## 2 Materials and Methods
### Reconstruction of Binary Patterns from the Empirical Model  
The goal of this section is to estimate how much the EM reliably describes a binary arbitrary pattern. To reach this goal, the original arbitrary pattern was converted into the EM, then reconstructed from the EM [19]. The patterns before and after this reconstruction were then compared. A binary pattern situated between \(R_{j-1}\), \(R_{j}\) consisted of black and white pixels. The black pixels formed a configuration of the pattern under study. The white pixels were used to construct G(N) There is a one-to-one correspondence between G(N) and the set of black pixels situated between \(R_{j-1}\) and \(R_{j}\). This means that if the colors are inverted, then G(\(R_{j-1}\), \(R_{j}\)) is converted into the pattern under study, and the pattern under study will be converted into G(\(R_{j-1}\), \(R_{j}\)). This feature of binary patterns was used to formalize the construction of G(N), and to measure the thickness and area of the layers [5,6]. The outputs of procedures for developing the EM are the following data sets:  
* The number of transects and their direction and position in the 2-D plane. * The coordinates of graph vertices situated along \(R_{j}\), \(j=1\), N.
* The adjacency matrix that describes the connectivity between graph vertices situated along \(R_{j-1}\) and \(R_{j}\), \(j=1\), N.
* The coordinates of points where the black pixels of the forming front and \(R_{j}\) intersect (Figure 5a). \(V_{j}\) denotes the set of points, i.e., vertices \(v_{1,j}\), \(v_{2,j}\),..., \(v_{p,j}\) of the intersection of the forming front and \(R_{j}\). Figure 5a shows the point of intersection of the forming front with transect R. A fragment of the pattern is presented in Figure 5a in the Comma Separate Values (CSV) file, where 1 represents black pixels. * The adjacency matrix (AM-P) describes the connectivity between \(V_{j-1}\) and \(V_{j}\) (Figure 5b). The AM-P describes the structure of the pattern under study. To reconstruct the pattern between \(R_{j-1}\) and \(R_{j}\), we construct a bipartite graph based on \(V_{j-1}\), \(V_{j}\) and the AM-P. The limitation of using the AM-P for pattern reconstruction lies in the fact that the AM-P contains no information about the configuration of edges between sets of vertices \(V_{j-1}\) and \(V_{j}\), nor does it contain information about the thickness of these edges. Thus, straight lines of identical thicknesses comprise the layers between  
Figure 5: Pattern reconstruction from EM. (**a**) Principal elements of binary layered patterns: (1) direction of the numbered layers; (2) forming front; (3) layer thickness; (4) point of intersection between the forming front and transect R. (**b**) Pattern before and after reconstruction and its adjacency matrix. nearby transects. Figure 5b shows the pattern before and after reconstruction and the corresponding AM-P. The accuracy of such pattern reconstruction increases as the distance between the transects decreases. The text pattern rendered in raster binary format is used as the test sample to illustrate the text's geometry before and after reconstruction (Figure 6a). The post-reconstruction text is readable but visually looks more complicated than the pre-reconstruction text because it has many structural irregularities. Charts DStr = f(number of transects) demonstrates (Figure 6b) two apparent facts in quantitative terms:  
* DStr (after reconstruction) > DStr (before reconstruction),
* The differences between DStr (after reconstruction) and DStr (before reconstruction) nonlinearly increase as the number of transects goes up. Thus, the parameter DStr could be applied to set up the correspondence between the number of transects used for the construction of the EM and the how well the EM compares to the pattern being studied. In other words, the DStr could potentially serve as the measure of how accurately the EM corresponds to the structure of the pattern being studied.