## 2 Materials and Methods
### Radial Fourier-Mellin Signatures through Hilbert Transform  
To generate the radial Fourier-Mellin signatures, first the image was split into its RGB channels and gray scale (Figure 3a). Then, the module of the Fourier-Mellin (FM) transforms of each skin digital image, named \(Im(x,y)\), was obtained using the next equation [24; 25; 26] (Figure 3b). \[\left|F_{M}(s,t)\right|=\iint_{0}^{\infty}\left|FT[Im(x,y)]\right|x^{s-1}y^{t-1 }dxdy=M\{\left|FT[Im(x,y)]\right|\} \tag{1}\]  
where \(\left|F_{M}(s,t)\right|\) is the module of the Mellin transform, which give us a scale invariance of an object in the image that is necessary as the skin lesion digital images were obtained from different lesion to camera distances. Thus, the lesion region is smaller for longer lesion to camera distances and the images contain a larger lesion region when the opposite is true. Moreover, \((s,t)\) represents the 2D coordinates of the transformed \((x,y)\) pixel coordinates on [PERSON]'s plane. Notice that these pixel coordinates \((x,y)\) correspond to the module Fourier transform of the image \((|FT[Im(x,y)]|)\), taking advantage of its translation invariance. Therefore, at this moment, the object (skin lesion) in the image is invariant to translation and scale. Now, using the Hilbert transform, we also achieve the skin lesion in the image being invariant to rotation. The Hilbert transform of the image is given by [22; 26; 27; 28; 29]:  
\[\mathcal{F}\{H_{r}[Im(x,y)]\}=e^{ip\theta}FT[Im(x,y)]=e^{ip\theta}F(u,v) \tag{2}\]  
where \(p\) is the order of the radial Hilbert transform, \(\theta\) is the angle on frequency domain/space of the pixel coordinates in the image \((x,y)\) after being transformed to the Fourier plane coordinates as \((u,v)\). Therefore, this angle is determined by \(\theta=\cos\left(u/\sqrt{u^{2}+v^{2}}\right)\). Then, using the Euler's formula, we calculated the binary ring masks of the RGB channels and gray-scale skin lesion digital image, using both the real (\(H_{R}\)) and imaginary (\(H_{I}\)) parts of the radial Hilbert transform of the image as follows (Figure 2), [22; 26; 27; 28; 29]. \[H_{R}=Re[H_{r}(u,v)]=\left\{ \begin{matrix}1,\;if\;\sin=(p\theta)>0\\ 0,\;\;otherwise\end{matrix} \right. \tag{3}\]  
\[H_{I}=Im[H_{r}(u,v)]=\left\{ \begin{matrix}1,\;if\;cos(p\theta)>0\\ 0,\;\;otherwise\end{matrix} \right. \tag{4}\]  
The binary ring masks obtained above were applied to filter the skin lesion digital images that were previously processed using the module of the Fourier-Mellin transform (Figure 3c). The results require the sum of the values in the pixels of each ring obtaining two unique signatures of each skin gray-scale lesion image \((Sgray_{H_{R}}\) and \(Sgray_{H_{I}})\) and its RGB channels given by: \(SR_{H_{R}}\), \(SR_{H_{I}}\), \(SG_{H_{R}}\), \(SG_{H_{I}}\), \(SB_{H_{R}}\), and \(SB_{H_{I}}\) (Figure 3d). Finally, each signature is normalized by its maximum value (Figure 3e,g). To include texture descriptors, we used the uniform local binary pattern (LBP) technique (Figure 3f). This is a texture analysis tool in computer vision and image processing [30]. It is a simple and efficient descriptor that describes textures, edges, corners, spots, and flat regions. Taking blocks of 3 \(\times\) 3 pixels, the intensity of each of the eight neighboring pixels is compared with the intensity of the central pixel (defined as the threshold). Suppose the intensity of the adjacent pixel is greater than or equal to the intensity of the central pixel; in this case, the position of the neighboring pixel is assigned a value of 1 (or otherwise a value of 0). Once all the pixels are compared, we obtain a set of zeros and ones that represent a binary number. Each position of the binary number is multiplied by its corresponding decimal value and then all the values are added. This sum is the LBP value that labels the central pixel. Figure 4 shows an example of the LBP calculation on any pixel for P = 8 neighborhood pixels.