# Direct Topology Tracking in Continuous Implicit Models

arXiv: https://arxiv.org/abs/2609.12157 · HTML: https://mlsanity.com/paper/2609.12157/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 63 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 30 of 30)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (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)
- [Learning Continuous Image Representation with Local Implicit Image Function](https://mlsanity.com/paper/2012.09161/) (2020)
- [NeRV: Neural Representations for Videos](https://mlsanity.com/paper/2110.13903/) (2021)
- [HNeRV: A Hybrid Neural Representation for Videos](https://mlsanity.com/paper/2304.02633/) (2023)
- [COIN++: Neural Compression Across Modalities](https://mlsanity.com/paper/2201.12904/) (2022)
- [Compressive Neural Representations of Volumetric Scalar Fields](https://mlsanity.com/paper/2104.04523/) (2021)
- [DNeRV: Modeling Inherent Dynamics via Difference Neural Representation for Videos](https://mlsanity.com/paper/2304.06544/) (2023)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Fast Neural Representations for Direct Volume Rendering](https://mlsanity.com/paper/2112.01579/) (2021)
- [ECNR: Efficient Compressive Neural Representation of Time-Varying Volumetric Datasets](https://mlsanity.com/paper/2311.12831/) (2023)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Adaptively Placed Multi-Grid Scene Representation Networks for Large-Scale Data Visualization](https://mlsanity.com/paper/2308.02494/) (2023)
- [Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization](https://mlsanity.com/paper/2106.00157/) (2021)
- [Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-aware Matchings](https://mlsanity.com/paper/1908.07841/) (2019)
- [Meta-INR: Efficient Encoding of Volumetric Data via Meta-Learning Implicit Neural Representation](https://mlsanity.com/paper/2502.09669/) (2025)
- [Neural Vector Fields for Implicit Surface Representation and Inference](https://mlsanity.com/paper/2204.06552/) (2022)
- [Lifted Wasserstein Matcher for Fast and Robust Topology Tracking](https://mlsanity.com/paper/1808.05870/) (2018)
- [Distributed Neural Representation for Reactive in situ Visualization](https://mlsanity.com/paper/2304.10516/) (2023)
- [Explorable INR: An Implicit Neural Representation for Ensemble Simulation Enabling Efficient Spatial and Parameter Exploration](https://mlsanity.com/paper/2504.00904/) (2025)
- [Topology-Based Feature Design and Tracking for Multi-Center Cyclones](https://mlsanity.com/paper/2011.08676/) (2020)
- [Geometry-Aware Merge Tree Comparisons for Time-Varying Data with Interleaving Distances](https://mlsanity.com/paper/2107.14373/) (2021)
- [Flexible and Probabilistic Topology Tracking with Partial Optimal Transport](https://mlsanity.com/paper/2302.02895/) (2023)
- [Scalable Volume Visualization for Big Scientific Data Modeled by Functional Approximation](https://mlsanity.com/paper/2312.15073/) (2023)
- [Efficient Neural Representation of Volumetric Data using Coordinate-Based Networks](https://mlsanity.com/paper/2401.08840/) (2024)
- [MFA-DVR: Direct Volume Rendering of MFA Models](https://mlsanity.com/paper/2204.11762/) (2022)
- [Critical Point Extraction from Multivariate Functional Approximation](https://mlsanity.com/paper/2408.13193/) (2024)
- [Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs](https://mlsanity.com/paper/2508.07637/) (2025)

## Abstract (arXiv)

We present a framework for tracking topological features directly within continuous implicit models. Such models, including implicit neural representations (INRs) and multivariate functional approximations (MFAs), are increasingly adopted to represent scientific data without the resolution constraints of discrete grids. They offer compact, smooth, and differentiable representations of complex fields, enabling new opportunities for high-performance data storage, reconstruction, and analysis. Given a continuous implicit model, our method tracks the evolution of critical points by querying the model and its derivatives, thereby eliminating the need to resample onto a grid. This approach enables faithful feature tracking while avoiding discretization-induced artifacts such as aliasing. We demonstrate the generality of our framework across a range of implicit representations, including analytic functions, MFAs, and INRs, and show that it produces smooth, coherent critical point trajectories. By enabling feature tracking directly on continuous representations, our method supports a new class of feature-driven visualization workflows centered on implicit models.