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.

翻译：我们建议采用一种灵活的算法,用于在超低信号到噪音比例的图像中进行特征探测和假设测试,使用单方持续同族元素。我们的主要应用是确定原子柱和传输电子显微镜(TEM)中的其他特征。在纳米粒子视频框架中,通过阴道持久性同族元素,在次区域中识别本地微型及其大小,这些微粒被虚度地与相关原子特征相对应。我们将我们的算法的性能与其他用于探测柱子及其强度的方法进行比较。此外,还开发并使用了利用光滑图像(来自位于图像真空区域的像素生成的)产生的粘度图的真值摘要进行蒙特卡洛美性试验,以确定我们算法产生的原子特征是否因噪音而产生。利用生成的耐久性图产生的这些摘要,可以产生纳米粒子视频的单向时间序列,从而为评估通性行为及其强度提供一种手段。此外,还建立了对蒙卡洛多次测试相同假设物的虚假发现率的保证。