# Factor Fields: A Unified Framework for Neural Fields and Beyond

arXiv: https://arxiv.org/abs/2302.01226 · HTML: https://mlsanity.com/paper/2302.01226/

## Key facts

- Cited by 58 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 73 references that are in the map.
- First posted on arXiv 2 Feb 2023.

## Builds on (most cited first, 30 of 59)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [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)
- [Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fields](https://mlsanity.com/paper/2111.12077/) (2021)
- [NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction](https://mlsanity.com/paper/2106.10689/) (2021)
- [Plenoxels: Radiance Fields without Neural Networks](https://mlsanity.com/paper/2112.05131/) (2021)
- [pixelNeRF: Neural Radiance Fields from One or Few Images](https://mlsanity.com/paper/2012.02190/) (2020)
- [D-NeRF: Neural Radiance Fields for Dynamic Scenes](https://mlsanity.com/paper/2011.13961/) (2020)
- [TensoRF: Tensorial Radiance Fields](https://mlsanity.com/paper/2203.09517/) (2022)
- [Efficient Geometry-aware 3D Generative Adversarial Networks](https://mlsanity.com/paper/2112.07945/) (2021)
- [Nerfies: Deformable Neural Radiance Fields](https://mlsanity.com/paper/2011.12948/) (2020)
- [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)
- [Direct Voxel Grid Optimization: Super-fast Convergence for Radiance Fields Reconstruction](https://mlsanity.com/paper/2111.11215/) (2021)
- [Neural Sparse Voxel Fields](https://mlsanity.com/paper/2007.11571/) (2020)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Neural Scene Flow Fields for Space-Time View Synthesis of Dynamic Scenes](https://mlsanity.com/paper/2011.13084/) (2020)
- [K-Planes: Explicit Radiance Fields in Space, Time, and Appearance](https://mlsanity.com/paper/2301.10241/) (2023)
- [MVSNeRF: Fast Generalizable Radiance Field Reconstruction from Multi-View Stereo](https://mlsanity.com/paper/2103.15595/) (2021)
- [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)
- [GRAF: Generative Radiance Fields for 3D-Aware Image Synthesis](https://mlsanity.com/paper/2007.02442/) (2020)
- [RegNeRF: Regularizing Neural Radiance Fields for View Synthesis from Sparse Inputs](https://mlsanity.com/paper/2112.00724/) (2021)
- [Google Scanned Objects: A High-Quality Dataset of 3D Scanned Household Items](https://mlsanity.com/paper/2204.11918/) (2022)
- [HexPlane: A Fast Representation for Dynamic Scenes](https://mlsanity.com/paper/2301.09632/) (2023)
- [Ref-NeRF: Structured View-Dependent Appearance for Neural Radiance Fields](https://mlsanity.com/paper/2112.03907/) (2021)

