Before revision:  We show that some well-known summaries, such as Betti number curves, the Euler Characteristic Curve, and Persistence Landscapes fall under the PC framework or are easily derived from it . Second, we provide a theoretical foundation for the stability analysis of PCs. In addition, we propose several new summaries based on PC framework and investigate their stability .
After revision:  We show that some well-known summaries, such as Betti number curves, the Euler Characteristic Curve, and Persistence Landscapes fall under the PC framework or are easily derived from it . Second, we propose several new summaries based on PC framework and investigate their stability .