Topological Function Optimization for Continuous Shape Matching

Abstract

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.

A Framework for Differential Calculus on Persistence Barcodes

“…Recall that Sing(F 0 ) is by definition the boundary of {θ ∈ M, ∃(v, v ) ∈ K 0 , F 0 (θ )(v) = F 0 (θ )(v )}, whose complement may not be generic (in fact it may even be empty, e.g., when F 0 = 0). This shows the interest of working with locally constant pre-orders on vertices, and not just with locally injective parametrizations as in the works of [3,14,29,30,43].…”

“…In many applications, F parametrizes lower-star filtrations, i.e., filter functions induced by their restrictions to the vertices of K [3,14,29,30,32,43]. In [43], the problem of shape matching is cast into an optimization problem involving the barcodes of the shapes. [14] uses the degree-0 persistent homology as a regularizer for classifiers.…”

“…In both scenarios, Proposition 4.14 gives a formula to compute a differential of B p at θ from the barcode template (P p , U p ). The optimization pipelines mentioned in Introduction [3,14,27,30,43] apply this strategy to compute differentials.…”

“…Parametrizations of lower star filtrations are involved in most practical scenarios [3,14,29,30,43]; here, we provide a common analysis of their differentiability.…”

Structured Regularization of Functional Map Computations

“…Similarly, several other regularizers have been proposed, including using robust norms and matrix completion techniques [KBB*13, KBBV15], exploiting the relation between functional maps in different directions [ERGB16], the map adjoint [HO17], and powerful cycle‐consistency constraints [HWG14] in the context of shape collections, among many others. More recently constraints on functional maps have been introduced to promote continuity of the recovered pointwise correspondence [PSO18], maps between curves defined on shapes [GBKS18], kernel‐based techniques aimed at extracting more information from given descriptor constraints [WGBS18], and an approach for incorporating orientation information into the functional map infererence pipeline [RPWO18] among others.…”