Confidence regions for a persistence diagram of a single image with one or more loops

Topological data analysis (TDA) uses persistent homology to quantify loops and higher-dimensional holes in data, making it particularly relevant for examining the characteristics of images of cells in the field of cell biology. In the context of a cell injury, as time progresses, a wound in the form of a ring emerges in the cell image and then gradually vanishes. Performing statistical inference on this ring-like pattern in a single image is challenging due to the absence of repeated samples. In this paper, we develop a novel framework leveraging TDA to estimate underlying structures within individual images and quantify associated uncertainties through confidence regions. Our proposed method partitions the image into the background and the damaged cell regions. Then pixels within the affected cell region are used to establish confidence regions in the space of persistence diagrams (topological summary statistics). The method establishes estimates on the persistence diagrams which correct the bias of traditional TDA approaches. A simulation study is conducted to evaluate the coverage probabilities of the proposed confidence regions in comparison to an alternative approach is proposed in this paper. We also illustrate our methodology by a real-world example provided by cell repair.