Feature Detection and Hypothesis Testing for Extremely Noisy Nanoparticle Images using Topological Data Analysis

We propose a flexible algorithm for feature detection and hypothesis testing in images with ultra-low signal-to-noise ratio using cubical persistent homology. Our main application is in the identification of atomic columns and other features in transmission electron microscopy (TEM). Cubical persistent homology is used to identify local minima and their size in subregions in the frames of nanoparticle videos, which are hypothesized to correspond to relevant atomic features. We compare the performance of our algorithm to other employed methods for the detection of columns and their intensity. Additionally, Monte Carlo goodness-of-fit testing using real-valued summaries of persistence diagrams derived from smoothed images (generated from pixels residing in the vacuum region of an image) is developed and employed to identify whether or not the proposed atomic features generated by our algorithm are due to noise. Using these summaries derived from the generated persistence diagrams, one can produce univariate time series for the nanoparticle videos, thus providing a means for assessing fluxional behavior. A guarantee on the false discovery rate for multiple Monte Carlo testing of identical hypotheses is also established.我们提出了一种基于立方持久同调（cubical persistent homology）的灵活算法，可用于超低信噪比图像的特征检测与假设检验。本算法的核心应用场景为透射电子显微镜（Transmission Electron Microscopy, TEM）图像中的原子柱与其他特征识别。我们利用立方持久同调识别纳米颗粒视频帧子区域内的局部极小值及其尺度，该局部极小值被假设对应于相关原子特征。我们将本算法的性能与其他已用于原子柱检测及其强度分析的现有方法进行了对比。此外，我们开发并采用了基于平滑图像（由图像真空区域内的像素生成）的持久同调图（persistence diagrams）实值汇总统计量的蒙特卡洛（Monte Carlo）拟合优度检验，用于判断本算法生成的候选原子特征是否由噪声产生。借助从生成的持久同调图中提取的汇总统计量，可针对纳米颗粒视频生成单变量时间序列，从而为评估其动态演化行为提供了有效手段。我们还建立了针对同一假设的多重蒙特卡洛检验的错误发现率（false discovery rate）保障准则。