# Fractal Dimension-Based Methodology for Discriminating Original Paintings from Replicas
###### Abstract  
Discriminating between original paintings and replicas is a challenging task. In recent years, the fractal dimension (FD) has been used as a quantitative measure of self-similarity to analyze differences between paintings. However, while the FD parameter has proven effective, previous studies often did not utilize all available image information, typically requiring binarization or grayscale analysis and the manual selection of painting regions. This study introduces a novel, color-FD-based method for differentiating original paintings from replicas. Our approach employs a sliding window approach combined with recent color-FD computation techniques. To assess the effectiveness of our FD methodology, we used two public datasets where originals and replicas were produced by the same artist under identical conditions, ensuring maximum similarity. Statistical comparisons were performed using the nonparametric Wilcoxon rank-sum test. Our method identified significant differences between original and replica paintings for 18 out of 19 pairs across both datasets, outperforming previous studies using the same datasets. As expected, our method discriminates more effectively between paintings by different artists (hit rate of 96.6%) than between originals and replicas by the same artist (hit rate of 91.7%). These findings indicate that combining the FD of color images with a sliding window approach is a promising tool for forgery detection.  
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Using a sliding window to process signals is a widely adopted technique across various fields [25; 26; 27]. In cultural heritage, this technique has been used to analyze different elements, including buildings, murals, ancient manuscripts, and paintings [28; 29; 30; 31]. The sliding window technique has recently been used to compute the fractal dimension of several types of signals, such as electroencephalograms [32], 4D point clouds [33], time series [34], and medical images [35]. However, to the best of our knowledge, the sliding window approach has not previously been used to analyze the fractal dimension of paintings.  
In the present study, we address the following research questions: (1) Is the FD computed from the color information of the images a useful metric for comparing original paintings to replicas? (2) Can the sliding window technique provide a parameterizable solution that avoids the need for the manual selection of painting regions for analysis? (3) Is it possible to obtain comparable or better results than state-of-the-art techniques for forgery detection by integrating color-FD computation into a sliding window approach?  
To address these questions, we introduce a novel FD-based methodology for differentiating original paintings from replicas. Recently, several new techniques have been proposed to accurately compute the FD of color images by jointly considering all color channels in the FD estimation process [36; 37; 38]. Our approach uses one such technique for FD computation [37] by integrating it into a sliding window process of the painting. To the best of our knowledge, this is the first color-FD-based method for distinguishing between original paintings and replicas. Moreover, our methodology does not require the manual selection of painting regions for analysis.