# A Level Set Theory for Neural Implicit Evolution under Explicit Flows

arXiv: https://arxiv.org/abs/2204.07159 · HTML: https://mlsanity.com/paper/2204.07159/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 72 references that are in the map.
- First posted on arXiv 14 Apr 2022.

## Builds on (most cited first, 30 of 39)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Volume Rendering of Neural Implicit Surfaces](https://mlsanity.com/paper/2106.12052/) (2021)
- [Accelerating 3D Deep Learning with PyTorch3D](https://mlsanity.com/paper/2007.08501/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [pi-GAN: Periodic Implicit Generative Adversarial Networks for 3D-Aware Image Synthesis](https://mlsanity.com/paper/2012.00926/) (2020)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [UNISURF: Unifying Neural Implicit Surfaces and Radiance Fields for Multi-View Reconstruction](https://mlsanity.com/paper/2104.10078/) (2021)
- [NICE-SLAM: Neural Implicit Scalable Encoding for SLAM](https://mlsanity.com/paper/2112.12130/) (2021)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [iMAP: Implicit Mapping and Positioning in Real-Time](https://mlsanity.com/paper/2103.12352/) (2021)
- [Deep Marching Tetrahedra: a Hybrid Representation for High-Resolution 3D Shape Synthesis](https://mlsanity.com/paper/2111.04276/) (2021)
- [Modular Primitives for High-Performance Differentiable Rendering](https://mlsanity.com/paper/2011.03277/) (2020)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [Soft Rasterizer: A Differentiable Renderer for Image-based 3D Reasoning](https://mlsanity.com/paper/1904.01786/) (2019)
- [Extracting Triangular 3D Models, Materials, and Lighting From Images](https://mlsanity.com/paper/2111.12503/) (2021)
- [Advances in Neural Rendering](https://mlsanity.com/paper/2111.05849/) (2021)
- [Neural Point-Based Graphics](https://mlsanity.com/paper/1906.08240/) (2019)
- [Deep Local Shapes: Learning Local SDF Priors for Detailed 3D Reconstruction](https://mlsanity.com/paper/2003.10983/) (2020)
- [Neural RGB-D Surface Reconstruction](https://mlsanity.com/paper/2104.04532/) (2021)
- [MetaSDF: Meta-learning Signed Distance Functions](https://mlsanity.com/paper/2006.09662/) (2020)
- [DIST: Rendering Deep Implicit Signed Distance Function with Differentiable Sphere Tracing](https://mlsanity.com/paper/1911.13225/) (2019)
- [SDFDiff: Differentiable Rendering of Signed Distance Fields for 3D Shape Optimization](https://mlsanity.com/paper/1912.07109/) (2019)
- [Modulated Periodic Activations for Generalizable Local Functional Representations](https://mlsanity.com/paper/2104.03960/) (2021)
- [A-SDF: Learning Disentangled Signed Distance Functions for Articulated Shape Representation](https://mlsanity.com/paper/2104.07645/) (2021)
- [MeshSDF: Differentiable Iso-Surface Extraction](https://mlsanity.com/paper/2006.03997/) (2020)

