## 2 Theoretical Background
### Classical and Fuzzy Hough Transforms
\tag{10}\]

Figure 2: The Hough transformed images of the edge detected picture in Figure 1c. The center coordinates \(a\) and \(b\) are shown in one plane for 4 different values of the radius \(r\). The number of the votes are indicated by colors. It can be seen in the colorbar beside each image. The rows mean the different radii, the 1 st row is \(r=42\), the 2 nd row is \(r=57\) (the radius of the final circle around the polyp), the 3 rd row is \(r=72\), and the 4 th, last row is \(r=115\) (the radius of the other final circle). The columns mean the following: the 1 st column: the classical Hough transform result, the 2 nd column: \(\sigma=5\) fuzzy Hough transform, and the 3 rd column: \(\sigma=15\) fuzzy Hough transform. \[Sobel_{x}= \begin{bmatrix}1&0&-1\\ 2&0&-2\\ 1&0&-1\end{bmatrix} ,\,Sobely= \begin{bmatrix}1&2&1\\ 0&0&0\\ -1&-2&-1\end{bmatrix} . \tag{11}\]

### The Implemented Edge Detection Algorithms

Edge detection methods can be used as mathematical techniques to identify particular locations in an image where the gray level intensities show discontinuities. The resulting edge maps serve as the basis for subsequent processing steps in numerous significant computer vision applications. In this section of the paper, a brief description of the four used edge detection methods is provided. #### 2.3.1 Roberts, Prewitt, and Sobel Edge Detection Algorithms

The well-known Roberts, Prewitt, and Sobel edge detection algorithms are widely used because of their simplicity and easiness of implementation. All of these algorithms have the same work mechanism, but with different kernels. Each kernel has the effect of calculating the gradient in the specified direction. However, the choice of algo

Figure 3: The resulting final circles from the inverse Hough transform of the picture in Figure 1. Similar to Figure 2., the columns indicate the fuzziness of the transform, i.e., the 1 st column: classical Hough transform, the 2 nd and 3 rd columns: fuzzy Hough transform for \(\sigma=5\) and \(15\), respectively. The rows in this case give the local maximum threshold compared to the global maximum. The 1 st row: 50%, the 2 nd row: 70%, and the 3 rd row: 90% of the global maximum. rithm to be used depends on the desired application and the characteristics of the image being processed. [PERSON] edge detection method uses convolutional filters to detect the variations in the image gray-level intensity in the diagonal directions [44], whereas [PERSON] and [PERSON] methods use convolutional matrices to detect the changes in both \(x\) and \(y\) directions [45; 46]. The previously mentioned kernels, (9), (10), and (11), are used by the [PERSON], [PERSON], and [PERSON] edge detection algorithms, consecutively. #### 2.3.2 Canny Edge Detection Algorithm

[PERSON] first presented Canny edge detection in 1986 [47] as a multistep algorithm. Canny algorithm is looking for the connectivity of the edge points as well as the high gradient image points which makes it the most popular edge detection technique in many computer vision and image processing applications. This technique produces very reliable and highly accurate edge maps that are close to the human perception of edges. The process of Canny algorithm consists of four main steps. First, the original image is refined using a Gaussian filter to remove unwanted noise. The applied Gaussian filter is defined as follows:

\[g(x,y)=G_{\sigma}(x,y)*f(x,y), \tag{12}\]

where

\[G_{\sigma}(x,y)=\frac{1}{2\pi\sigma^{2}}\mathrm{exp}\left(-\frac{x^{2}+y^{2}} {2\sigma^{2}}\right) \tag{13}\]

The convolution operator is represented by the symbol \(*\), and the indices \(x\) and \(y\) are used to identify a pixel's location within an image. The two-dimensional function \(G_{\sigma}(x,y)\) is a Gaussian function with the variance of \(\sigma^{2}\). The smoothened image's gradient magnitude and direction are then calculated using a certain gradient operator, i.e., Roberts, Prewitt, or Sobel.