# Cross-Correlation and Fractal Analysis in the Images Diatoms Symmetry
## 3 Results and Discussion  
### Self-Similarity and Symmetry  
Most of the self-similarity from the natural structures occurs only at a finite number of stages with certain imperfections [39]. In fact, it is unlikely that a smaller cluster could be exactly the same as a larger cluster, i.e., the self-similarity is approximate [40]. Two parameters were considered in order to measure de self-similarity: Nf and the projection (\(\theta\)). The projection (\(\theta\)) is a parameter calculated from two CC using Equation (1); the first CC is measured for the same image with and without a symmetry operation (plane of rotation), and the second CC is measured for the same image itself without symmetry operation, it is named autocorrelation (_CCauto_). The Nf parameter was calculated by the Hausdorff Box-Counting [34], which is widely used in biological systems [41] and \(\theta\) was calculated as shown in the following equation:  
\[\theta=\cos^{-1}\left(\frac{CC[I,Im]*CCauto[I]}{\|CC[I,Im]\|\|CCauto[I]\|}\right) \tag{1}\]  
where, \(I\) is the diatom image, \(Im\) is the diatom imagen with the symmetry operation, \(CC[I,Im]\) is the CC calculated between \(I\) and \(Im\), and \(CCauto[I]\) is CC calculated for \(I\) (where the CC matrices became vectors). Basically, the projection between these two vectors is obtained by the dot product divided by the product of the norms of these. If \(CC[I,Im]\) is exactly equal to \(CCauto[I]\), the \(\theta\) value is zero indicating that the symmetry operation is typical of the image. However, in natural or biological structures it is not possible because there is always some imperfection. Therefore, the values of \(\theta\) could be approximately zero or close to zero which indicates a high symmetry with respect to the operation. If the value of \(\theta\) is ninety or close to ninety, it indicates a low symmetry for the operation. Three symmetry operations to calculate \(\theta\) were used: a reflection in the vertical plane (PSv), a reflection in the horizontal plane (PSO) and a rotation of 180 degrees (R180). All the preprocessing steps to calculate \(\theta\) are outlined in Figure 2. For each symmetry operation was necessary the complete preprocessing steps, i.e., a \(\theta\) value was obtained for each symmetry operation. In Figure 2, the first step is to convert the image to grayscale, followed by a Canny Edge detection to generate an outline image. Next, this outline image is centered. Then, the CCauto is calculated, and at the same time, one of the three symmetry operations is applied, followed by the calculation of CC. Finally, the matrices CCauto and CC are converted to vectors in order to find the projection with Equation (1). Two diatom images, I3 and I34, were used how examples using all the preprocessing steps to calculate the \(\theta\) projection (see Figure 3a,b, respectively). Figure 3 shows the original images after edge detection and centering (B), and each symmetry operation applied (PSv, R180, PSO) for the two diatom images. Moreover, the CC for each symmetry operation is shown in Figure 3a,b (see blue circles in the bottom left), where the red line represents the CCauto for each image (I3 and I34). Additionally, three lineal correlation plots for the  
Figure 2: Scheme of all preprocessing steps to calculate \(\theta\). CCauto vs. CC calculated for each symmetry operation are shown in Figure 3a,b (bottom right). Lastly, the Nf and \(\theta\) values are reported on the top right of Figure 3a,b. Figure 4a-c represents the polar plots of the Nf and \(\theta\) parameters for all diatom images selected, where Nf was designated strategically as the radius and \(\theta\) as the polar variable. This polar plot shows the difference in self-similarity for the diverse diatoms images after the symmetry operations were applied. From Figure 4a-c, the applied symmetry operations were PSv, R180, and PSO, respectively. Nevertheless, a complete separation between the diverse species of diatoms could not be achieved in each of the polar plots.