## 4 Results and Discussion
### Number of Circles Found by the Algorithm
The next, middle segment corresponds to the gradient-weighted voting with wider threshold, and the lowermost segment belongs to the gradient-weighted voting with a thinner threshold. Each line in each segment belongs to one image, in the order of Table 4 (except for the first two lines in each database, i.e., the \"bad\" images). The plots have segments along the horizontal axes, too: the upper subplot has the results grouped by the peak percentage \(P_{p}\) (i.e., threshold for the local maxima of the voting map \((a,b,r)\) to be considered as the circles), and each segment lists the \(\sigma\)s from classical (i.e., \(\sigma=0\)) till the widest fuzziness parameter \(\sigma=15\). The lower subplot orders the values the opposite way: the segments belong to various fuzziness parameters \(\sigma\), while the columns within the segments belong to the peak percentages \(P_{p}=0.5,\ 0.6,\ 0.7,\ 0.8,\ 0.9\). The colored squares mean the number of circles found in the ring mask versus the total number of circles for a given image, for a given \(\big{(}P_{p},\sigma\big{)}\) pair. The darker red the little square is, the fewer circles outside of the polyp contour domain are found. If the little square is bright green, then no circle is found in the polyp contour's ring mask. From Figure 13 we can conclude that for database CVC-Clinic the fuzzy Hough transforms can mostly find circles belonging to polyp contours (most of the little squares in the maps are red). The classical transform's results are either very good (dark red) or very bad (green). Mostly, increasing the threshold improves the results, the circles will be more in the area of the polyp contour, but the 90% threshold might be too much. It loses the circles in the ring mask for some cases. Thus, the most ideal value for \(P_{p}\) is 70-80% of the global maximum of the votes. The fuzziness has a kind of optimal value around \(\sigma=5\) (there the images have the largest number of dark red points and darkest red points), but this optimum is not very sharp, the neighboring \(\sigma\) values have very similar results in the domain of \(\sigma=\{3,5,7\}\). Even for those images that have no circle found in the ring mask for the classical case, the results can be improved by the fuzzy version of the transform. Regarding the gradient-weighted voting, and the gradient-based thresholding, the results improve compared to the original voting scheme, but the difference is not extremely large. There is one sample image where the circles found are not around the polyp contour for the classical transform: image No. 475. However, it can be seen that this particular polyp has barely visible circular contour segments. The results improve even for this case for the gradient-weighted voting processes (except for the classical Hough transform in the thin threshold case). Here, we have to note that in order to have the transform calculable and also to remove the very large and very small circles from the results, we limited our search for the \(r\) domain between 1% and 25% of the longer image dimension, so the longer edges of the very elongated polyps, like the one on image No. 475 could not be detected. From Figure 15, for database CVC-Colon, it can be seen that more circles are found outside of the polyp contour ring mask, i.e., the little squares in the plots tend towards the yellow and greenish part of the color palette. The value of the ideal sigma seems to be a little bit higher, around \(\sigma=7\) or \(\sigma=9\), but there is much more variation than in the case of the previous database. For some images the higher \(\sigma\) domains give better results, for others the lower \(\sigma\) domains. The thin threshold loses the circles in the ring mask for more images, so it is not a good option for this database. For the peak percentage \(P_{p}\) this database behaves similarly to the previous one. Again, there is an image that does not show circles in the ring mask for most of the three cases: image No. 230, which has a brightly lighted front part and the polyp is at the background.