We introduce ellipsoidal filtration, a novel method for persistent homology, and demonstrate its effectiveness in denoising recurrent signals. Unlike standard Rips filtrations, which use isotropic neighbourhoods and ignore the signal's direction of evolution, our approach constructs ellipsoids aligned with local gradients to capture trajectory flow. The death scale of the most persistent H_1 feature defines a data-driven neighbourhood for averaging. Experiments on synthetic signals show that our method achieves better noise reduction than both topological and moving-average filters, especially for low-amplitude components.

翻译：本文提出了一种用于持续同调的新方法——椭球滤波，并验证了其在周期信号去噪中的有效性。与采用各向同性邻域且忽略信号演化方向的标准Rips滤波不同，本方法通过构建与局部梯度对齐的椭球来捕捉轨迹流。最具持续性的H_1特征的消亡尺度定义了用于平均的数据驱动邻域。在合成信号上的实验表明，本方法在降噪效果上优于拓扑滤波器和移动平均滤波器，尤其对低振幅信号成分具有显著优势。