## 4 Results and Discussion
### Roundness Metrics Evaluation  
In order to test how tolerant the classical and fuzzy Hough transforms for the deviation from the circle, we used a test image of five ellipses in an image of size 200 \(\times\) 200 pixels. The larger axis of the ellipse was the same for each of the objects in one image, and the ratio of the two axes were 1:1, 5:6, 4:6, 3:6, and 2:6. Three images were used, one with larger axis of 30 pixels, one with 60 pixels and one with 90 pixels. The properties of the ellipses are given in Table 5. In Figure 19, the number of circles that were found by the Hough transforms are summarized for each of the ellipses separately. As the centers of the ellipses were rather close to each other especially on the largest axis case, for lower thresholds one or more unphysical \"ghost\" circles appeared, typically at the regions between 2 or 3, almost overlapping ellipses. The number of these ghost circles are also given under the numbers belonging to the real, physical ellipses. It can be seen from Figure 19 that of the artificial image with objects of the smallest size, the classical Hough transform could only detect the precise circle, even the roundness error \(\delta_{roundness}=0.135\) was not tolerable. As the fuzziness parameter increased, the peaks belonging to more and more elongated ellipses grew above the half of the global maximum of the votes, i.e., above \(P_{p}=50\%\). Already for \(\sigma=5\) all the ellipses were found in the \(P_{p}=50\%\) case. The corresponding radii were smaller than the larger axis of the ellipse. As the size of the ellipses increased, the limit for finding all the ellipses went higher. Additionally, due to the closeness of the ellipses compared to the full width at half maximum of the voting matrix, overlaps arose between the individual ellipses and thus, false peaks appeared in the accumulator space, either in the area at the center of the image, which \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Artificial Image** & **Axes (Pixels)** & \(\mathbf{r_{\mathrm{avg}}}\) & \(\mathbf{\Delta_{radial}}\) & \(\delta_{roundness}\) \\ \hline \multirow{4}{*}{\(\mathbf{\Theta}\)} & 30, 30 & 15.5 & 0.6508 & 0.042 \\  & 30, 25 & 14 & 1.8902 & 0.135 \\  & 30, 20 & 13 & 2.8758 & 0.221 \\  & 30, 15 & 11.75 & 4.7322 & 0.403 \\  & 30, 10 & 10.5 & 5.0081 & 0.477 \\ \hline \multirow{4}{*}{\(\mathbf{\Theta}\)} & 60, 60 & 30.5 & 0.7931 & 0.026 \\  & 60, 50 & 28 & 2.8902 & 0.103 \\  & 60, 40 & 25.5 & 5.2002 & 0.204 \\  & 60, 30 & 23 & 7.6023 & 0.331 \\  & 60, 20 & 20.5 & 10.0369 & 0.490 \\ \hline \multirow{4}{*}{\(\mathbf{\Theta}\)} & 90, 90 & 45.5 & 1.6081 & 0.035 \\  & 90, 75 & 41.5 & 4.2220 & 0.102 \\  & 90, 60 & 38 & 8.4958 & 0.224 \\  & 90, 45 & 34.25 & 12.1989 & 0.356 \\  & 90, 30 & 30.5 & 15.0686 & 0.494 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Properties of the ellipses of the artificial test images of size 200 \(\times\) 200 pixels. The order of the ellipses is the same as the order in the vertical axis of Figure 19, except for the ghost circles. is circumvented by the ellipses, or at the two most elongated ellipses, that sometimes appeared as two twin peaks at the vote maps. It can be seen that for the larger sized ellipse, even the classical Hough transform has a small roundness error tolerance: \(\delta_{roundness}=0.1\) is still found, although as two circles. If the roundness of the tested colonoscopy images is studied in the last two columns of Table 4, we can see that very much elongated polyps also exist with \(\delta_{roundness}>0.9\) roundness error, but the typical roundness error is between 0.15 and 0.5. This is the reason we selected this domain for the ellipses to be studied in our artificial examples. Similar to the case of the artificially generated test images, the polyps with very much elongated shapes \(\delta_{roundness}>0.4\), the polyps were very often not found if their sizes were large, or found only for very low threshold \(P_{p}\), like in the cases of ETIS 1 st, 3 rd and 4 th images, or CVC-Clinic 4 th.