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

arXiv: https://arxiv.org/abs/2508.07637 · HTML: https://mlsanity.com/paper/2508.07637/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 70 references that are in the map.
- First posted on arXiv 11 Aug 2025.

## Builds on (most cited first, 17 of 17)

- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [GIRAFFE: Representing Scenes as Compositional Generative Neural Feature Fields](https://mlsanity.com/paper/2011.12100/) (2020)
- [MonoSDF: Exploring Monocular Geometric Cues for Neural Implicit Surface Reconstruction](https://mlsanity.com/paper/2206.00665/) (2022)
- [Compressive Neural Representations of Volumetric Scalar Fields](https://mlsanity.com/paper/2104.04523/) (2021)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization](https://mlsanity.com/paper/2106.00157/) (2021)
- [Neural Vector Fields for Implicit Surface Representation and Inference](https://mlsanity.com/paper/2204.06552/) (2022)
- [Edge, Ridge, and Blob Detection with Symmetric Molecules](https://mlsanity.com/paper/1901.09723/) (2019)
- [Scalable Volume Visualization for Big Scientific Data Modeled by Functional Approximation](https://mlsanity.com/paper/2312.15073/) (2023)
- [Shearlet-Based Detection of Flame Fronts](https://mlsanity.com/paper/1511.03753/) (2015)
- [MFA-DVR: Direct Volume Rendering of MFA Models](https://mlsanity.com/paper/2204.11762/) (2022)
- [Jacobi Set Driven Search for Flexible Fiber Surface Extraction](https://mlsanity.com/paper/2208.06723/) (2022)
- [Reduced Connectivity for Local Bilinear Jacobi Sets](https://mlsanity.com/paper/2208.07148/) (2022)
- [Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields](https://mlsanity.com/paper/2407.03348/) (2024)
- [Critical Point Extraction from Multivariate Functional Approximation](https://mlsanity.com/paper/2408.13193/) (2024)

## Built on by (most cited first, 1 of 1)

- [Direct Topology Tracking in Continuous Implicit Models](https://mlsanity.com/paper/2609.12157/) (2026)

## Abstract (arXiv)

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.