# A Fast Method of Visually Lossless Compression of Dental Images
## 2 Some Assumptions on Image/Noise Properties  
The dental images acquired by the modern imaging systems are large in size, at least one megapixel data [12; 13; 31]. In particular, this relates to the images acquired by the system Morita, panoramic X-ray (Veraviewepocs 3D R100 J) [37]. Such images usually have a square (in general, rectangular) shape where the ratio of dimensions depends on imaging system and on its operation mode. Meanwhile, one may need to compress an image fragment in a region of interest that can be square or rectangular and of different context. Examples of such fragments are given in Figure 1.  
There is a noise present in dental images, as one can see from these fragments, that might partly mask important diagnostic information. This noise arises due to a limited time of imaging the object, and properties of noise are complex [29; 31; 38]. In the simplest case, one can model noise in dental images as Poisson-like signal dependent [29]. However, more detailed analysis shows that it is, in general, signal dependent but has more complex nature. It has a signal-independent (SI) component and a signal-dependent (SD) component, variance of which can be proportional to a signal intensity [31]. For example, a variance can be expressed as \(\delta^{2}=\delta_{\mu}^{2}\cdot\mathrm{I}_{\mathrm{ij}}^{2}+\hat{\mathrm{k}} \cdot\mathrm{I}_{\mathrm{ij}}+\delta_{\mathrm{a}}^{2}\) (Model 3), where \(\mathrm{I}_{\mathrm{ij}}\) is a true intensity value of an ij-th image pixel (\(\mathrm{i}=\overline{1,N},\ \mathrm{j}=\overline{1,M},\mathrm{N}\) and \(\mathrm{M}\) denote vertical and horizontal image sizes, respectively); \(\delta_{\mathrm{a}}^{2}\) is an estimate of multiplicative noise relative variance; \(\delta_{\mathrm{a}}^{2}\) is an estimate of additive noise variance; \(\hat{\mathrm{k}}\) is a quasi-Poisson noise parameter. Experiments in [31] show that the simple models \(\delta^{2}=\delta_{\mu}^{2}\cdot\mathrm{I}_{\mathrm{ij}}^{2}+\delta_{\mathrm{a }}^{2}\) (Model 1) and \(\delta^{2}=\hat{\mathrm{k}}\cdot\mathrm{I}_{\mathrm{ij}}+\delta_{\mathrm{a}}^{2}\) (Model 2) often fit real data better than the complex general model given above. Figure 1: The 512 \(\times\) 512 pixel fragments of different context (**a**,**b**) and scatterplot of local variance estimates (**c**) for homogeneous regions ((**d**), gray color) of the fragment in (**a**). Experiments carried out with Morita system show that an equivalent noise variance for a most general model depends on both image and noise properties and is approximately equal to  
\[\hat{\sigma}_{\text{eq}}^{2}=\ \hat{\sigma}_{\text{a}}^{2}+\ \hat{\sigma}_{\mu}^{2} \sum_{\text{i=1}}^{\text{N}}\sum_{\text{j=1}}^{\text{M}}I_{\text{ij}}^{2}/ \text{NM}+\hat{\kappa}\sum_{\text{i=1}}^{\text{N}}\sum_{\text{j=1}}^{\text{M}}I _{\text{ij}}/\text{NM} \tag{1}\]  
Figure 1c presents the scatterplot of local variance estimates on local mean estimates for the fragment in Figure 1a. These estimates have been obtained only in locally passive (quasi-homogeneous) blocks of this fragment--the corresponding map of block locations is given in Figure 1d, the curves for all three models are fitted (Figure 1c) and it is seen that Models 1 and 3 provide almost the same approximations that are considerably better than for Model 2 (it is not a problem to determine the best fit, see [31] for more details). The determined parameters for Model 1 for the fragment in Figure 1c are the following: \(\hat{\sigma}_{\mu}^{2}=0.0020,\ \hat{\sigma}_{\text{a}}^{2}=2.27\) and equivalent variance is, respectively, equal to 15.7. Similar operations give \(\hat{\sigma}_{\text{eq}}^{2}=80.7\) for the fragment in Figure 1b (\(\hat{\sigma}_{\mu}^{2}=0.0043,\ \hat{\sigma}_{\text{a}}^{2}=37.49\)). Note that the noise characteristics for Morita system dental images have been analyzed for two modes of its operation. For one mode, \(\hat{\sigma}_{\text{eq}}^{2}\) varies in the range 5-20 for images of low mean intensity (Figure 1a) and in the range 50-70 for images with a high mean intensity.