Before revision:  First, we develop a general  framework of vectorizing diagrams that we call the Persistence Curves (PCs) . 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.
After revision:  First, we develop a general framework of vectorizing diagrams that we call the Persistence Curves (PCs) , and show that several 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.