## Materials and Methods
### On Quantifying the Attributes
#### 3.2.6 Homogeneity  
Based on discussions with the experts, the most complicated value is homogeneity. As even a healthy liver has a fine-grained inhomogeneity (which the medical experts consider as homogeneous), the classical homogeneity measures are not applicable. The radiologist experts understand the larger grained inhomogeneity or gradient within the lesion as inhomogeneity. Several quantities were measured: the mean, median, standard deviation, and the difference between the minimum and maximum density values of the healthy liver and the lesion, as well as the raster-averaged lesion (the raster-averaged lesion arose in the following way: the lesion's bounding box was divided into 10 by 10 smaller rectangles (for smaller lesions, this number was smaller) and for those small rectangles, which contained part of the lesion, the statistical parameters, mean, median, and standard deviation were calculated). As we wanted to use easily computable and system independent quantities, the following experimental formulae were introduced to represent the experts' classification. \[\text{Homogeneity}_{\text{SD}}=\frac{\text{Standard deviation of the density of the lesion}}{\text{Standard deviation of the density of the healthy liver}}, \tag{13}\]  
\[ \begin{array}{l}\text{Homogeneity}_{\text{median}}=\\ =\frac{\text{Difference between the mean and median of the density of the lesion}}{\text{Difference between the mean and median of the density of the healthy liver}},\end{array}  \tag{14}\]  
\[ \begin{array}{l}\text{Homogeneity}_{\text{ raster mean diff}}=\\ =\frac{\text{Difference of the minimum and maximum of the raster mean values}}{\text{Mean value of the density of the nodule}},\end{array}  \tag{15}\]  
\[ \begin{array}{l}\text{Homogeneity}_{\text{ raster median diff}}=\\ =\frac{\text{Difference of the minimum and maximum of the raster median values}}{\text{Median value of the density of the nodule}},\end{array}  \tag{16}\]  
\[ \begin{array}{l}\text{Homogeneity}_{\text{ raster mean SD}}=\frac{\text{Standard deviation of the raster mean values}}{\text{Mean value of the density of the nodule}},\end{array}  \tag{17}\]  
\[ \begin{array}{l}\text{Homogeneity}_{\text{ raster median SD}}=\frac{\text{Standard deviation of the raster median values}}{\text{Median value of the density of the nodule}},\end{array}  \tag{18}\]  
\[ \begin{array}{l}\text{Homogeneity}_{\text{ raster SD diff}}=\frac{\text{Median of the standard deviation of the raster mean values}}{\text{Standard deviation of the density of the nodule}}.\end{array}  \tag{19}\]  
According to Figure 8, however, the quantities combined from these parameters fail to characterize the homogeneity, so a deeper study dedicated to this question is required, though the standard deviation seems to be somewhat distinguishable for the four expert categories. It seems that the rastering, though it seems plausible, does not solve the problem of the distinguishing of the fine grained and coarse-grained homogeneity. Other methods, such as edge detection and edge counting might result in a better agreement with the experts' opinion. During the discussions with the medical experts, however, a very interesting conclusion could be drawn. Even though they used the expression homogeneity and inhomogeneity, they meant different things by inhomogeneity. There are subcategories, such as spotted or puddle-like inhomogeneity, and ring-like inhomogeneity, and these categories have different roles in the diagnostic process, so these subcategories should also be represented for a more accurate decision support system. Figure 8: Expert opinion vs. computer calculated homogeneity values (10) to (16). The expert categories are represented as different colors, according to the legend. The upper part of the images shows the histograms of the distribution of the machine calculated quantities for the different experts’ categories, the lower part of the images shows their mean value by a marker \(\star\), and their standard deviation as the interval around the mean value.