The panels in the figure below show, from left to right:
- a piecewise affine function with support equal to a bounded interval and an indication of its superlevel filtration;
- the corresponding persistence diagram;
- a different function in the same spirit as the first, and with the same persistence diagram.
This example shows that there is a many-to-one mapping between functions and persistence diagrams, even after accounting for an obvious parity symmetry.
However, not every ordering of the blue line segments indicating lifetimes of components is consistent with a filtration of some function. While it's probably not too hard to sort this all out, it seems likely that somebody already has.
So: has the equivalence class of [some class of] functions having the same persistence diagram been considered in the literature? 1D is fine for my considerations.