# 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{6}{c}{**Sequential Processing of Paintings**} \\ \hline \multirow{2}{*}{**Dataset**} & \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**} \\  & **T (s)** & **MP/s** & **T (s)** & **MP/s** & **T (s)** & **MP/s** & **T (s)** & **MP/s** & **T (s)** & **MP/s** \\ \hline Princeton & 2.32 & 172.35 & 2.96 & 237.73 & 7.42 & 206.97 & 21.22 & 286.28 & 124.26 & 294.47 \\ Cleveland & 0.59 & 131.52 & 0.95 & 148.38 & 1.85 & 159.33 & 7.11 & 161.71 & 44.42 & 167.47 \\ \hline All Paintings & 2.90 & 164.11 & 3.92 & 215.98 & 9.26 & 197.48 & 28.33 & 255.02 & 168.69 & 261.02 \\ \hline \multicolumn{10}{c}{**Parallel Processing of Paintings**} \\ \hline \multirow{2}{*}{**Dataset**} & \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**} \\  & **T (s)** & **MP/s** & **T (s)** & **MP/s** & **T (s)** & **MP/s** & **T (s)** & **MP/s** \\ \hline Princeton & 0.84 & 473.35 & 1.10 & 677.50 & 1.78 & 861.94 & 6.60 & 920.24 & 38.34 & 954.51 \\ Cleveland & 0.41 & 188.90 & 0.72 & 197.71 & 0.64 & 463.19 & 2.51 & 458.07 & 16.47 & 451.63 \\ \hline All Paintings & 1.12 & 425.14 & 1.25 & 641.17 & 2.50 & 732.28 & 9.57 & 754.99 & 54.21 & 812.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Execution times (T, in seconds) and megapixels per second (MP/s) for dataset processing with window overlap (wo) configurations. The difference in megapixels processed per second (MP/s in Table 8) between the two datasets is due to the fact that the Princeton dataset used a patch size of 512 \(\times\) 512, while the Cleveland dataset used a patch size of 256 \(\times\) 256. This means that each patch in the Princeton dataset processed four times as many pixels as in the Cleveland dataset. Therefore, the selected patch size affects the execution time; however, in our study, the patch sizes were chosen based on the characteristics of the images, following the same criteria as in [2,5]. ## 5 Discussion and Conclusions

In this study, we introduced a novel, color-FD-based method for differentiating original paintings from replicas. By using a sliding window approach and recent color-FD computation techniques [37], our method effectively analyzed two public datasets, identifying significant differences between original and replica paintings. These findings suggest that combining the FD of color images with a sliding window approach is a promising tool for forgery detection. Previous multifractal analysis of the Princeton dataset [5] revealed significant differences in only three of seven original-replica painting comparisons, whereas our method, employing the same statistical test, detected significant differences in all seven. This enhanced performance required a more intensive image analysis, using a 90% window overlap, in contrast to the expert-driven manual patch selection employed by the multifractal approach in [5]. The analysis of the Princeton dataset revealed that most replica paintings had higher average FD values compared to their original counterparts (see Table 4 and Figure 5). This finding aligns with the increased multifractal parameter values reported in [5] for cases where significant differences between originals and replicas were detected. In the case of original paintings, the artist's hand movements, brush strokes, and color application may be more spontaneous and natural, leading to a lower FD value. Replicas, on the other hand, might be created with a more deliberate and controlled approach, as the artist attempted to replicate the original work. This more methodical process could result in a higher FD value due to the presence of more intricate and detailed patterns. In contrast, in the Cleveland dataset, we did not observe a similar pattern in the average FD values.