# Fractal Dimension-Based Methodology for Discriminating Original Paintings from Replicas
## 3 Material and Methods
### The DBC-RGB Algorithm
Since the present study has proven the effectiveness of our FD-based methodology for differentiating between original paintings and replicas, further research could also be directed towards incorporating the FD value computed at each individual patch through the DBC-RGB algorithm as a feature in machine learning models used in previous studies [1; 2; 3; 6]. Among them, only the model described in [6] incorporated the FD as one of the features describing each patch. However, they used an FD computation approach based on the pre-binarization of the patches. We think that processing the full color information using algorithms such as DBC-RGB could further enhance the results of these machine learning classifiers. Our methodology requires establishing two key parameters: patch size and window overlap percentage. To ensure fair comparisons in the present work, we selected the same patch sizes recommended by the studies that analyzed the same datasets [2; 5]. In general, small patches allow for a detailed analysis of the painting but at the cost of longer computation times and potentially overlooking key painting characteristics such as the size of the brushstrokes. Conversely, large patch sizes accelerate the calculations, but the number of FD values for statistical comparisons of the paintings could be insufficient, and the portion of the painting analyzed in each patch could be too large, including too much heterogeneity (painting techniques and combinations of brushes). According to our experiments, increasing the window overlap percentage leads to the detection of a greater number of significant differences between paintings within the datasets. While a higher window overlap parameter value allows for more detailed analysis, it also demands longer computation times. Nevertheless, establishing a high window overlap does not ensure finding significant differences when comparing every pair of paintings, as seen in the case of paintings A and C by artist 2 in the Cleveland dataset. This pair of paintings was further examined using extremely high window overlaps of 95% and 99%, yet no significant differences were found. Unfortunately, we cannot provide general rules or criteria for automatically determining the optimal patch size and window overlap parameters. Therefore, some empirical experimentation is required depending on the characteristics of the dataset of paintings being analyzed. In order to use our approach on other datasets, the following considerations must be taken into account: (1) High-resolution images are needed to capture as much as possible the subtle details of the paintings that can differentiate between originals and replicas. (2) Since the FD computation method in our methodology is based on the color intensities, the acquisition of images for each original-replica pair must be performed under identical conditions, and (3) the patch size in sliding window processing must be large enough to obtain a number of scales that ensure an accurate linear regression computation in the FD estimation through the DBC-RGB algorithm (at least 128 \(\times\) 128 pixels as suggested in [37]). Images of the paintings at a low resolution could imply that the patch size does not meet this requirement. The primary limitation of our sliding window FD algorithm is its high computational demand, particularly when processing high-resolution images with a large window overlap. Unfortunately, prior related studies have not reported the computation times of their respective algorithms, making it impossible to make a direct performance comparison. Nonetheless, given that our methodology is based on a fast algorithm executed in parallel on a GPU [37], we believe that our computation times would not be significantly worse than those obtained in similar studies, particularly those that also process grids of patches within the painting [1; 2; 3; 6]. As we demonstrated with our OpenMP implementation, the overall dataset processing can be significantly optimized through the parallel processing of multiple paintings.