Characterising Epithelial Tissues Using Persistent Entropy

Abstract

In this paper, we apply persistent entropy, a novel topological statistic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geometric information encoded by α-complexes and persistent homology. After using some statistical tests, we can guarantee the existence of significant differences in the studied tissues.

Stable Topological Summaries for Analyzing the Organization of Cells in a Packed Tissue

“…Again, stability was not guaranteed, since the number of bars of each barcode was not fixed, as required in [19] for a stable result. Finally, another approach was introduced in the conference paper [20]. In that case, instead of using the contact graph of the cells, spatial distribution of their centroids was studied, using the alpha filtration (or alpha-complex), which was constructed over the Delaunay complex generated by the set of centroids [21].…”

“…In that case, instead of using the contact graph of the cells, spatial distribution of their centroids was studied, using the alpha filtration (or alpha-complex), which was constructed over the Delaunay complex generated by the set of centroids [21]. Nevertheless, keeping the infinity bar in the barcodes, as in [20], made the summaries depend on the original scale of the image, introducing bias in the analysis. One of the major motivations of this paper was to solve that problem.…”

“…This paper can be seen as a continuation of the conference papers [18,20], where the experiments have been extended and improved, adding more variables and combining different filtrations. In addition, an exhaustive theoretical analysis is presented to avoid the bias present in those preliminary studies.…”

“…In Section 2, we will modify the methods appearing in [8,20] to guarantee the stability in the entire procedure. Mathematical proofs to support correctness are provided.…”

“…Then, we will fix N = 187, as it is the second minimum number of cells appearing in Table 1. We follow Algorithm 1 [20] to select the desired number of cells in each image. In Figure 1, an intuitive idea of how the algorithm works is provided.…”