# Method of Multi-Angle Transmission Radiowave Tomography of Dielectric Objects
## 3 Method of Inverse Task Solution  
The solution of the inverse problem is based on the inverse propagation of waves from the measurement region towards the emitter. We calculate the field in the sounding region on the basis of the formula:  
\[U(x,y,z,\alpha,\omega)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\overline{A }\Big{(}k_{x},k_{y},\alpha,\omega\Big{)}exp\Big{(}ik_{x}x+ik_{y}y\Big{)}(exp( ik_{z}(Z-z)))^{*}dk_{x}dk_{y} \tag{4}\]  
where \(Z=R\) is the position of the measurement plane; * represents complex conjugation; and \(\overline{A}\Big{(}k_{x},k_{y},\alpha,\omega\Big{)}=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}A(x,y,\alpha,\omega)exp\Big{(}-ik_{x}x-ik_{y}y\Big{)}dxdy\) is the spatial field spectrum in the measurement domain. In this solution, only the fields of homogeneous waves are restored. We propose calculating the time delay of the signal in the medium on the basis of the calculated field, \(U(x,y,z,\alpha,\omega)\). The time delay, \(\Delta T(x,y,z,\alpha)\) (delay function), will be calculated with respect to a  
Figure 4: Field in the measurement area at 20 GHz for 256 different angles of rotation of the test object. (**a**) Without noise; (**b**) Additive noise, 30%; (**c**) Additive noise, 100%. plane wave, propagating in a homogeneous background medium. To obtain the value of \(\Delta T(x,y,z,\alpha)\), we multiply \(U(x,y,z,\alpha,\omega)\) on the conjugated plane wave and move to the time domain:  
\[W(x,y,z,\alpha,t)=\int_{-\infty}^{\infty}U(x,y,z,\alpha,\omega)exp(-ik_{z}z-i \omega t)d\omega \tag{5}\]  
where \(t\) = the time of equivalent pulse propagation. If dispersion effects are not taken into account, then at all frequencies, the delay in matter occurs identically, so after returning to the time domain function, \(W(x,y,z,\alpha,t)\), it will have the form of a short pulse with a maximum magnitude at \(t=\Delta T(x,y,z,\alpha)\). Thus, finding the maximum magnitude of \(W(x,y,z,\alpha,t)\) in time, we can find the value of \(\Delta T(x,y,z,\alpha)\), which describes the time delay of the pulse in a heterogeneous medium relative to that in the homogeneous medium in the entire space. Figure 4 shows the result of calculating the delay function, \(\Delta T(x,0,z,\alpha)\), for the test objects shown in Figure 3a. In Figure 5, darker areas correspond to larger values of delay. Additionally, in Figure 5, it can be seen that an object with a large refractive index corresponds to a darker color in the figure, or larger delay values. This delay is proportional to the product of the refractive index minus 1 and a distance (\(2R(n-1)\)) and in the ray approximation, without allowance for refraction, is described by the integral:  
\[\Delta T(x,y,z,\alpha)\sim\int_{-R}^{+R}(n-1)(x\cos\alpha-z^{\prime}\sin \alpha,y,x\sin\alpha+z^{\prime}\cos\alpha)dz^{\prime}, \tag{6}\]  
where \(n(x,y,z)\) is a function describing the distribution of the refractive index in a medium. The transformation of the rotation of the coordinate system for different probing angles is taken into account under the integral sign. In this approximation, the delay does not depend on z, but the computation of the delay in accordance with Equations (4) and (5) will have an insignificant dependence on z. In fact, the function \(\Delta T(x,y,z,\alpha)\) is a linearized back shadow projection of the objects under investigation. Figure 5: The obtained delay function for two dielectric objects with refractive indices of 1.5 and 2 at a rotation angle of 0. Numerically, the operations in Equations (4) and (5) are implemented using the fast Fourier transform algorithm, which allows us to speed up the solution of the inverse problem many times. Next, we use the analogy with the method of shadow projections used in X-ray tomography [14; 15]. According to the method of shadow projections, it is necessary to summarize \(\Delta T(x,y,z,\alpha)\) for different angles:  
\[P^{\prime}(x,y,z)=\int_{-\pi}^{\pi}\Delta T(x,y,z,\alpha)d\alpha, \tag{7}\]  
where \(P^{\prime}(x,y,z)\) is the cumulative shadow projection.