Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs
This enables topological data analysis and visualization for implicit continuous models, which are increasingly used for scientific data representation.
The paper tackles the problem of extracting complex topological features from continuous implicit models, introducing the first framework to directly extract contours, Jacobi sets, and ridge-valley graphs from multivariate functional approximation (MFA) without reverting to discrete representations.
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.