A novel topological-data-analytical (TDA) method is proposed to distinguish, from noise, small holes surrounded by high-density regions of a probability density function. The proposed method is robust against additive noise and outliers. Traditional TDA tools, like those based on the distance filtration, often struggle to distinguish small features from noise, because both have short persistences. An alternative filtration, called the Robust Density-Aware Distance (RDAD) filtration, is proposed to prolong the persistences of small holes of high-density regions. This is achieved by weighting the distance function by the density in the sense of Bell et al. The concept of distance-to-measure is incorporated to enhance stability and mitigate noise. The persistence-prolonging property and robustness of the proposed filtration are rigorously established, and numerical experiments are presented to demonstrate the proposed filtration's utility in identifying small holes.

翻译：提议采用一种新的地形数据分析法(TDA),以区分由概率密度函数高密度区域环绕的小孔,而不是噪音。拟议方法对添加性噪音和外缘具有很强的抗力。传统的TDA工具,如那些以距离过滤为基础的工具,往往因为小特征和噪音都有短持久性,而努力将小特征与噪音区分开来。还提议采用另一种过滤法,即称为强密度-软件距离(RDAD)过滤法,以延长高密度区域小洞的持久性。这是通过用Bell等人的感知密度加权远距离功能来实现的。远程测量概念被纳入以提高稳定性和减少噪音。拟议的过滤法的持久性延长性能和稳健性得到了严格确立,并进行了数字实验,以证明拟议的过滤在识别小洞方面的效用。