cylinder with a refractive index of 1.5; (**d**) Stepped object with an index of refraction of 1.5. ```

Algorithm 1 shows the steps for reconstructing the real part of the refractive index. ## 4 Conclusions

The authors developed a method for the broadband multi-angle transmission radiowave tomography of dielectric objects based on the backward wave propagation model. A comparison is made between the time delay of the signal calculated from the measurement results in the entire sounding region and the propagation time in a homogeneous medium. Studies were conducted on the modeling of this method at various objects. On the basis of the results of these numerical simulations, it can be concluded that the method allows the obtaining of a resolution in the order of a wavelength in the theoretical limit. The combination of time delays obtained at different probing angles makes it possible to reconstruct the refractive index distribution in the medium. However, eight projections are sufficient to distinguish between the dielectric permittivities of several objects. Note that the proposed method does not take into account the refraction of waves in dielectric objects and multiple reflections in the probed region. However, as shown by the results of the numerical experiments, this method is highly resistant to additive noise. Even when adding noise with a level of 100%, it is still possible to visualize objects in the study area. In the future, the results of numerical simulations will be checked by the results of field experiments. However, even now, we can say that the proposed method and the studies carried out are of great interest for radio tomography. The novelty of the proposed method consists of taking into account the wave propagation of the probe radiation, and the time delays acquired by the wave when passing through dielectric inhomogeneities throughout the probed area are used as shadow projections. The proposed method is high-speed because of its non-iterative approach.