Stability of critical points with interval persistence

Abstract

Scalar functions defined on a topological space Ω are at the core of many applications such as shape matching, visualization and physical simulations. Topological persistence is an approach to characterizing these functions. It measures how long topological structures in the sub-level sets {x ∈ Ω: f(x) ≤ c} persist as c changes. Recently it was shown that the critical values defining a topological structure with relatively large persistence remain almost unaffected by small perturbations. This result suggests that topological persistence is a good measure for matching and comparing scalar functions. We extend these results to critical points in the domain by redefining persistence and critical points and replacing sub-level sets {x ∈ Ω: f(x) ≤ c} with interval sets {x ∈ Ω: a ≤ f(x) < c}. With these modifications we establish a stability result for critical points. This result is strengthened for maxima that can be used for matching two scalar functions. © 2007 Springer.