Homology of filtrations
Broadly, this is my view on persistent homology.
The current, shortened, version is called Robustness of topology of digital images and point clouds.
Abstract. Such modern applications of topology as digital image analysis and data analysis have to deal with noise and other uncertainty. In this environment, the data structures often appear "filtered" into a sequence of cell complexes. We introduce the homology group of the filtration as the group of all possible homology classes of all elements of the filtration, without double count. The second step of analysis is to discard the features that lie outside the user’s choice of the acceptable level of noise.
The work started in 2006 with Homology of parametric images, aka Homology of gray scale images.