## 3 Results and Discussion
### Physics of X-ray imaging
From the statistical point of view, the contrast value obtained according to eq. (60) entails that the number of photons per unit surface employed in a conventional radiological examination is quite excessive. Consequently, a detector with near-unity efficiency and capable of single photon counting would require less photons for the same quality of radiological images and, therefore, a lower radiation dose for the subject as well. The program GEANT4 [39], a toolkit originally designed for high energy physics experiments, was employed to simulate the passage and scattering of X-rays through various breast tissues in order to define the imaging system response. The \(SNR\) in the case of soft tissues like normal breast is a function of energy; Monte-Carlo simulations [37] indicated a maximum of the \(SNR\) in the range of \(20\) to \(26\) keV. Inhomogeneous breast-like tissue of 5 cm thickness with calcification inclusions and various opacities were considered in simulations, whereas the X-ray energy covered the range from 18 to 30 keV in steps of 2 keV [38]. The parameters used throughout in simulations were selected close to the corresponding values encountered in common radiological practice and the characteristics of our incoming silicon-based receptor: (i) tissue thickness of \(5\) cm (homogeneous and inhomogeneous); (ii) total number of events for each run: \(10\) millions; (iii) Dirac-distributed X-ray beam shape with orthogonal and central incidence; (iv) receptor area of \(400\times 100\) pixels, each square detector of \(0.2\) mm \(\times 0.2\) mm; (v) recorded data for an area of \(31\times 31\) pixels around the incident point of the X-ray beam (Table 1). The simulations of scattering at 24 keV suggested a symmetric space-invariant \(5\times 5\) pixel PSF matrix. The indexes of the matrix elements of the receptor are defined in Table 2 and blurring area produced by \begin{table}
\begin{tabular}{|c|c|c|} \hline X-ray beam energy (keV) & Central recorded photons & Total recorded photons \\ \hline
18 & 15,660 \(\pm\) 125 & 18,390 \(\pm\) 135 \\ \hline
20 & 59,130 \(\pm\) 240 & 70,900 \(\pm\) 270 \\ \hline
22 & 184,070 \(\pm\) 430 & 226,290 \(\pm\) 475 \\ \hline
24 & 337,400 \(\pm\) 580 & 424,340 \(\pm\) 650 \\ \hline
26 & 531,130 \(\pm\) 730 & 683,330 \(\pm\) 825 \\ \hline
28 & 738,230 \(\pm\) 860 & 976,540 \(\pm\) 990 \\ \hline
30 & 890,710 \(\pm\) 940 & 1,204,780 \(\pm\) 1100 \\ \hline \end{tabular}
\end{table} 
Table 1: Average recorded quanta around the incident point of the X-ray beam as resulted from simulations using GEANT for the most common X-ray energies. a simulated Dirac distribution of the X-ray beam centered on the _k_-th pixel is presented in Table 3. The convergence of the image restoration algorithm was ensured by simulations of various inhomogeneities embedded in breast-like tissue and setting the value of the model parameter, \(\alpha\), in such a way as to achieve the best average reproduction of the simulated opacities in terms of the mean energy of the reconstruction error [40]. The optimal experimental values of the model parameter were found to be within the range of \(\alpha=10^{-5}\div 10^{-4}\). As for instance, an opacity of \(2\times 2\) pixel area embedded in a \(31\times 31\) pixel area of simulated inhomogeneous tissue is shown before and after processing (Figure 6). In addition, a test pattern designed to disclose any artifacts eventually introduced by entropic processing was designed. The pattern consisted of marble grains of various sizes embedded in organic materials with mammalian-like consistency. Neither spurious patterns nor any suspicious forms were detected (Figure 7) for the range of X-ray energy under study. These observations were sustained in a similar approach, by employing more elaborated phantoms in conjunction with MEM image processing to deblur mammographic images [41]. Simulations concluded that an X-ray beam is scattered around \(20\)% at worst and about \(98\)% of quanta are recorded on a \(5\times 5\) pixel area centered on its incident direction.