首次从连续模型中直接提取复杂拓扑结构，无需转为离散表示。
Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs
- 基于多变量函数逼近（MFA）直接计算拓扑特征。
- 支持函数值与高阶导数查询，实现无离散化拓扑提取。
- 适用于各类连续隐式模型，助力科学数据可视化分析。
隐式连续模型（如函数模型和隐式神经网络）正日益成为替代离散数据表示的流行方法，提供更高效的数据存储、传输与分析新视角。本文首次提出框架，直接从多变量函数逼近（MFA）这一类连续隐式模型中提取复杂拓扑特征——等值线、Jacobi集与脊谷图。MFA以分段光滑连续函数替代离散数据。给定MFA模型作为输入，本方法可直接从中提取拓扑特征，无需还原为离散表示。该方法可推广至任何支持函数值与高阶导数查询的连续隐式模型。工作奠定了在隐式连续模型上开展拓扑数据分析与可视化的基础。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。