  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.
