## 2 Test Objects and Methods
### Method of Projection Formation in X-ray CT  
The basis of the method for generating projections in X-ray CT is the model proposed in [14; 37]. In accordance with the task under consideration, the TO contains one or more fragments, the materials of which may differ from the surrounding material by higher values of density and (or) effective atomic number. Figure 1 shows a diagram of the formation of projections in the X-ray CT in the parallel beam geometry for the TO slice described by the set \(\mathbf{S}{\subset}\Re^{2}\). Traditionally, fixed \(XOY\) and moving \(X^{\prime}OY^{\prime}\) Cartesian coordinate systems are used for the convenience of describing the process of projection formation. The finiteness of the set \(\mathbf{S}\) allows it to be inscribed in a circle \(\mathbf{R}{\subset}\Re^{2}\) with radius \(R\). The coordinates of a point in the fixed coordinate system \((x,y)\) are related to the coordinates of the same point in the moving coordinate system \((x^{\prime},y^{\prime})\) by the rotation transformation by the angle \(\theta\)[14; 15]  
\[\left\{ \begin{array}{c}x^{\prime}=x\cos\theta+y\sin\theta\\ y^{\prime}=-x\sin\theta+y\cos\theta\end{array} \right.,\ \ \left\{ \begin{array}{c}x=x^{\prime}\cos\theta-y^{ \prime}\sin\theta\\ y=x^{\prime}\sin\theta+y^{\prime}\cos\theta\end{array} \right.. \tag{13}\]  
Angle \(\theta\) characterizes one projection. The point with coordinates \((x^{\prime},y^{\prime})\) belongs to the front surface of the detector's array if \(\,\left|\,x^{\prime}\,\right|\,\leq A\) and \(y^{\prime}=F\). Let the shadow of the TO slice \(\mathbf{S}\) and the circumscribed circle \(\mathbf{R}\) for any projection \((0\leq\theta\leq 2\pi)\) does not go beyond the boundaries of the detector's array, then \(R<A\). The key element of the descriptive model of the process of generating projections in X-ray CT is the set of rays \(\mathbf{L}\left(\theta,x^{\prime}\right)\) passing through the points \(x^{\prime},\,\left|\,x^{\prime}\,\right|\,\leq A\), at an angle \(\theta\). The range and direction of the beam \(\mathbf{L}\left(\theta,x^{\prime}\right)\) when generating the projection \(\mathbf{P_{\theta}}\) are indicated in Figure 1 red dotted arrow. The projection \(\mathbf{P_{\theta}}\) is understood as a set of integrals of the informative parameter over the entire set of rays \(\mathbf{L}\) (\(\theta\), \(x^{\prime}\)), \(\mid x^{\prime}\mid\leq A\). The introduction lists the main informative parameters of X-ray CT and notes that density and effective atomic number are preferable for X-ray tomography as a means of measurement. This statement is obvious since it is these two parameters at fixed photon (gamma-ray) energy that uniquely determine the LAC of gamma radiation and all other informative parameters of the X-ray CT. In accordance with the approach outlined in [14], we present an expression for estimating the TO thickness along the beam \(L\) (\(\theta\), \(x^{\prime}\)) in mean free paths \(p(E,x^{\prime},\theta)\) for gamma radiation with energy \(E\)[14; 15]  
\[p(E,x^{\prime},\theta)=\int\limits_{L(\theta,x^{\prime})}m(E,Z \big{(}x^{\prime},y^{\prime}\big{)})\rho\big{(}x^{\prime},y^{\prime}\big{)} \mathrm{d}y^{\prime}. \tag{14}\]  
The formula for calculating the estimate of radioscopic transparency \(T_{N}^{*}(E_{\mathrm{max}},x^{\prime},\theta)\) along the beam \(L\) (\(\theta\), \(x^{\prime}\)), taking into account expressions (9) for the counting registration mode, will take the form  
\[T_{N}^{*}(E_{\mathrm{max}},x^{\prime},\theta)=\frac{\left[N_{0 }\int\limits_{0}^{E_{\mathrm{max}}}f_{N}^{*}(E,E_{\mathrm{max}},Z_{f},\rho_{f} H_{f},Z_{d},\rho_{d}H_{d})\exp(-p(E,x^{\prime},\theta))dE\right]}{\left[N_{0 }\int\limits_{0}^{E_{\mathrm{max}}}f_{N}^{*}(E,E_{\mathrm{max}},Z_{f},\rho_{ f}H_{f},Z_{d},\rho_{d}H_{d})dE\right]}. \tag{15}\]  
Figure 1: Scheme of projection formation in X-ray CT in parallel beam geometry: 1—photon radiation source; 2—test object; 3—detector array; 4—a circle describing the test object section.