Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs

TL;DR

This paper introduces a novel framework for directly extracting complex topological features such as contours, Jacobi sets, and ridge-valley graphs from multivariate functional approximations, enabling advanced analysis of implicit continuous models.

Contribution

It is the first to provide a method for extracting topological features directly from MFA models without converting to discrete data, supporting broader analysis of implicit models.

Findings

Enables direct extraction of topological features from MFA models

Supports analysis without reverting to discrete representations

Generalizable to any implicit model with function and derivative queries

Abstract

Implicit continuous models, such as functional models and implicit neural networks, are an increasingly popular method for replacing discrete data representations with continuous, high-order, and differentiable surrogates. These models offer new perspectives on the storage, transfer, and analysis of scientific data. In this paper, we introduce the first framework to directly extract complex topological features -- contours, Jacobi sets, and ridge-valley graphs -- from a type of continuous implicit model known as multivariate functional approximation (MFA). MFA replaces discrete data with continuous piecewise smooth functions. Given an MFA model as the input, our approach enables direct extraction of complex topological features from the model, without reverting to a discrete representation of the model. Our work is easily generalizable to any continuous implicit model that supports the queries of function values and high-order derivatives. Our work establishes the building blocks for performing topological data analysis and visualization on implicit continuous models.