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.

翻译：拓扑数据分析（TDA）利用持续同调量化数据中的环路和高维空洞，使其特别适用于细胞生物学领域中细胞图像特征的检测。在细胞损伤情境下，随时间推移，细胞图像中会出现环形伤口并逐渐消失。由于缺乏重复样本，对单张图像中此类环状模式进行统计推断极具挑战性。本文提出一种新框架，利用TDA估计单张图像中的潜在结构，并通过置信区域量化相关不确定性。该方法将图像分割为背景区域和受损细胞区域，然后利用受损细胞区域内的像素在持续图（拓扑汇总统计量）空间中建立置信区域。该方法建立的持续图估计修正了传统TDA方法的偏差。通过模拟研究，将所提置信区域的覆盖概率与本文提出的替代方法进行比较。最后，通过细胞修复的实际案例验证了该方法的有效性。