Fig. 1. An illustration of feature extraction using TDA.

a Three filtration functions are illustrated. In the upper row, the density of white pixels is calculated using the kNN density estimator, and elevated regions in the 3D map represent a higher density. As the threshold decreases from t₁ to t₃, the superlevel set (the domain above the threshold) grows from Xt₁ to Xt₃. In the middle and lower rows, the signed distance and the 8-bit grayscale are, respectively, used as filtration functions. As the threshold decreases, the superlevel set (white region) expands. b The flowchart of our method. An original image is transformed into a grayscale image. For the kNN density estimator or signed distance as the filtration function, the image is binarized with Otsu’s method. Wavelet transformation and morphological operations can be applied. Following TDA, the persistence diagram is obtained. c A schematic diagram of thresholds and superlevel sets. In the upper row, as the threshold decreases, the superlevel set (white region) grows larger. The middle row shows the birthplace of connected components (0-dim topological features), and the lower row shows the holes (1-dim topological features) in the corresponding superlevel set. d The information on the detected topological features of each image is summarized in two diagrams called the persistence diagrams. The axes represent the log-scale thresholds where the feature appears (“birth”) and disappears (“death”). The 0-dim persistence diagram represents information of the connected components, and the 1-dim persistence diagram represents that of holes. The color scale represents the density of points in the persistence diagram calculated by the function “kde2d” of the R package MASS. e For interpretability and handiness, we transform “birth” and “death” into their mean (mid-life) and their difference (life-time) and plot them. f The persistence diagrams with the axes mid-life and life-time. The color scale represents the density of points in the persistence diagram. The three small panels on the right show the holes in each corresponding domain of the 1-dim persistence diagram. The life-time roughly corresponds to the size of holes.