## 3 Results and Discussion
### Physics of X-ray imaging
Therefore, if in a unit time and per unit area an average number \(\overline{n}\) of photons is recorded, the standard deviation of measurements will be \(\sqrt{n}\). Such fluctuation in single measurements is intrinsic to the photon counting, so that one may speak of a _quantum_ limited image. An ideal imaging device can be thought as a spatial array of individual photon receptors and counters, each having identical properties. It is assumed that each receptor can receive light quanta (photons) independent of its neighboring receptors. The image state of a detector is completely determined by the number of quanta it receives and records, and each detector can be in one of a finite number of distinguishable image states. Since the possible number of states of a detector is finite, after a detector attains the final state, all other additional incident quanta will remain unrecorded, that is the detector gets saturated. In case of a digital image system, each pixel can be viewed as a receptor. The spatial resolution of the system depends on the spatial size of the pixel, whereas each pixel can have only a finite number of states until reaching its saturation level. The observed grey level of a pixel is the consequence of the received quanta by the corresponding receptor. Up to a certain level, the larger the number of the recorded quanta, the larger is the grey value. The knowledge of the dose and the energy spectrum of the X-rays delivered to the subject during exposure allows in principle the computation of photon number per unit surface of the image system. Let us consider a sample to be imaged by means of a detector whose surface \(A\) is subdivided in identical square pixels of side \(d\)[42]. Let us denote by \(N_{b}\) the number of photons incident on a pixel looking at the \"background\", and \(N_{d}\) the number of photons incident on a pixel containing some texture to be imaged assuming that a percentage \(p\) of the original incident photons has been absorbed, that is \(N_{d}=(1-p)N_{b}\). Further, define the _signal_ as \(N_{b}-N_{d}=pN_{b}\) and the noise as \(\sqrt{N_{b}+N_{d}}=\sqrt{N_{b}(2-p)}\). Consequently, it appears natural to express the _contrast_\(C\) as:  
\[C=\frac{N_{b}-N_{d}}{(N_{b}+N_{d})/2}=\frac{2p}{2-p} \tag{58}\]  
while the signal-to-noise ratio (\(SNR\)) to be given by:  
\[SNR=\frac{pN_{b}}{\sqrt{N_{b}(2-p)}} \tag{59}\]  
Hence the number of background photons required to reveal a certain target detail with a given contrast \(C\) is \(N_{b}=2(SNR)^{2}/pC\). If \(p<2\) then \(C\approx p\) resulting in \(N_{b}\approx 2(SNR/C)^{2}\). The result holds for an imaging system that is completely noise-free, i.e., the target is visible against a smooth background. Image noise degrades the overall quality of an image making difficult to reveal certain details of interest. The major source of noise in X-ray and nuclear imaging is the random distribution of photons over the surface of the image. The amount of noise, or variation in photon concentration from area to area (e.g., from one detector cell to another) is inversely related to the number of photons used to form the image. Therefore, by increasing the total number of photons, the noise of an image can be reduced. It is important that the \(SNR\) to be large enough so that in a set of pixels the probability of any pixel giving accidentally a false signal to be negligible. In the case of \(10^{5}\div 10^{6}\) pixels, the condition is accomplished by setting \(SNR=5\)[36]. If \(N\) photons are incident on the surface \(A\) of the detector:  
\[\frac{N}{N_{b}}=\frac{A}{d^{2}}\Rightarrow C=\frac{\sqrt{2}(SNR)}{d\sqrt{N/A}} \tag{60}\]  
Inserting typical numerical values in eq. (60), such as \(N/A=4\times 10^{6}\) photons/\(\mathrm{mm}^{2}\), \(SNR=5\), and the pixel side of the square detector \(d=0.2\) mm, leads to \(C=1.7\times 10^{-2}\). Such a contrast value is extremely low since the practical rule is currently setting the minimum contrast perceivable by the human eye to be of about \(2.5\%\) over a smooth background.