# Natural Hydrocarbon Samples Classification by Topological Analysis Method
## 3 Results  
We used the proposed algorithm to find and plot the distances between persistent diagrams of selected samples:  
**An**--the A Formation type (A), \(n\) refers to different binarization threshold and resolution as follows:  
* **A1**--resolution is 16.5 \(\upmu\)m/pixel, binarization threshold (t) = 103
* **A2**--16.5 \(\upmu\)m/px, t = 113
* **A3**--16.5 \(\upmu\)m/px, t = 123
* **A4**--16.5 \(\upmu\)m/px, t = 133
* **A5**--16.5 \(\upmu\)m/px, t = 143
* **A6**--33 \(\upmu\)m/px, t = 113  
**S**--Sandstone Bentheimer (Sandstone), 20 \(\upmu\)m/px, different subsamples with same threshold;  
**C**--carbonate rock (Carbonate), 20 \(\upmu\)m/px, different subsamples with same threshold;  
**T**--T formations, 25 \(\upmu\)m/px, different subsamples with same threshold. From the computed persistent diagrams for \(H_{0}\), we can see that point clouds for CH and T samples (Figure 2) are on average closer to the diagonal line than that of the CL sample. That means CL has more topologically significant connected components (more long-living topological features) and was formed from bigger fractions. As for \(H_{1}\) diagrams in Figure 3, we can see that point cloud for CH sample is more concentrated along with the diagonal than those of CL and T samples. That means CL and T samples have on average wider tunnels passing through them than CH samples. From \(H_{2}\) diagrams in Figure 4, we can note that the cloud point for the CH sample has more distance from the diagonal line points. That means the CH sample has on average more enclosed cavities than CL and T samples. Figure 2: \(H_{0}\) persistent diagrams for CH, CL, and T samples. Then we calculated distances between the \"base\" sample and each other. As the \"base\" sample we chose \(S_{1}\). Figure 1 presents distances between \"base\" sample and others in Bottleneck metric in the 3D plot, as we described above, and in Figure 5 are presented projections of this plot on \(H_{0}\)-\(H_{1}\) and \(H_{1}\)-\(H_{2}\) planes. We can see that samples are not well clustered in this metric. Figures 6 and 7 present distances between \"base\" sample and others in Heat Kernel metric, as well as projections of this plot on \(H_{0}\)-\(H_{1}\), \(H_{1}\)-\(H_{2}\), and \(H_{0}\)-\(H_{2}\) planes. Figure 4: \(H_{2}\) persistent diagrams for CH, CL, and T samples. Figure 5: Common distances in Bottleneck metric in \(H_{0}\)–\(H_{1}\) and \(H_{1}\)–\(H_{2}\) coordinates. Figure 3: \(H_{1}\) persistent diagrams for CH, CL, and T samples. Figure 6: Common distances in Heat Kernel metric in 3D and \(H_{0}\)–\(H_{1}\) coordinates. Figure 7: Common distances in Heat Kernel metric in \(H_{1}\)–\(H_{2}\) and \(H_{0}\)–\(H_{2}\) coordinates. In Figures 9-12, the influence of the binarization threshold of the A sample on the distance values from \(S_{1}\) sample (\"base\" sample) in the Bottleneck, Heat Kernel, and Sliced Wasserstein metrics is shown (threshold value is marked as label for dots). In addition, five different subsamples of the C sample were taken for reference and marked as \(C_{x}\). It can be seen that changing the binarization threshold makes the A sample more similar to S sample in terms of surveyed metrics. It can be explained by eliminating tiny features and elements and by changing the considered scale. While transiting the inversion point of a binarized image, the differences start to arise--the details of microstructure appear. This effect is the same as while we change the resolution of the original CT image. This mechanism allows us to search for common on different scales or for different image quality. In this case, the C sample differs significantly from other samples on all considered scales because of fundamental differences in structure. In Figure 14, CT slices with different binarization thresholds are presented.