# 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 \multirow{2}{*}{**Comparison**} & \multirow{2}{*}{**\#Pairs**} & \multicolumn{2}{c}{**0\% wo**} & \multicolumn{2}{c}{**25\% wo**} & \multicolumn{2}{c}{**50\% wo**} & \multicolumn{2}{c}{**75\% wo**} & \multicolumn{2}{c}{**90\% wo**} \\  & & **\#SD** & **\% SD** & **\% SD** & **\% SD** & **\% SD** & **\% SD** & **\#SD** & **\% SD** & **\#SD** & **\% SD** \\ \hline Same artist & 12 & 5 & 41.7\% & 5 & 41.7\% & 7 & 58.3\% & 9 & 75.0\% & 11 & 91.7\% \\ Diff. artists & 58 & 37 & 63.8\% & 40 & 69.0\% & 45 & 77.6\% & 52 & 89.7\% & 56 & 96.6\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Significant differences in FD between pairs of paintings in the Cleveland dataset when comparing pairs by the same artist and pairs painted by different artists for each sliding window configuration. #Pairs: number of pairs compared, #SD: number of pairs showing significant differences in FD, % SD: percentage of pairs with significant differences, wo: window overlap. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Artist** & **Painting** & **FD** & **Std** & \(p\)**-Value** \\ \hline \multirow{4}{*}{1} & A & 3.40 & 0.15 & **0.000**1  \\  & B & 3.44 & 0.16 & **0.0000**2  \\  & C & 3.41 & 0.16 & **0.0030**3  \\ \hline \multirow{4}{*}{2} & A & 3.38 & 0.17 & **0.0000**1  \\  & B & 3.37 & 0.18 & **0.0000**2  \\  & C & 3.38 & 0.16 & 0.2079  \\ \hline \multirow{4}{*}{3} & A & 3.57 & 0.12 & **0.0000**1  \\  & B & 3.66 & 0.16 & **0.0000**2  \\  & C & 3.54 & 0.13 & **0.0000**3  \\ \hline \multirow{4}{*}{4} & A & 3.38 & 0.14 & **0.0000**1  \\  & B & 3.32 & 0.13 & **0.0000**2  \\  & C & 3.27 & 0.17 & **0.0000**3  \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Detailed results of the window sliding FD algorithm with a 90% window overlap for the Cleveland dataset. FD: average FD, Std: standard deviation, \(p\)-value: \(p\)-value for the Wilcoxon rank-sum test. 1: \(p\)-value for the A-B comparison, 2: \(p\)-value for the B-C comparison, 3: \(p\)-value for the C-A comparison. Statistically significant differences are indicated with a bold \(p\)-value. ### Results on Comparing Artists

We also compared the FD distributions between the four artists in the Cleveland dataset. In this case, the values compared between artists correspond to the pooled set of FD values from all three of their respective paintings. This comparison is not about differentiating originals from replicas, but rather about distinguishing between artists based on their styles or \"artist hand\". Table 7 shows the statistical comparisons of FD values between the four artists in the Cleveland dataset for each tested window overlap percentage in the sliding window configuration used to process each painting. These results show that all comparisons found significant differences between the artists. This indicates that our proposed FD methodology can distinguish not only between original paintings and replicas by the same or different artists but also between sets of similar paintings created under identical conditions by different artists. Figure 7 shows boxplots comparing the FD values between the artists in the Cleveland dataset using the sliding window configuration with a 0% window overlap. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Compared Artists**} & \multicolumn{5}{c}{_p-Value for Wilcoxon Test_} \\  & **0\% wo** & **25\% wo** & **50\% wo** & **75\% wo** & **90\% wo** \\ \hline Artist 1 vs. Artist 2 & 0.0460 & 0.0060 & 0.0000 & 0.0000 & 0.0000 \\ Artist 1 vs. Artist 3 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Artist 1 vs. Artist 4 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Artist 2 vs. Artist 3 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Artist 2 vs. Artist 4 & 0.0010 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Artist 3 vs. Artist 4 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: \(p\)-values for the Wilcoxon test comparing the FD values between the artists in the Cleveland dataset.