Local, Smooth, and Consistent Jacobi Set Simplification

TL;DR

This paper presents a new framework for simplifying Jacobi sets derived from Morse functions, enabling smooth, local, and consistent reductions that improve manageability and applicability in noisy or discretized data.

Contribution

It introduces an atomic operation-based algorithm for Jacobi set simplification via smooth approximations, extending critical point cancellations to two dimensions.

Findings

The algorithm correctly simplifies Jacobi sets to their minimal form on simply connected domains.

It guarantees termination when no further smooth, local simplifications are possible.

The framework generalizes existing critical point cancellation techniques to Jacobi sets.

Abstract

The relation between two Morse functions defined on a common domain can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces have shown to be useful in various applications. Unfortunately, in practice functions often contain noise and discretization artifacts causing their Jacobi set to become unmanageably large and complex. While there exist techniques to simplify Jacobi sets, these are unsuitable for most applications as they lack fine-grained control over the process and heavily restrict the type of simplifications possible. In this paper, we introduce a new framework that generalizes critical point cancellations in scalar functions to Jacobi sets in two dimensions. We focus on simplifications that can be realized by smooth approximations of the corresponding functions and show how this implies simultaneously simplifying contiguous subsets of the Jacobi set. These extended cancellations form the atomic operations in our framework, and we introduce an algorithm to successively cancel subsets of the Jacobi set with minimal modifications according to some user-defined metric. We prove that the algorithm is correct and terminates only once no more local, smooth and consistent simplifications are possible. We disprove a previous claim on the minimal Jacobi set for manifolds with arbitrary genus and show that for simply connected domains, our algorithm reduces a given Jacobi set to its simplest configuration.