## Built on by (most cited first, 30 of 58)

- [Scaffold-GS: Structured 3D Gaussians for View-Adaptive Rendering](https://mlsanity.com/paper/2312.00109/) (2023)
- [Real-time Photorealistic Dynamic Scene Representation and Rendering with 4D Gaussian Splatting](https://mlsanity.com/paper/2310.10642/) (2023)
- [GS-LRM: Large Reconstruction Model for 3D Gaussian Splatting](https://mlsanity.com/paper/2404.19702/) (2024)
- [LVSM: A Large View Synthesis Model with Minimal 3D Inductive Bias](https://mlsanity.com/paper/2410.17242/) (2024)
- [Single-Stage Diffusion NeRF: A Unified Approach to 3D Generation and Reconstruction](https://mlsanity.com/paper/2304.06714/) (2023)
- [Periodic Vibration Gaussian: Dynamic Urban Scene Reconstruction and Real-time Rendering](https://mlsanity.com/paper/2311.18561/) (2023)
- [DynMF: Neural Motion Factorization for Real-time Dynamic View Synthesis with 3D Gaussian Splatting](https://mlsanity.com/paper/2312.00112/) (2023)
- [NeuRBF: A Neural Fields Representation with Adaptive Radial Basis Functions](https://mlsanity.com/paper/2309.15426/) (2023)
- [Tetra-NeRF: Representing Neural Radiance Fields Using Tetrahedra](https://mlsanity.com/paper/2304.09987/) (2023)
- [MesonGS: Post-training Compression of 3D Gaussians via Efficient Attribute Transformation](https://mlsanity.com/paper/2409.09756/) (2024)
- [LaRa: Efficient Large-Baseline Radiance Fields](https://mlsanity.com/paper/2407.04699/) (2024)
- [AutoSplat: Constrained Gaussian Splatting for Autonomous Driving Scene Reconstruction](https://mlsanity.com/paper/2407.02598/) (2024)
- [GenFusion: Closing the Loop between Reconstruction and Generation via Videos](https://mlsanity.com/paper/2503.21219/) (2025)
- [Binary Radiance Fields](https://mlsanity.com/paper/2306.07581/) (2023)
- [Strivec: Sparse Tri-Vector Radiance Fields](https://mlsanity.com/paper/2307.13226/) (2023)
- [NeuManifold: Neural Watertight Manifold Reconstruction with Efficient and High-Quality Rendering Support](https://mlsanity.com/paper/2305.17134/) (2023)
- [ZeroRF: Fast Sparse View 360deg Reconstruction with Zero Pretraining](https://mlsanity.com/paper/2312.09249/) (2023)
- [IntrinsicAvatar: Physically Based Inverse Rendering of Dynamic Humans from Monocular Videos via Explicit Ray Tracing](https://mlsanity.com/paper/2312.05210/) (2023)
- [4D Gaussian Splatting: Modeling Dynamic Scenes with Native 4D Primitives](https://mlsanity.com/paper/2412.20720/) (2024)
- [HOSC: A Periodic Activation Function for Preserving Sharp Features in Implicit Neural Representations](https://mlsanity.com/paper/2401.10967/) (2024)
- [JointRF: End-to-End Joint Optimization for Dynamic Neural Radiance Field Representation and Compression](https://mlsanity.com/paper/2405.14452/) (2024)
- [PanopticNeRF-360: Panoramic 3D-to-2D Label Transfer in Urban Scenes](https://mlsanity.com/paper/2309.10815/) (2023)
- [Efficient Dynamic-NeRF Based Volumetric Video Coding with Rate Distortion Optimization](https://mlsanity.com/paper/2402.01380/) (2024)
- [Curvature-Aware Training for Coordinate Networks](https://mlsanity.com/paper/2305.08552/) (2023)
- [Canonical Factors for Hybrid Neural Fields](https://mlsanity.com/paper/2308.15461/) (2023)
- [View-Consistent Hierarchical 3D Segmentation Using Ultrametric Feature Fields](https://mlsanity.com/paper/2405.19678/) (2024)
- [3D Gaussian Editing with A Single Image](https://mlsanity.com/paper/2408.07540/) (2024)
- [Optimizing Rank for High-Fidelity Implicit Neural Representations](https://mlsanity.com/paper/2512.14366/) (2025)
- [3D Motion Magnification: Visualizing Subtle Motions with Time Varying Radiance Fields](https://mlsanity.com/paper/2308.03757/) (2023)
- [Gear-NeRF: Free-Viewpoint Rendering and Tracking with Motion-aware Spatio-Temporal Sampling](https://mlsanity.com/paper/2406.03723/) (2024)

## Abstract (arXiv)

We present Factor Fields, a novel framework for modeling and representing signals. Factor Fields decomposes a signal into a product of factors, each represented by a classical or neural field representation which operates on transformed input coordinates. This decomposition results in a unified framework that accommodates several recent signal representations including NeRF, Plenoxels, EG3D, Instant-NGP, and TensoRF. Additionally, our framework allows for the creation of powerful new signal representations, such as the "Dictionary Field" (DiF) which is a second contribution of this paper. Our experiments show that DiF leads to improvements in approximation quality, compactness, and training time when compared to previous fast reconstruction methods. Experimentally, our representation achieves better image approximation quality on 2D image regression tasks, higher geometric quality when reconstructing 3D signed distance fields, and higher compactness for radiance field reconstruction tasks. Furthermore, DiF enables generalization to unseen images/3D scenes by sharing bases across signals during training which greatly benefits use cases such as image regression from sparse observations and few-shot radiance field reconstruction.