## 2 Theoretical Background
### Classical and Fuzzy Hough Transforms
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66 When selecting the final circles, i.e., the local maxima in the transformed image, a threshold \(P_{p}\) is to be set within the Hough transform. The arising circles (their number and location) were studied for different local maximum thresholds, namely for \(P_{p}=50\%\), \(60\%\), \(70\%\), \(80\%\), and \(90\%\) of the global maximum of the votes. The goal of this step is to determine, how many of the final circles are around the polyp for the various \(\sigma\)s and local maximum thresholds \(P_{p}\)s. For this purpose, the total number of circles \(N_{total}\) was counted. A circle was considered to be around the polyp by testing, if its center was inside the ground truth mask, and it had points within the ring mask around the polyp contour. The number of such circles within the ring mask \(N_{ring}\) was also counted. The ratio  
Figure 2.1  
was used as a metric for effectiveness of finding circles related to the polyp. If \(A_{r}\) is 0, then the polyp is not found, if \(A_{r}\) is too small, then too many other circles are found, and if \(A_{r}\) is around 1, then the circle(s) around the polyp are found, but not many other circles can be seen in the final results. To test the different types of Hough transforms' tolerance degree to the deviation from the circle, it is necessary to know the size of the polyps as well as their roundness. To calculate the average radius of the polyp \(r_{avg}\), the maximum and minimum coordinates of the polyp mask in \(x\) and \(y\) direction were used as follows,  
Figure 2.2  
The center of the polyp was determined as the average of these coordinates,  
Figure 2.3  
This center point and the average radius were used to determine the radial displacement for each mask contour point \(\left(m_{x},m_{y}\right)\) by  
Figure 2.4  
The roundness error can be measured by the maximum of the ratio of the radial displacements and the average radius:  
Figure 2.5  
The larger the roundness error \(\delta_{roundness}\), the more the shape deviates from circle. Please note, the roundness error of a circle with diameter of 30 pixels can still be as large as \(\delta_{roundness,30}=0.04\), which decreases for a circle twice as large (i.e., of diameter 60 pixels) to \(\delta_{roundness,60}=0.03\).