Stability of critical points with interval persistence

Scalar functions defined on a topological space Omega 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 is an element of Omega: 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 is an element of Omega: f(x) <= c} with interval sets {x is an element of Omega: a <= f(x) < b}. 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.