## 2 Theoretical Background
### Classical and Fuzzy Hough Transforms
As we are interested not only in decreasing the number of edge points scanned by the Hough transform, but also in increasing the efficiency of finding the colorectal polyp, we relied on the definition of the previous two evaluation metrics \(R\_calc\) and \(R\_edge\) as a basic for selecting the most appropriate edge detection technique as following. * **Selection of the most appropriate edge detection technique using _R_calc_:** We tested two different selection strategies using metric _R_calc_. Based on the definition of _R_calc_, the nearest the value _R_calc_ to 1 is, the better the filter is. According to the 1 st strategy, we can select the filter, that has the mean of the metric _R_calc_ values closest to 1. For each database and each type of the four edge detection methods we calculated the mean of the _R_calc_ values. However, for the values of metric _R_calc_, in all databases, for all the four edge detection techniques, there was many samples between [0, 0.1], as it is visible in Figure 8. This is the reason why another strategy has to be considered as well. According to the 2 nd strategy, we can select the filter, which has the most samples close to the ideal value. For this purpose, a goodness interval can be defined, and the number of samples within that interval can be calculated. In

Figure 1.0 1 st row: (**a**) The Canny edge detected version of Figure 1.1 image, and (**b**) the Canny edge masked gradient magnitude image. The 2 nd, 3 rd, and 4 th rows are exactly like the 1 st row, but for [PERSON] (**c**,**d**), [PERSON] (**e**,**f**), and Sobel (**g**,**h**) edge detection methods consecutively. our case, for every edge detection technique, we checked how many samples in each database had a \(R\_calc\) value greater than 0.1 as a measure of the filter suitability. Accordingly, the higher the number of resulting samples, the better the filter is. Of course, the percentage of this number within the total number of images in each database has to be considered; database CVC-Clinic [9] has 612 images, database CVC-Colon [42] has 379 images, and database ETIS-Larib [34] has 196 images). Table 3 lists the total results of this step. 2. **Selection of the most appropriate edge detection technique using \(R\_edge\)**: We also tested two different choosing strategies based on metric \(R\_edge\). According to the 1 st strategy, similar to what we executed for \(R\_calc\), we can test how similar the results to their ideal value are. However, instead of calculating the mean value, we calculated the mean absolute error (MAE) value for metric \(R\_edge\) from its ideal value 1. For this criterion, the smallest MAE value nominates the better filter. According to the 2 nd strategy, as metric \(R\_edge\) should be as close to 1 as possible, we suggested finding the goodness interval around \(R\_edge=1\), i.e., considering each sample that has \(R\_edge\) value within [0.5, 1.5] among the good samples. Consequently, the higher the number of samples within the goodness interval, the better the filter is. Table 2 arranges the total results of this step. Figure 1.2 The total number of edge pixels resulting from Canny, Prewitt, Roberts, and Sobel techniques for the three databases. Figure 1.3 Metric \(R\_calc\) values resulting from [PERSON], Prewitt, Roberts, and Sobel techniques for the three databases. Figure 1.4 Metric \(R\_edge\) values resulting from [PERSON], Prewitt, Roberts, and Sobel techniques for the three databases. The highlighted results presented in the above two tables show that the Prewitt filter is the most appropriate edge detection technique based on our proposed selection strategies using both metrics _R_calc_ and _R_edge_ in most cases. It should be noted from Table 3 (2 nd strategy) that the results of Sobel filter are very close to the Prewitt ones, especially for database CVC-Clinic: they are the same. Moreover, for database CVC-Colon, the number of samples that had an _R_calc_ value greater than 0.1 is three samples more in the Sobel filter than in the Prewitt case.