## 2 Materials and Methods
### Radial Fourier-Mellin Signatures Through Hilbert Transform  
To create the Radial Fourier-Mellin signatures, the image was first separated into its RGB channels and grayscale (Figure 3a). After that, the Hermite transform of order \((1,1)\) is obtained from each image component, RGB channels, and grayscale (Figure 3b). The Hermite transform was implemented using order (1,1) to balance computational efficiency and spatial detail capture. Preliminary experiments with higher orders (e.g., (2,2), (3,3)) did not yield significant improvements in classification performance but increased the feature vector dimensionality and computational cost. Therefore, order (1,1) was selected to effectively capture first-order spatial derivatives relevant to lesion boundary and texture features. Next, the magnitude of the Fourier-Mellin (FM) transform for each skin lesion digital image, denoted as \(Im(x,y)\), was computed using the following equation (Figure 3c). \[c\left|F_{M}(s,t)\right|=\] \[\iint_{0}^{\infty}\left|FT[Im(x,y)]\right|x^{(s-1)}y^{(t-1)}dxdy \tag{11}\] \[=M\{|FT[Im(x,y)]|\},\]  
Here, \(\left|F_{M}(s,t)\right|\) represents the magnitude of the Mellin transform, which provides scale invariance for objects in the image. This is essential because the skin lesion digital images were captured at varying distances between the lesion and the camera. As a result, lesions appear smaller at greater distances and larger when the distance is shorter. The coordinates \((s,t)\) represent the 2D positions of the transformed \((x,y)\) pixel coordinates on Mellin's plane. These original \((x,y)\) coordinates correspond to the magnitude of the Fourier transform of the image, \(|FT[Im(x,y)]|\), leveraging its translation invariance. Consequently, at this stage, the object (skin lesion) in the image becomes invariant to both translation and scale. By applying the Hilbert transform, rotational invariance of the skin lesion in the image is also achieved. The Hilbert transform of the image is expressed as:  
\[\mathcal{F}\{H_{r}[Im(x,y)]\}=\] \[e^{ip\theta}FT[Im(x,y)]= \tag{12}\] \[e^{ip\theta}F(u,v),\]  
Here, \(p\) represents the order of the radial Hilbert transform, and \(\theta\) is the angle in the frequency domain corresponding to the pixel coordinates \((x,y)\) after their transformation into Fourier plane coordinates \((u,v)\). This angle is calculated as \(\theta=\text{acos}\left(u/\sqrt{u^{2}+v^{2}}\right)\). Using [PERSON]'s formula, binary ring masks were generated for the RGB channels and the grayscale skin lesion digital image. These masks utilized both the real \(H_{R}\) and the imaginary \(H_{I}\) components of the radial Hilbert transform of the image (Figure 2). \[H_{R}=Re[H_{r}(u,v)]=\left\{ \begin{array}{cl}&1,\quad if\quad sin(p\theta)>0 \\ &0,\quad\quad otherwise\end{array} \right., \tag{13}\]  
\[H_{I}=Im[H_{r}(u,v)]=\left\{ \begin{array}{cl}&1,\quad if\quad cos(p\theta)> 0\\ &0,\quad\quad otherwise\end{array} \right.. \tag{14}\]  
The binary ring masks generated earlier were used to filter the skin lesion digital images that had been processed with the magnitude of the Fourier-Mellin transform (Figure 3c). The process involved summing the pixel values within each ring, resulting in two distinct signatures for each grayscale skin lesion image \(Sgray_{H_{R}}\) and \(Sgray_{H_{I}}\); andfor its RGB channels: \(SR_{H_{R}}\), \(SR_{H_{I}}\), \(SG_{H_{R}}\), \(SG_{H_{I}}\), \(SB_{H_{R}}\), and \(SB_{H_{I}}\) (Figure 3d). Finally, each signature is normalized by its maximum value (Figure 3e). To incorporate texture descriptors, we applied the uniform Local Binary Pattern (_LBP_) technique (Figure 3f), a widely used tool in computer vision and image processing for texture analysis. _LBP_ is a simple yet effective descriptor that captures textures, edges, corners, spots, and flat regions. For each \(3\times 3\) pixel block, the intensity of the eight surrounding pixels is compared to the intensity of the central pixel, which serves as the threshold.