Topological Function Optimization for Continuous Shape Matching

We present a novel approach for optimizing real-valued functions based on a wide range of topological criteria. In particular, we show how to modify a given function in order to remove topological noise and to exhibit prescribed topological features. Our method is based on using the previously-proposed persistence diagrams associated with real-valued functions, and on the analysis of the derivatives of these diagrams with respect to changes in the function values. This analysis allows us to use continuous optimization techniques to modify a given function, while optimizing an energy based purely on the values in the persistence diagrams. We also present a procedure for aligning persistence diagrams of functions on different domains, without requiring a mapping between them. Finally, we demonstrate the utility of these constructions in the context of the functional map framework, by first giving a characterization of functional maps that are associated with continuous point-to-point correspondences, directly in the functional domain, and then by presenting an optimization scheme that helps to promote the continuity of functional maps, when expressed in the reduced basis, without imposing any restrictions on metric distortion. We demonstrate that our approach is efficient and can lead to improvement in the accuracy of maps computed in practice.