# Method of Multi-Angle Transmission Radiowave Tomography of Dielectric Objects
## 3 Method of Inverse Task Solution
For the simulated scene, a function, \(P^{\prime}(x,0,z)\), was calculated, which is shown in Figure 6b. In this image, one can already distinguish test objects. It can be seen that an object with a large refractive index has a more intense image (a darker color). To compare our method and the ray approximation shadow projection technique, we have calculated the same image with ray approximation (Figure 6a). Visually, our approach gives a better quality. We performed data processing with additive noise of 30% and 100% (Figure 4b,c). As a result of data processing, images of given inhomogeneities in noise conditions were obtained (Figure 7a,b). One can see that with an additive noise of 30%, the recovery result is little different from the restored image without noise. With an additive noise of 100%, a significant distortion of the image occurs; however, it is still possible to distinguish the presence of simulated objects. Moreover, an object with a large refractive index is visualized more clearly. We normalize the function \(P^{\prime}(x,0,z)\), believing that \(P^{\prime}(x,0,z)=0\) at a refractive index \(n=1\) and maximum values are taken at \(n=2\). Figure 6 shows graphs of the cross-sections of functions \(P^{\prime}(x,0,z)\) for the first and second objects. In the graph (Figure 8), the values of the normalized function \(P^{\prime}(x,0,z)\) approximately correspond to a given distribution of the refractive index. Figure 6: The sum of shadow projections \(P^{\prime}(x,0,z)\) for two objects with refractive indices of 1.5 and 2 without noise in the input signal. (**a**) Ray approximation; (**b**) Our approach on the basis of backward wave propagation. Furthermore, according to the method of shadow projections for X-ray radiation, it is necessary to calculate the inverse convolution with Tikhonov regularization from a function \(P(x,y,z)\) to a function of the form \(1/\sqrt{x^{2}+z^{2}}\) in the following way:  
\[P(x,y,z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\widetilde{P}^{ \prime}(k_{x},y,k_{z})\widetilde{G^{\prime}}\big{(}k_{x},k_{y}\big{)}}{ \widetilde{G}\big{(}k_{x},k_{y}\big{)}\widetilde{G^{\prime}}\big{(}k_{x},k_{y} \big{)}+\gamma}dk_{x}dk_{z}, \tag{8}\]  
where \(\widetilde{P}^{\prime}(k_{x},y,k_{z})=\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}P^{\prime}(x,y,z)exp(-ik_{x}x-ik_{z}z)dxdz\) is the flat spatial spectrum of the sum of the shadow projections, \(\widetilde{G}(k_{x},k_{z})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{exp(-ik_{x}x-ik_{z}z)}{\sqrt{x^{2}+z^{2}}}dxdz\) is the flat spatial spectrum of the reaction to a pointwise heterogeneity in the summation of the shadow projections. The regularization parameter is selected interactively based on the noise level and artifacts in the restored image; the larger this  
Figure 8: Cross-sections of a normalized function \(P^{\prime}(x,0,z)\). Figure 7: The sum of shadow projections \(P^{\prime}(x,0,z)\) for two objects with refractive indices of 1.5 and 2 on the basis of backward wave propagation. (**a**) Noise of input data, 30%; (**b**) Noise of input data, 100%. parameter, the lower the noise level but the more blurry the image is. Value \(P(x,y,z)\) is proportional to the values of the refractive index in the medium. As a result of applying Equation (8), the image of dielectric heterogeneities was calculated (Figure 9). In this case, the regularization parameter, \(\gamma\), is chosen to be equal to \(3\cdot 10^{-3}\cdot max\left\langle\left|\overline{G}\right|^{2}\right\rangle\), where \(max\left(\left|\overline{G}\right|^{2}\right)\) is the maximum modulus of the square of the function \(\overline{G}(k_{x},k_{z})\). The refraction coefficient could be calculated by the following formula:  
Figure 4:0  
The application of Equation (8) allows an increase in the sharpness of the image; however, since the occurring wave effects are not equivalent to the ray approximation as in X-ray tomography, it is necessary to set the regularization parameter to be relatively large, which leads to the blurring of the resulting image.