## 2 Theoretical Background
### Classical and Fuzzy Hough Transforms
The flowchart in Figure 9 summarizes the overall edge detection method selection procedure. ### Gradient-Based Thresholding for Prewitt Edge Detection Results

In this part of the proposed method, the dynamic distribution of the normalized gradient-weighted edges pixels resulting from the Prewitt method of both the ring mask (i.e., the area surrounding the ground truth mask contour) and the full image was studied by generating the individual histograms of all images in all databases. Four individual histograms of four different samples can be found in Figure 1.5 Please note that as in most of the cases the gradient intensities were in the first 20%, the density of the histogram bins in that domain were made to be larger. This dividing of the normalized gradient range resulted in a more detailed tendency view in the domain with dense bin distribution. Additionally, as the column height in the dense bin distribution domain became relatively smaller, thus, the visibility of the small column magnitudes in the other parts of the histogram (i.e., in the higher normalized gradient domain) became better. In most of the cases, the distributions of the full image edges (cyan) and the ring mask edges (yellow) showed a very similar tendency with different magnitude, like in the case of the first example, subplot (a) of Figure 1.6 Very often beside the similar tendency in the low gradient domain, the higher gradient parts were missing in the case of the ring mask edges, like in the 2 nd and 3 rd example (subplots (b) and (c)). In some other cases, the distributions of the full picture and the ring mask had different tendencies, like in the case of the 4 th example, in subplot (d). To summarize the results of the individual histograms, 3D-histograms were created for the three databases with the picture number being the 3 rd dimension. These total histograms are plotted in Figures 11 and 12. Figure 11 shows the perspective view in linear

\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & \\ \hline \multirow{3}{*}{1 st strategy: _R_edge_ MAE} & 0.459919 & **0.447193** & 0.579451 & 0.44937 & CVC-Clinic \\ \cline{2-6}  & 0.779827 & **0.582246** & 0.683746 & 0.588328 & CVC-Colon \\ \cline{2-6}  & 2.609909 & **0.44718** & 0.536933 & 0.451125 & ETIS-Larib \\ \hline \multirow{3}{*}{2 nd strategy: Num. of samples with \(0.5<R\scale, while Figure 12 gives the top view of the same histograms in logarithmic scale. It is worth mentioning that the linear and logarithmic scale plots of the same histogram are both given, because the smaller valued histogram parts at the lowest and the highest normalized gradient bins cannot be observed well in the linear scale plots of the total histograms. On the other hand, the top view of the logarithmic scale plots makes the entire set of data viewable without columns blocking the ones behind them. Figure 1.7 Histograms of the normalized gradient-weighted edge pixels resulting from Prewitt method for four different samples (examples from Database CVC-Clinic (**a**,**c**), CVC-Colon (**d**), and ETIS-Larib (**b**), are selected so, that typical behaviors would be visible). The interval [0, 0.2] has narrower bins, as most of the values in most of the samples were concentrated there, and we needed higher resolution results in this domain. (The yellow histograms for the ring masks are semitransparent, plotted in front of the teal columns for the full images.) Figure 1.8 Flowchart of the edge detection method selection procedure. Based on results in Figures 11 and 12, it is visible that the edges with very high gradient values do not belong to the polyp edge, i.e., most of the ring mask edge results have 0 values over 0.3. Additionally, they have zero values below 0.06. Moreover, in many cases this lower limit can be brought up to 0.08, and the upper limit can be as low as 0.2.