# Fractal Dimension-Based Methodology for Discriminating Original Paintings from Replicas
## 2 Related Work  
FD has already been used to differentiate between original paintings and replicas. [PERSON] et al. [8] analyzed the FD of [PERSON]'s paintings, comparing them to similar poured paintings created by art students. In their study, the FD was calculated separately for specific color channels within each painting (black, aluminum, gray, light yellow, and dark yellow) using the box-counting algorithm for binary images [22]. [PERSON] et al. demonstrated that the FD values of [PERSON]'s original works exhibited a distinct pattern, meeting specific criteria that were not observed in the FD values of the student-generated paintings.  
[PERSON] et al. [5] applied multifractal analysis [39] to differentiate original paintings and replicas created by the same artist under identical conditions (paints, grounds, and brushes). They manually selected corresponding patches from both the original and replica, ensuring consistent spatial location. Subsequently, they calculated the multifractal parameters (\(h_{m}\), \(c_{1}\), and \(c_{2}\)) for these selected patches and statistically compared the resulting values between originals and replicas. This multifractal analysis was conducted independently for several image channels (red, green, blue, saturation, lightness, and gray-level intensity). Their findings revealed significant differences in multifractal parameters for five of the seven original-replica pairs.  
Recently, [PERSON] et al. [7] demonstrated that the FD provides the strongest discriminatory power between authentic and non-authentic [PERSON] paintings, outperforming a wide range of other image metrics (including texture descriptors, color features, edge statistics, object statistics, distribution of pixel intensity values, multi-scale histograms, and Chebyshev statistics). [PERSON] et al. estimated the FD using the box-counting algorithm applied to the grayscale images of each painting at multiple resolutions [40].  
The FD can also be incorporated as a feature within machine learning algorithms for classifying paintings as originals or replicas, as demonstrated by [PERSON] et al. [6]. In their approach, the painting was divided into a regular grid of patches; then, the image in each patch was binarized using a segmentation algorithm, after which the FD was computed in each binarized patch using the box-counting algorithm [41].  
Although the FD parameter has proven its efficacy as a metric for distinguishing between original paintings and replicas, previous studies have not fully utilized all the available information within each painting. Methodologies described in [6, 8] require the binarization of the paintings, omitting color information. In [5,7], a broader spectrum of data from each image channel is incorporated into the fractal analysis, but only at the grayscale level. Moreover, several of these methodologies require experts to manually select painting regions for analysis [5], preventing the automated processing of the datasets.  
Alternatively, other methodologies [1,3,6] use machine learning classifiers trained using features extracted from each individual patch and tested by classifying individual patches as well. Various types of features are used in these studies, such as those based on the hidden Markov tree multiresolution model in [1], the histogram of oriented gradients technique in [3], and the FD combined with other grayscale and texture features in [6].