# Fractal Dimension-Based Methodology for Discriminating Original Paintings from Replicas
## 3 Material and Methods
### The DBC-RGB Algorithm
All differences between artists are statistically significant. wo: window overlap. Figure 7: Boxplots showing significant differences in FD between the artists in the Cleveland dataset. Paintings for each artist were processed with a window overlap percentage of 0%. *: \(p\)-value < 0.05, **: \(p\)-value < 0.01 and ***: \(p\)-value < 0.0001. ### Performance Analysis

Finally, we conducted a performance analysis of our algorithm. For time measurements, we used the _now_ method of the _high_resolution_clock_ class in the _chrono_ C++ library, providing millisecond precision. We sequentially processed the two datasets (one painting after the other) on a computing server with the following configuration: Debian Linux 5.10 operating system, Intel Xeon Silver 4210 @ 2.20 GHz (10 cores and 20 threads), 96 GB of RAM, three GPUs, an NVIDIA GeForce RTX 3090 with 10,496 CUDA cores and 24 GB of global memory, and two NVIDIA GeForce RTX 2080 Ti GPUs, each with 4352 CUDA cores and 11 GB of global memory. Additionally, we processed the paintings in both datasets in parallel using OpenMP technology [46] within our C++ code. Table 8 shows the times required to process each dataset for each window overlap configuration in two scenarios: sequential processing of paintings and parallel processing of paintings. Although the sequential processing times for high window overlaps may appear lengthy (approximately averaging 6.5 s per image with a 90% window overlap), this is primarily due to the large number of patches processed (between 9500 and 13,500 patches per image at a 90% window overlap). Nonetheless, processing such a large number of patches per image provides a very detailed analysis. Furthermore, without the recently released ultra-fast GPU algorithm for computing the FD of each patch, as provided in [37], the processing times would have been up to eight times slower, in comparison to other parallel GPU algorithms for FD computation [37]. Compared to sequential CPU implementations of color-FD algorithms with similar accuracy [36], the processing times required for the two datasets would have been up to nine hundred times slower [37], resulting in approximately two days of computation. Parallel processing of the paintings significantly improved performance. The multi-threaded OpenMP execution of the experiment on a server equipped with three GPUs greatly reduced the time required to process all the paintings in both datasets. As shown in Table 8, with a window overlap of 90%, the total processing time for all paintings decreased from 168.69 s (sequential) to 54.21 s (parallel). With a 0% window overlap, the total processing time decreased from 2.90 s (sequential) to 1.12 s (parallel). This speedup implies that our software would be able to process up to 1725 images with a 90% window overlap or 83,720 images with a 0% window overlap, where each image has an average high-resolution size of 4092 \(\times\) 4369 pixels, all within just one hour. This demonstrates the efficiency of our methodology in processing extensive image datasets.