## 3 Results and Discussion  
### Physics of X-ray imaging  
The production of X-ray images is largely based on the assumption that the X-ray photons are passing through the imaged object (tissue) along rectilinear paths. As such, the refractive index of the X-rays is implicitly assumed practically one, which is reasonably true in a first approximation only. The interactions remove some of the photons from the beam in a process known as attenuation. The _linear attenuation coefficient_\(\mu\) is defined as the relative decrease of the number of photons \(N\) in the incident beam while crossing the tissue by the rate \(\mathrm{d}N/N=-\mu\,\mathrm{d}x\). Linear attenuation coefficient values indicate the rate at which photons interact as they move through the tissue and are inversely related to the average distance photons travel before interacting. Furthermore, only a fraction of the impinging photons are assumed to be absorbed while crossing the imaged object. The removal of photons from the beam is supposed to occur only because of the interactions with the atoms of the object (actually with the electrons). The total attenuation depends on the individual rates of all interactions that may occur while the photons are passing through a sample. In the case of a photon beam with energy less than \(100\) keV, the main interactions are the photoelectric effect, Compton (or incoherent) scattering, and Rayleigh (or coherent) scattering. The breast has very low physical contrast being composed of soft tissues: essentially a background of fat surrounded by slightly more dense glandular structures. Even though calcium is somewhat more dense and has a higher atomic number, typical breast calcifications are quite small and produce low physical contrast. As such, relatively low energy photons are employed in mammography, most of the interactions of the X-ray photons with tissue being photoelectric processes. The main limiting contrast factors between soft tissues and between soft tissue and fluids are the small differences between their physical characteristics (density and atomic number) and the relatively low number of photoelectric interactions because of the tissue low atomic numbers. The rate of photoelectric interactions generally decreases with increasing energy of the individual photons. Since dose and contrast decrease with increasing photon energy with different rates, it is possible to adjust the X-ray beam spectrum for optimized imaging with respect to contrast and dose. The quantity that affects attenuation rate is not the total mass of an object but the _area mass_, which is the amount of material under the unit surface area. Assuming a homogeneous imaged object, the attenuation is obtained by integration:  
\[\frac{N(E)}{N_{0}}=-\exp[-\mu(E)]\mathrm{d}x=\exp\left[-\frac{\mu}{\rho}(E)\, \rho\,\mathrm{d}x\right] \tag{56}\]  
where \(N_{0}\) is the number of photons in the incident beam and the dependence on the X-ray energy \(E\) was explicitly displayed. The _mass attenuation coefficient_, \(\mu/\rho\), which is defined as the rate of photons per unit area mass, depends on the material composition and the energy of the incident photons. A simple empirical approximation of the mass absorption coefficient for low \(Z\) matrices (tissue) was considered in the present experiments:  
\[\frac{\mu}{\rho}(E)=20.64\cdot E^{-3.28}\cdot\overline{Z}^{4.62}+\sigma_{KN}( E)\cdot\overline{Z}+2.8\cdot E^{-2.02}\cdot\overline{Z}^{4.86} \tag{57}\]  
where \(\sigma_{KN}(E)\) is the Klein-Nishina cross section and \(\overline{Z}\) is the effective atomic number of the sample. The basic unit of information in an imaging system is the detected photon. The physical image corresponds to the recorded number of photons impinging on some photosensitive device at specified locations. Photon emission is a Poissonian process so that the number of detected photons will be distributed according to Poisson statistics.