Stability of Critical Points with Interval Persistence

Scalar functions defined on a topological space Ω are at the core of many ap-plications such as shape matching, visualization and physical simulations. Topo-logical persistence is an approach to characterizing these functions. It measureshow long topological structures in the sub-level sets {x ∈ Ω : f(x) ≤ c} persistas c changes. Recently it was shown that the critical values defining a topologicalstructure with relatively large persistence remain almost unaffected by small per-turbations. This result suggests that topological persistence is a good measure formatching and comparing scalar functions. We extend these results to critical pointsin the domain by redefining persistence and critical points and replacing sub-levelsets {x ∈ Ω : f(x) ≤ c} with interval sets {x ∈ Ω : a ≤ f(x) < b}. Withthese modifications we establish a stability result for critical points. This result isstrengthened for maxima that can be used for matching two scalar functions.