## Built on by (most cited first, 30 of 42)

- [Flexible Isosurface Extraction for Gradient-Based Mesh Optimization](https://mlsanity.com/paper/2308.05371/) (2023)
- [DrapeNet: Garment Generation and Self-Supervised Draping](https://mlsanity.com/paper/2211.11277/) (2022)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [NeuManifold: Neural Watertight Manifold Reconstruction with Efficient and High-Quality Rendering Support](https://mlsanity.com/paper/2305.17134/) (2023)
- [TetSphere Splatting: Representing High-Quality Geometry with Lagrangian Volumetric Meshes](https://mlsanity.com/paper/2405.20283/) (2024)
- [Tuning the Frequencies: Robust Training for Sinusoidal Neural Networks](https://mlsanity.com/paper/2407.21121/) (2024)
- [Neural Geometry Processing via Spherical Neural Surfaces](https://mlsanity.com/paper/2407.07755/) (2024)
- [NEAT: Distilling 3D Wireframes from Neural Attraction Fields](https://mlsanity.com/paper/2307.10206/) (2023)
- [DMesh++: An Efficient Differentiable Mesh for Complex Shapes](https://mlsanity.com/paper/2412.16776/) (2024)
- [Level-S2fM: Structure from Motion on Neural Level Set of Implicit Surfaces](https://mlsanity.com/paper/2211.12018/) (2022)
- [DMesh: A Differentiable Mesh Representation](https://mlsanity.com/paper/2404.13445/) (2024)
- [HotSpot: Signed Distance Function Optimization with an Asymptotically Sufficient Condition](https://mlsanity.com/paper/2411.14628/) (2024)
- [NeRFs: The Search for the Best 3D Representation](https://mlsanity.com/paper/2308.02751/) (2023)
- [A Simple Approach to Differentiable Rendering of SDFs](https://mlsanity.com/paper/2405.08733/) (2024)
- [Birth and Death of a Rose](https://mlsanity.com/paper/2412.05278/) (2024)
- [4Deform: Neural Surface Deformation for Robust Shape Interpolation](https://mlsanity.com/paper/2502.20208/) (2025)
- [A Theory of Topological Derivatives for Inverse Rendering of Geometry](https://mlsanity.com/paper/2308.09865/) (2023)
- [PHRIT: Parametric Hand Representation with Implicit Template](https://mlsanity.com/paper/2309.14916/) (2023)
- [Implicit-ARAP: Efficient Handle-Guided Neural Field Deformation via Local Patch Meshing](https://mlsanity.com/paper/2405.12895/) (2024)
- [Understanding Sinusoidal Neural Networks](https://mlsanity.com/paper/2212.01833/) (2022)
- [Neural Implicit Shape Editing using Boundary Sensitivity](https://mlsanity.com/paper/2304.12951/) (2023)
- [Sparse 3D Reconstruction via Object-Centric Ray Sampling](https://mlsanity.com/paper/2309.03008/) (2023)
- [McGrids: Monte Carlo-Driven Adaptive Grids for Iso-Surface Extraction](https://mlsanity.com/paper/2409.06710/) (2024)
- [Geometric implicit neural representations for signed distance functions](https://mlsanity.com/paper/2511.07206/) (2025)
- [Text-guided Image-and-Shape Editing and Generation: A Short Survey](https://mlsanity.com/paper/2304.09244/) (2023)
- [PBIR-NIE: Glossy Object Capture under Non-Distant Lighting](https://mlsanity.com/paper/2408.06878/) (2024)
- [FLOWING: Implicit Neural Flows for Structure-Preserving Morphing](https://mlsanity.com/paper/2510.09537/) (2025)
- [Self-Supervised Dual Contouring](https://mlsanity.com/paper/2405.18131/) (2024)
- [SDFs from Unoriented Point Clouds using Neural Variational Heat Distances](https://mlsanity.com/paper/2504.11212/) (2025)
- [Low-Rank Adaptation of Neural Fields](https://mlsanity.com/paper/2504.15933/) (2025)

## Abstract (arXiv)

Coordinate-based neural networks parameterizing implicit surfaces have emerged as efficient representations of geometry. They effectively act as parametric level sets with the zero-level set defining the surface of interest. We present a framework that allows applying deformation operations defined for triangle meshes onto such implicit surfaces. Several of these operations can be viewed as energy-minimization problems that induce an instantaneous flow field on the explicit surface. Our method uses the flow field to deform parametric implicit surfaces by extending the classical theory of level sets. We also derive a consolidated view for existing methods on differentiable surface extraction and rendering, by formalizing connections to the level-set theory. We show that these methods drift from the theory and that our approach exhibits improvements for applications like surface smoothing, mean-curvature flow, inverse rendering and user-defined editing on implicit geometry.