## 5 Conclusions  
A novel voting method was suggested for circular Hough transform analysis of colonoscopy images. As the colorectal polyp contours have neither too large nor very small gradients, a gradient-based thresholding was also introduced. Based on statistical data, thresholds of \([0.06,0.3]\) and \([0.08,0.2]\) of the normalized gradient domain were suggested to improve the ratio of the circles in the surroundings of  
Figure 20: The ratio of the classical and fuzzy Hough transformations’ total running time for the wide and thin gradient thresholds compared to the original, full Hough transform times. The two subplots have different horizontal axes: the first subplot just lists the images according to their order in Table 4, the second subplot’s horizontal axis is the number of pixels to be transformed in the original image (i.e., the 5 th column of Table 4). the polyp among all the circles that are found by the Hough transform. The first, wider threshold was proven to give more reliable results. Using this method, the running time of the algorithms decreased to less than 50% compared to the full Hough transforms. The thin threshold removes so many valuable edge points from the polyp contour edges for some of the images that the circles there cannot be found by Hough transforms, so even though the threshold \([0.08,0.2]\) further improves the running time, it is safer to keep the wider threshold \([0.06,0.3]\). The main strength of the gradient-weighted voting mechanism is that it improves the ratio of the circles near the polyp among all the circles for the Hough transforms, especially for the fuzzy Hough transform. The ideal threshold for the local maxima of the voting map \((a,b,r)\) to be considered as circles is 70-80% of the global maximum for colonoscopy images. The fuzzy Hough transforms perform better in finding circles in the polyp contour ring mask than the classical transform, but the best performing fuzziness parameter varies with the image size. For the smaller images (CVC-Clinic has image size of \(384\times 288\)) the small \(\sigma=3\), 5, or 7 voting matrices are the most ideal, for the medium image size (CVC-Colon has image size of \(574\times 500\)) the \(\sigma=7\) or 9 is better, while for the largest images (ETIS-Larib, of size \(1225\times 966\)) the larger, \(\sigma\) values are the most fitting for the purpose of finding polyp contours in colonoscopy images. However, this sensitivity for the values of \(\sigma\) becomes less and less expressed for the larger images. The detection success also depends on the size of the polyp and its roundness. To measure the tolerance for shape deviation of the classical and fuzzy Hough transform, the roundness error was introduced together with the average radius of the polyp. For objects larger in size, smaller tolerance is allowed in roundness, while for smaller objects, the tolerance is larger, which is rather advantageous, as in colonoscopy, mostly the smaller lesions are the ones that might be missed by humans. The main strength of applying fuzzy Hough transform and gradient-weighted voting together is that fuzzy Hough transform has higher tolerance for shape deviations. Although fuzzy voting needs more calculations than the classical one, the introducing of the gradient threshold and the gradient-weighted voting and thus, decimating the edge points to be transformed helps to keep the calculation time lower. The weaknesses of our methods include that it is hard to determine a universal gradient threshold that is valid for all databases and image sizes. Additionally, the non-conventional angles, when the polyps are in front of dark background, need to be treated separately, though these polyps are usually hard to miss for a human expert. In addition, many polyps with extremely large, irregular shapes are not found by Hough transforms. As computer aided diagnosis is meant to aid the human expert and not to substitute them, finding the smaller polyps that are easy to miss with naked eye is the most important task in colorectal polyp localization.