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

Abstract

We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.

Persistent Homology of Coarse Grained State Space Networks

“…In this example, we take the state sequence S on the left side of Fig. 2 with symbols in the alphabet A = [1,2,3,4] and create the network shown network in the middle of the figure. This network is represented as a directed and weighted adjacency matrix, as shown on the right side of Fig.…”