# Deep Medial Fields

arXiv: https://arxiv.org/abs/2106.03804 · HTML: https://mlsanity.com/paper/2106.03804/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 49 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 25 of 25)

- [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)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [UNISURF: Unifying Neural Implicit Surfaces and Radiance Fields for Multi-View Reconstruction](https://mlsanity.com/paper/2104.10078/) (2021)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Local Implicit Grid Representations for 3D Scenes](https://mlsanity.com/paper/2003.08981/) (2020)
- [Deep Local Shapes: Learning Local SDF Priors for Detailed 3D Reconstruction](https://mlsanity.com/paper/2003.10983/) (2020)
- [Neural Unsigned Distance Fields for Implicit Function Learning](https://mlsanity.com/paper/2010.13938/) (2020)
- [Learning Shape Templates with Structured Implicit Functions](https://mlsanity.com/paper/1904.06447/) (2019)
- [MetaSDF: Meta-learning Signed Distance Functions](https://mlsanity.com/paper/2006.09662/) (2020)
- [Texture Fields: Learning Texture Representations in Function Space](https://mlsanity.com/paper/1905.07259/) (2019)
- [DIST: Rendering Deep Implicit Signed Distance Function with Differentiable Sphere Tracing](https://mlsanity.com/paper/1911.13225/) (2019)
- [DeRF: Decomposed Radiance Fields](https://mlsanity.com/paper/2011.12490/) (2020)
- [Neural-Pull: Learning Signed Distance Functions from Point Clouds by Learning to Pull Space onto Surfaces](https://mlsanity.com/paper/2011.13495/) (2020)
- [LSMAT Least Squares Medial Axis Transform](https://mlsanity.com/paper/2010.05066/) (2020)

## Built on by (most cited first, 13 of 13)

- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](https://mlsanity.com/paper/2112.05124/) (2021)
- [3D Gaussian as a New Era: A Survey](https://mlsanity.com/paper/2402.07181/) (2024)
- [Latent Partition Implicit with Surface Codes for 3D Representation](https://mlsanity.com/paper/2207.08631/) (2022)
- [3D Concept Grounding on Neural Fields](https://mlsanity.com/paper/2207.06403/) (2022)
- [GEM3D: GEnerative Medial Abstractions for 3D Shape Synthesis](https://mlsanity.com/paper/2402.16994/) (2024)
- [Coverage Axis++: Efficient Inner Point Selection for 3D Shape Skeletonization](https://mlsanity.com/paper/2401.12946/) (2024)
- [NIFT: Neural Interaction Field and Template for Object Manipulation](https://mlsanity.com/paper/2210.10992/) (2022)
- [CrossSDF: 3D Reconstruction of Thin Structures From Cross-Sections](https://mlsanity.com/paper/2412.04120/) (2024)
- [Deep Medial Voxels: Learned Medial Axis Approximations for Anatomical Shape Modeling](https://mlsanity.com/paper/2403.11790/) (2024)
- [CDM: Contact Diffusion Model for Multi-Contact Point Localization](https://mlsanity.com/paper/2502.06109/) (2025)
- [MIND: Material Interface Generation from UDFs for Non-Manifold Surface Reconstruction](https://mlsanity.com/paper/2506.02938/) (2025)
- [Medial Axis Aware Learning of Signed Distance Functions](https://mlsanity.com/paper/2604.16512/) (2026)

## Abstract (arXiv)

Implicit representations of geometry, such as occupancy fields or signed distance fields (SDF), have recently re-gained popularity in encoding 3D solid shape in a functional form. In this work, we introduce medial fields: a field function derived from the medial axis transform (MAT) that makes available information about the underlying 3D geometry that is immediately useful for a number of downstream tasks. In particular, the medial field encodes the local thickness of a 3D shape, and enables O(1) projection of a query point onto the medial axis. To construct the medial field we require nothing but the SDF of the shape itself, thus allowing its straightforward incorporation in any application that relies on signed distance fields. Working in unison with the O(1) surface projection supported by the SDF, the medial field opens the door for an entirely new set of efficient, shape-aware operations on implicit representations. We present three such applications, including a modification to sphere tracing that renders implicit representations with better convergence properties, a fast construction method for memory-efficient rigid-body collision proxies, and an efficient approximation of ambient occlusion that remains stable with respect to viewpoint variations.