## 3 Results and Discussion
### X-ray image quality assessment  
The evaluation of the X-ray images was based on contrast-detail (CD) phantoms as fostered by the Department of Radiology, University Hospital Nijmegen [44]. The CDRAD phantoms consist of square plexiglas tablets of \(265\times 265\) mm with thickness of \(10\) mm. Each tablet contains cylindrical holes of exact diameter and depth with tolerances less than \(0.02\) mm arranged in \(15\times 15\) columns\(\times\)rows small squares. First, CDRAD 2.0 phantoms were employed to plotting a CD-curve based on the threshold contrast as a function of object diameter. A better image should reveal smaller details or quite the same details at lower contrasts. All processed phantom images displayed a shift of the CD-curve to the lower left part of the phantom relatively to the CD-curve of the input images. Qualitatively and quantitatively, the meaning was a better overall visibility of details. Secondly, the image quality was expressed in a  
Figure 6: Artificial-generated opacity of \(2\times 2\) pixel size (left) embedded in a simulated background breast-like tissue of \(31\times 31\) pixel area (middle) and subsequently restored by the entropic restoration algorithm.  
Figure 7: Arrangement of marble grains (left), raw radiography (middle), and processed radiography by the entropic algorithm (right). Average dimension of individual and cluster grains are specified in millimeters.  
figure by the calculation of the ratio of the correctly identified hole-positions to the total number of the phantom squares. The correct observation ratios were higher for the processed images with \(5\div 12\)%. Thirdly, the CDRAD 2.0 phantoms were employed to quantify the visibility of details at various low-contrast values.  
To estimate the restoration quality in case of \(M=N\), the mean energy of reconstruction error:  
\[D=\frac{1}{N}\sum_{n=1}^{N}\left(g_{n}-\widetilde{f}_{n}\right)^{2} \tag{63}\]  
where \(\widetilde{\mathbf{f}}\) is the best statistical estimation of the correct solution \(\mathbf{f}\), can be used as a factor of merit of the reconstruction algorithm. Yet too high a value for \(D\) may put the restored image too far away from the original one and raise questions about introducing spurious features for which there is no clear evidence in the acquired data.  
A more realistic degradation measure of image blurring by additive noise can be formulated in terms of blurred signal-to-noise ratio (_BSNR_) metric redefined here by using the noise variance, \(\sigma\), in each pixel:  
\[BSNR=10\,\log_{10}\frac{1}{N}\sum_{n=1}^{N}\frac{(y_{n}-\overline{y}_{n})^{2} }{\sigma^{2}} \tag{64}\]  
where \(\mathbf{y}=\mathbf{g}-\mathbf{e}\) is the difference between the measured data, \(\mathbf{g}\), and the noise (error), \(\mathbf{e}\).  
In simulations, where the original image, \(\mathbf{f}\), of the measured data \(\mathbf{g}\) is available, the quality of image restoration algorithms may be assessed by the improvement in signal-to-noise metric:  
\[ISNR=10\,\log_{10}\frac{\sum_{n=1}^{N}(f_{n}-g_{n})^{2}}{\sum_{n=1}^{N}(f_{n} -\widetilde{f}_{n})^{2}} \tag{65}\]  
While mean squared error metrics like _ISNR_ do not always reflect the perceptual properties of the human visual system, they may provide an objective standard in comparing different image processing techniques. It is nevertheless of crucial significance that various algorithms behavior should be analyzed from the point of view of ringing and noise amplification, which can be a key indicator of improvement in quality for subjective comparisons of restoration protocols [45].