# Fractal Dimension-Based Methodology for Discriminating Original Paintings from Replicas
## 3 Material and Methods
### The DBC-RGB Algorithm
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Sliding Window FD**} & \multicolumn{6}{c}{**Multifractal ([PERSON] et al. [5])**} \\  & **Painting** & **FD** & **Std** & \(p\)**-Value** & \(p\)**-Value** & **All Channels** & \multicolumn{3}{c}{\(p\)**-Value Luminance**} \\  & & & & & & \(\mathbf{h_{m}}\) & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & \(\mathbf{h_{m}}\) & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) \\ \hline \multirow{3}{*}{1} & Original & 4.17 & 0.14 & **0.0000** & **0.00** & **0.00** & **0.02** & **0.03** & 0.14 & 0.28 \\  & Replica & 4.16 & 0.15 & **0.0000** & **0.00** & **0.00** & **0.00** & **0.00** & **0.02** & **0.03** \\  & Original & 3.94 & 0.10 & **0.0000** & **0.00** & **0.00** & **0.00** & **0.02** & **0.03** \\  & Replica & 3.96 & 0.11 & **0.0000** & **0.01** & **0.03** & 0.53 & 0.08 & 0.21 & 0.54 \\  & Original & 3.92 & 0.13 & **0.0000** & **0.01** & **0.03** & 0.53 & 0.08 & 0.21 & 0.54 \\  & Replica & 3.94 & 0.09 & 0.17 & **0.0000** & 0.58 & 0.90 & 0.21 & 0.87 & 1.00 & 0.87 \\  & Original & 4.09 & 0.18 & **0.0091** & 0.38 & 0.24 & 0.49 & 0.72 & 0.87 & 0.98 \\  & Original & 3.70 & 0.19 & **0.0091** & 0.38 & 0.24 & 0.49 & 0.72 & 0.87 & 0.98 \\  & Replica & 3.72 & 0.19 & **0.0000** & 0.39 & 0.79 & 0.37 & 0.94 & 0.94 & 0.61 \\  & Original & 3.84 & 0.12 & **0.0000** & 0.29 & 0.13 & 0.26 & 0.75 & 0.77 & 0.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between our results (Sliding window FD) and multifractal results from [PERSON] et al. for the Princeton dataset [5]. The sliding window FD algorithm was executed with a 90% window overlap. FD: average FD, Std: standard deviation. All \(p\)-values correspond to Wilcoxon rank-sum tests. Statistically significant differences are indicated with a bold \(p\)-value. Figure 6: Boxplots showing significant differences between paintings in the Cleveland dataset. **: \(p\)-value < 0.01 and **: \(p\)-value < 0.0001. For the Cleveland dataset, a direct comparison between our results and those obtained in [2] is not possible because their goal was to classify the paintings according to the artist, rather than to differentiate between paintings by the same artist. As shown in Table 5, our results for this dataset show more variability in the average FD values among paintings by the same artist. This explains why a lower percentage of window overlap in the algorithm (implying a less detailed analysis of the paintings) was sufficient to obtain significant differences (see Table 3). Given that the Cleveland dataset includes works from four artists, it was also possible to compare the FD between pairs of paintings where one painting was by one artist and the other was by a different artist. In this case, 58 pairs of paintings were compared (e.g., 1-A to 2-A, 1-A to 2-B, 1-A to 2-C, 1-B to 2-A, etc.) using the same five sliding window configurations. Table 6 presents the results showing the number of pairs of paintings and the percentage of the total pairs that exhibit significant differences in the FD. These results are also compared in Table 6 with those obtained from comparing pairs of paintings by the same artist (results extracted from Table 3). More significant differences in FD were found when comparing pairs of paintings by different artists. Additionally, high percentages of significant differences were observed with smaller window overlap percentages when the compared pairs were by different artists. Therefore, as expected, the method discriminated more effectively between paintings by different artists than between paintings by the same artist, requiring a lower window overlap to achieve significant differentiation.