# Implicit-PDF: Non-Parametric Representation of Probability Distributions on the Rotation Manifold

arXiv: https://arxiv.org/abs/2106.05965 · HTML: https://mlsanity.com/paper/2106.05965/

## Key facts

- Cited by 65 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 53 references that are in the map.
- First posted on arXiv 10 Jun 2021.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [T-LESS: An RGB-D Dataset for 6D Pose Estimation of Texture-less Objects](https://mlsanity.com/paper/1701.05498/) (2017)
- [Implicit 3D Orientation Learning for 6D Object Detection from RGB Images](https://mlsanity.com/paper/1902.01275/) (2019)
- [Modelling Uncertainty in Deep Learning for Camera Relocalization](https://mlsanity.com/paper/1509.05909/) (2015)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Render for CNN: Viewpoint Estimation in Images Using CNNs Trained with Rendered 3D Model Views](https://mlsanity.com/paper/1505.05641/) (2015)
- [An Analysis of SVD for Deep Rotation Estimation](https://mlsanity.com/paper/2006.14616/) (2020)
- [Discovery of Latent 3D Keypoints via End-to-end Geometric Reasoning](https://mlsanity.com/paper/1807.03146/) (2018)
- [Explaining the Ambiguity of Object Detection and 6D Pose From Visual Data](https://mlsanity.com/paper/1812.00287/) (2018)
- [Viewpoints and Keypoints](https://mlsanity.com/paper/1411.6067/) (2014)
- [Deep Directional Statistics: Pose Estimation with Uncertainty Quantification](https://mlsanity.com/paper/1805.03430/) (2018)
- [Deep Bingham Networks: Dealing with Uncertainty and Ambiguity in Pose Estimation](https://mlsanity.com/paper/2012.11002/) (2020)
- [Spherical Regression: Learning Viewpoints, Surface Normals and 3D Rotations on n-Spheres](https://mlsanity.com/paper/1904.05404/) (2019)
- [Probabilistic orientation estimation with matrix Fisher distributions](https://mlsanity.com/paper/2006.09740/) (2020)
- [On Object Symmetries and 6D Pose Estimation from Images](https://mlsanity.com/paper/1908.07640/) (2019)
- [A Mixed Classification-Regression Framework for 3D Pose Estimation from 2D Images](https://mlsanity.com/paper/1805.03225/) (2018)
- [Learning Orientation Distributions for Object Pose Estimation](https://mlsanity.com/paper/2007.01418/) (2020)
- [Pose Estimation for Objects with Rotational Symmetry](https://mlsanity.com/paper/1810.05780/) (2018)
- [Cross-Domain 3D Equivariant Image Embeddings](https://mlsanity.com/paper/1812.02716/) (2018)

