# Representing 3D Shapes with Probabilistic Directed Distance Fields

arXiv: https://arxiv.org/abs/2112.05300 · HTML: https://mlsanity.com/paper/2112.05300/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 70 of its 96 references that are in the map.
- First posted on arXiv 10 Dec 2021.

## Builds on (most cited first, 30 of 70)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [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)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction](https://mlsanity.com/paper/2106.10689/) (2021)
- [pixelNeRF: Neural Radiance Fields from One or Few Images](https://mlsanity.com/paper/2012.02190/) (2020)
- [Volume Rendering of Neural Implicit Surfaces](https://mlsanity.com/paper/2106.12052/) (2021)
- [PlenOctrees for Real-time Rendering of Neural Radiance Fields](https://mlsanity.com/paper/2103.14024/) (2021)
- [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)
- [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)
- [GRAF: Generative Radiance Fields for 3D-Aware Image Synthesis](https://mlsanity.com/paper/2007.02442/) (2020)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [KiloNeRF: Speeding up Neural Radiance Fields with Thousands of Tiny MLPs](https://mlsanity.com/paper/2103.13744/) (2021)
- [Pixel2Mesh: Generating 3D Mesh Models from Single RGB Images](https://mlsanity.com/paper/1804.01654/) (2018)
- [UNISURF: Unifying Neural Implicit Surfaces and Radiance Fields for Multi-View Reconstruction](https://mlsanity.com/paper/2104.10078/) (2021)
- [3D-R2N2: A Unified Approach for Single and Multi-view 3D Object Reconstruction](https://mlsanity.com/paper/1604.00449/) (2016)
- [FastNeRF: High-Fidelity Neural Rendering at 200FPS](https://mlsanity.com/paper/2103.10380/) (2021)
- [Baking Neural Radiance Fields for Real-Time View Synthesis](https://mlsanity.com/paper/2103.14645/) (2021)
- [Neural 3D Mesh Renderer](https://mlsanity.com/paper/1711.07566/) (2017)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)

## Built on by (most cited first, 21 of 21)

- [Pointersect: Neural Rendering with Cloud-Ray Intersection](https://mlsanity.com/paper/2304.12390/) (2023)
- [DDF-HO: Hand-Held Object Reconstruction via Conditional Directed Distance Field](https://mlsanity.com/paper/2308.08231/) (2023)
- [RayDF: Neural Ray-surface Distance Fields with Multi-view Consistency](https://mlsanity.com/paper/2310.19629/) (2023)
- [The Sky's the Limit: Re-lightable Outdoor Scenes via a Sky-pixel Constrained Illumination Prior and Outside-In Visibility](https://mlsanity.com/paper/2311.16937/) (2023)
- [HORT: Monocular Hand-held Objects Reconstruction with Transformers](https://mlsanity.com/paper/2503.21313/) (2025)
- [FIRe: Fast Inverse Rendering using Directional and Signed Distance Functions](https://mlsanity.com/paper/2203.16284/) (2022)
- [NeuralODF: Learning Omnidirectional Distance Fields for 3D Shape Representation](https://mlsanity.com/paper/2206.05837/) (2022)
- [Unsigned Orthogonal Distance Fields: An Accurate Neural Implicit Representation for Diverse 3D Shapes](https://mlsanity.com/paper/2403.01414/) (2024)
- [Reconstructive Latent-Space Neural Radiance Fields for Efficient 3D Scene Representations](https://mlsanity.com/paper/2310.17880/) (2023)
- [CoFie: Learning Compact Neural Surface Representations with Coordinate Fields](https://mlsanity.com/paper/2406.03417/) (2024)
- [Adaptive Local Basis Functions for Shape Completion](https://mlsanity.com/paper/2307.08348/) (2023)
- [TransPoser: Transformer as an Optimizer for Joint Object Shape and Pose Estimation](https://mlsanity.com/paper/2303.13477/) (2023)
- [Neural directional distance field object representation for uni-directional path-traced rendering](https://mlsanity.com/paper/2306.16142/) (2023)
- [Mesh Compression with Quantized Neural Displacement Fields](https://mlsanity.com/paper/2504.01027/) (2025)
- [RayletDF: Raylet Distance Fields for Generalizable 3D Surface Reconstruction from Point Clouds or Gaussians](https://mlsanity.com/paper/2508.09830/) (2025)
- [Text2shape Deep Retrieval Model: Generating Initial Cases for Mechanical Part Redesign under the Context of Case-Based Reasoning](https://mlsanity.com/paper/2302.06341/) (2023)
- [Probabilistic Directed Distance Fields for Ray-Based Shape Representations](https://mlsanity.com/paper/2404.09081/) (2024)
- [COBRA -- COnfidence score Based on shape Regression Analysis for method-independent quality assessment of object pose estimation from single images](https://mlsanity.com/paper/2404.16471/) (2024)
- [Learning Scene-Level Signed Directional Distance Function with Ellipsoidal Priors and Neural Residuals](https://mlsanity.com/paper/2503.20066/) (2025)
- [Directed Distance Fields for Constant-Time Ray Queries on Gaussian Splatting](https://mlsanity.com/paper/2606.00817/) (2026)
- [Generalist Open-World Temporal Perception](https://mlsanity.com/paper/2609.06823/) (2026)

## Abstract (arXiv)

Differentiable rendering is an essential operation in modern vision, allowing inverse graphics approaches to 3D understanding to be utilized in modern machine learning frameworks. Explicit shape representations (voxels, point clouds, or meshes), while relatively easily rendered, often suffer from limited geometric fidelity or topological constraints. On the other hand, implicit representations (occupancy, distance, or radiance fields) preserve greater fidelity, but suffer from complex or inefficient rendering processes, limiting scalability. In this work, we endeavour to address both shortcomings with a novel shape representation that allows fast differentiable rendering within an implicit architecture. Building on implicit distance representations, we define Directed Distance Fields (DDFs), which map an oriented point (position and direction) to surface visibility and depth. Such a field can render a depth map with a single forward pass per pixel, enable differential surface geometry extraction (e.g., surface normals and curvatures) via network derivatives, be easily composed, and permit extraction of classical unsigned distance fields. Using probabilistic DDFs (PDDFs), we show how to model inherent discontinuities in the underlying field. Finally, we apply our method to fitting single shapes, unpaired 3D-aware generative image modelling, and single-image 3D reconstruction tasks, showcasing strong performance with simple architectural components via the versatility of our representation.