## 2 Methodology
### Handcrafted Fractal Features
#### 2.1.3 Multidimensional and Multiscale Higuchi Fractal Dimension  
The fractal dimension, proposed by [PERSON] (\(DF_{H}\)) in 1988 [71], enables the analysis of time series in the 1D domain. The method proved relevant results for biological signal analysis [27; 72; 73]. The approach explored here considered the approach presented in [28], which expanded the analysis process by including a multidimensional and multiscale strategy. The applied model considered sets of 1D series from an image \(I\), with dimensions \(h\) x \(w\). The multiscale step was defined using different observation scales. The procedure involved sliding vectors to analyze each pixel series. The initial position of the sliding vector was determined from the first position of the pixel series under analysis, and it was incremented by one position to the right until its last position coincided with the last position in the series. The vector lengths were represented as \(l\), such that \(3\leq l\leq min(h,w)\), given the need for each sliding vector to have a central pixel. In this process, as previously indicated, each pixel of the series under analysis was represented as a 5D vector \((x,y,r,g,b)\)[28; 41], allowing an analysis between the pixels involved in each transition of the sliding vector by the series. This procedure characterized the multidimensional approach, which is a useful strategy to quantify the patterns in color images. In each iteration in the multidimensional approach, the central pixel of the sliding vector was compared to every other pixel inside it based on the Manhattan distance (Equation (3)). If \(\Delta\) was less than or equal to the current sliding vector length, the central pixel composed an auxiliary subseries referring to the observation scale. The Higuchi fractal dimension (\(DF_{H}\)) was defined through each finite auxiliary subseries extracted from the input series of an image \(I\). In this approach, a finite series of discrete points can be defined as \(X=x(1),x(2),x(3),\ldots,x(N_{DH})\), such that \(x\) indicates an element of the image \(I\) and \(N_{DH}\) indicates the maximum number of points available in the series under analysis. Thus, the method generated \(d\) new series \(X_{dj}\), considering that the starting point occurred from \(j\), and \(d\) is a granularity factor:  
\[X_{dj}:x(j),x(j+d),x(j+2d),\ldots,x\left(j+\lfloor\frac{N_{DH}-j}{d}\rfloor \cdot d\right), \tag{7}\]  
where \(j=1,2,\ldots,d\). \begin{table}
\begin{tabular}{c c c c c} \hline  & **3** & **5** & **...** & \(L_{max}\) \\ \hline
1 & \(P(1,3)\) & \(P(1,5)\) &... & \(P(1,L_{max})\) \\
2 & \(P(2,3)\) & \(P(2,5)\) &... & \(P(2,L_{max})\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\ddots\) & \(\vdots\) \\ \(L^{2}\) & \(P(L^{2},3)\) & \(P(L^{2},5)\) & \(\vdots\) & \(P(L^{2},L_{max})\) \\ \hline \end{tabular}
\end{table} 
Table 1: Illustration of a probability matrix \(P\). For each \(X_{dj}\), the curve length, \(Len_{j}(d)\), was calculated such that:  
\[Len_{j}(d)=\frac{1}{d}\Bigg{\{}\Bigg{(}\sum_{i=1}^{\frac{N_{DH}-j}{d}}\lvert x(j+ id)-x(j+(i-1)\cdot d)\rvert\Bigg{)}\frac{N_{DH}-1}{\lfloor\frac{N_{DH}-j}{d} \rfloor\cdot d}\Bigg{\}}. \tag{8}\]  
Thus, \(Len\) was defined as the length of the curve for the interval \(d\), which was calculated as the average value across the \(d\) sets of \(Len_{j}(d)\):  
\[Len(d)=\frac{1}{d}\sum_{j=1}^{d}Len_{j}(d). \tag{9}\]  
The \(DF_{H}\) value for a subseries was given by the angular coefficient obtained from the linear regression of a \(\ln Len(d)\times\ln d\) plot, which was obtained through linear least squares fitting. The \(t\) auxiliary pixel subseries from each initial input set was employed to calculate the descriptor \(DF_{H}\)[28]. Consequently, there are \(t\) values of \(DF_{H}\). The fractal dimension value of the input series, \(DF_{H\text{{Serrie}}}\), was defined as the average of these \(t\) values:  
\[DF_{H\text{{Serrie}}}=\frac{1}{t}\sum_{i=1}^{t}DF_{H_{i}}, \tag{10}\]  
where \(t\) indicates the number of values that \(l\) can take.