## Built on by (most cited first, 30 of 65)

- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [UniDexGrasp: Universal Robotic Dexterous Grasping via Learning Diverse Proposal Generation and Goal-Conditioned Policy](https://mlsanity.com/paper/2303.00938/) (2023)
- [MagicPony: Learning Articulated 3D Animals in the Wild](https://mlsanity.com/paper/2211.12497/) (2022)
- [RelPose: Predicting Probabilistic Relative Rotation for Single Objects in the Wild](https://mlsanity.com/paper/2208.05963/) (2022)
- [RelPose++: Recovering 6D Poses from Sparse-view Observations](https://mlsanity.com/paper/2305.04926/) (2023)
- [GenPose: Generative Category-level Object Pose Estimation via Diffusion Models](https://mlsanity.com/paper/2306.10531/) (2023)
- [Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3D](https://mlsanity.com/paper/2401.12046/) (2024)
- [Shelving, Stacking, Hanging: Relational Pose Diffusion for Multi-modal Rearrangement](https://mlsanity.com/paper/2307.04751/) (2023)
- [MatchU: Matching Unseen Objects for 6D Pose Estimation from RGB-D Images](https://mlsanity.com/paper/2403.01517/) (2024)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)
- [Image to Sphere: Learning Equivariant Features for Efficient Pose Prediction](https://mlsanity.com/paper/2302.13926/) (2023)
- [SC6D: Symmetry-agnostic and Correspondence-free 6D Object Pose Estimation](https://mlsanity.com/paper/2208.02129/) (2022)
- [Confronting Ambiguity in 6D Object Pose Estimation via Score-Based Diffusion on SE(3)](https://mlsanity.com/paper/2305.15873/) (2023)
- [Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation](https://mlsanity.com/paper/2311.16199/) (2023)
- [FisherMatch: Semi-Supervised Rotation Regression via Entropy-based Filtering](https://mlsanity.com/paper/2203.15765/) (2022)
- [SpyroPose: SE(3) Pyramids for Object Pose Distribution Estimation](https://mlsanity.com/paper/2303.05308/) (2023)
- [Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modeling](https://mlsanity.com/paper/2304.03937/) (2023)
- [Rectified Point Flow: Generic Point Cloud Pose Estimation](https://mlsanity.com/paper/2506.05282/) (2025)
- [A Probabilistic Framework for Visual Localization in Ambiguous Scenes](https://mlsanity.com/paper/2301.02086/) (2023)
- [AnyPlace: Learning Generalized Object Placement for Robot Manipulation](https://mlsanity.com/paper/2502.04531/) (2025)
- [A Laplace-inspired Distribution on SO(3) for Probabilistic Rotation Estimation](https://mlsanity.com/paper/2303.01743/) (2023)
- [Ki-Pode: Keypoint-based Implicit Pose Distribution Estimation of Rigid Objects](https://mlsanity.com/paper/2209.09659/) (2022)
- [Learning Distributions on Manifolds with Free-Form Flows](https://mlsanity.com/paper/2312.09852/) (2023)
- [Normalizing Flows on the Product Space of SO(3) Manifolds for Probabilistic Human Pose Modeling](https://mlsanity.com/paper/2404.05675/) (2024)
- [Temporal Score Rescaling for Temperature Sampling in Diffusion and Flow Models](https://mlsanity.com/paper/2510.01184/) (2025)
- [Equivariant Single View Pose Prediction Via Induced and Restricted Representations](https://mlsanity.com/paper/2307.03704/) (2023)
- [Simulation-based Bayesian inference for multi-fingered robotic grasping](https://mlsanity.com/paper/2109.14275/) (2021)
- [Multi-view object pose estimation from correspondence distributions and epipolar geometry](https://mlsanity.com/paper/2210.00924/) (2022)
- [Towards Robust Probabilistic Modeling on SO(3) via Rotation Laplace Distribution](https://mlsanity.com/paper/2305.10465/) (2023)
- [Shape Completion with Prediction of Uncertain Regions](https://mlsanity.com/paper/2308.00377/) (2023)

## Abstract (arXiv)

Single image pose estimation is a fundamental problem in many vision and robotics tasks, and existing deep learning approaches suffer by not completely modeling and handling: i) uncertainty about the predictions, and ii) symmetric objects with multiple (sometimes infinite) correct poses. To this end, we introduce a method to estimate arbitrary, non-parametric distributions on SO(3). Our key idea is to represent the distributions implicitly, with a neural network that estimates the probability given the input image and a candidate pose. Grid sampling or gradient ascent can be used to find the most likely pose, but it is also possible to evaluate the probability at any pose, enabling reasoning about symmetries and uncertainty. This is the most general way of representing distributions on manifolds, and to showcase the rich expressive power, we introduce a dataset of challenging symmetric and nearly-symmetric objects. We require no supervision on pose uncertainty -- the model trains only with a single pose per example. Nonetheless, our implicit model is highly expressive to handle complex distributions over 3D poses, while still obtaining accurate pose estimation on standard non-ambiguous environments, achieving state-of-the-art performance on Pascal3D+ and ModelNet10-SO(3) benchmarks.