## 3 Materials and Methods
To improve recognition quality, additional processing methods may be required, such as restoring blurred images, analyzing shape and texture, extracting image fragments, and matching two images based on key points, among others. Most image recognition methods are based on keypoint detection, image hashing, stochastic geometry, Markov chains, perceptual hashing methods, and others. These methods are highly effective in identifying near-image duplicates. All image recognition methods can be categorized: methods based on key point extraction and methods based on pixel-level image analysis. Before applying recognition methods, it is necessary to filter graphical data, which simplifies and increases the sensitivity of analysis methods for detecting similarities in these data. The characteristic feature of filtering methods is that they allow the extraction of specific characteristics of local image regions. Filtering involves applying transformations to the entire image, such as binarization with a given threshold, high-frequency filtering (Gabor filter), low-frequency filtering (Gaussian filter), Fourier filtering, and more. A specific type of image filtering involves identifying elementary mathematical functions within the image (such as lines, parabolas, circles, etc.). Filters of this type include Hough, Radon, and other filters. In the case of complex images with many fragments, it is often sufficient to analyze not the entire image but only its contours. To formally cover all possible cases and data types for identifying near duplicates, the table format will be considered. Since tables may contain textual information, numerical data, mathematical formulas, and images, if only some of these data types need to be analyzed, the other types are simply disregarded. Let B be some input table, and \(\overline{B}=\left\{B_{1},B_{2},\ldots,B_{p}\right\}\) be a set of tables extracted from the text and stored in a common table database, where p is the number of tables in the database. The task is to identify a set of tables \(\widetilde{B}=\left\{\widetilde{B}_{1},\widetilde{B}_{2},\ldots,\widetilde{B }_{1}\right\}\) where \(\widetilde{B}\subset\overline{B},1<p\), for which the condition holds: \(F\left(B,\widetilde{B}_{i}\right)<\lambda\), \(i=\overline{1,1}\), where \(\lambda\) is threshold value, and F is the distance between the tables. Distance as a metric must satisfy the properties of non-negativity, the identity of indiscernible factors, symmetry, and triangle inequality. The presence in the database of at least one table from the set \(\overline{B}\), for which the condition \(F\left(B,\widetilde{B}_{i}\right)<\lambda\) is satisfied, indicates that table B is borrowed [12]. A cell \(K_{ij}\) of some table B is an element of the table that is formed at the intersection of the i-th row and the j-th column of this table. The cell content \(K_{ij}\) of table B is the value (numeric, text, date type, etc.) corresponding to this cell. Let us denote the content of cell \(K_{ij}\) as \(C\left(K_{ij}\right)\), where \(i=\overline{1,r_{1}},j=\overline{1,r_{2}}\) where \(r_{1}\) is the number of rows of table B, and \(r_{2}\) is the number of columns of table B. As previously defined, the content of cells may include the following types: numeric, text, date, image, formula, combined data. If the table contains images and formulas, they are allocated for separate analysis. The content of the \"date\" type after conversion to a single format may be considered as a regular text string. Let \(I=\left\{1,2,\ldots,r_{1}\right\}\) be a set of table row numbers, and \(J=\left\{1,2,\ldots,r_{2}\right\}\) be a set of column numbers of a table B. We iterate through all the cells of the table and determine their data type. Thus, simplifying the possible variants of cell content types, we can consider that the content of the table is presented as \(B=\left\langle\overline{N},\overline{S}\right\rangle\).