# 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} \hline \hline
**Dataset** & **Painting** & **Size (Pixels)** &
\begin{tabular}{c} **Window Size** \\ **(Pixels)** \\ \end{tabular} & \multicolumn{4}{c}{**Number of Patches**} \\  & & & **(Pixels)** & **0\% wo** & **25\% wo** & **50\% wo** & **75\% wo** & **90\% wo** \\ \hline \multirow{5}{*}{Princeton} & 1, 2 and 3 & 6272 \(\times\) 6528 & & 144 & 256 & 552 & 2208 & 13,334 \\  & 4 & 3200 \(\times\) 6144 & & 72 & 120 & 252 & 990 & 5882 \\ \cline{1-1}  & 5 & 3328 \(\times\) 4608 & 512 \(\times\) 512 & 54 & 88 & 204 & 758 & 4536 \\ \cline{1-1}  & 6 & 3456 \(\times\) 5504 & & 60 & 112 & 240 & 960 & 5684 \\ \cline{1-1}  & 7 & 6400 \(\times\) 6528 & & 144 & 256 & 576 & 2256 & 13,688 \\ \hline Cleveland & All & 3000 \(\times\) 2400 & 256 \(\times\) 256 & 98 & 180 & 374 & 1462 & 9460 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of patches generated for each painting using each sliding window configuration. wo: window overlap. Once the window size and overlap percentage are chosen, our methodology operates automatically, eliminating the need for manual patch selection and ensuring a detailed analysis of the entire painting. The C++ source code for computing the FD of a painting and processing the two datasets using our sliding window methodology is publicly available at [[https://www.ugr.es/~demiras/swfd](https://www.ugr.es/~demiras/swfd)]([https://www.ugr.es/~demiras/swfd](https://www.ugr.es/~demiras/swfd)) (accessed on 1 May 2025). ### Statistical Analysis

To study the differences between the distributions of FD values from patches in original and replica paintings, we performed the nonparametric Wilcoxon rank-sum test (see step 4 in Figure 4). Note that this is the same test used for non-pairwise comparisons in [5]. Since several groups (paintings) were compared simultaneously, the Wilcoxon test was configured with a Bonferroni post hoc correction for multiple comparisons to control the overall probability of making a Type I error. Statistical test results were considered significant when the \(p\)-value obtained was below 0.05. All statistical analyses were performed using R version 4.2.1 with the Stats package. ## 4 Results

### Results on Comparing Paintings

Table 3 shows the statistical comparisons of FD values between pairs of paintings for each tested percentage of window overlap in the sliding window configuration. These results show that as the percentage of window overlap increases, more statistically significant differences (\(p\)-value \(<\) 0.05) between original and replica paintings are observed across both datasets. This finding indicates that a higher window overlap percentage provides a more detailed analysis of the painting. Our method identified significant differences between original and replica paintings for all pairs in the Princeton dataset. Nevertheless, it did not detect differences between the FD distributions of paintings A and C for artist 2 in the Cleveland dataset. As shown later, the FD distributions for these two paintings are indeed almost identical. Figure 5 shows boxplots comparing the FD values between the pairs of original and replica paintings in the Princeton dataset using the sliding window configuration with a 90% window overlap. Table 4 shows the detailed results from our algorithm at 90% window overlap, along with a comparison with the results obtained in [5] for the same dataset using multifractal analysis. The multifractal parameters (\(l_{lm}\), \(c_{1}\), and \(c_{2}\)) showed significant differences in only three of the seven comparisons between pairs of original and replica paintings [5], whereas our method found significant differences in all seven comparisons. Notably, both methodologies used the same statistical test (Wilcoxon rank-sum test) to compare the distributions of measures obtained from the paintings. Regarding the average FD values shown in Table 4 for the Princeton dataset, most replica paintings (five out of seven) exhibit higher FD values than their corresponding originals.