# Defining the Pose of any 3D Rigid Object and an Associated Distance

arXiv: https://arxiv.org/abs/1612.04631 · HTML: https://mlsanity.com/paper/1612.04631/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2016.

## Built on by (most cited first, 18 of 18)

- [Deep Regression on Manifolds: A 3D Rotation Case Study](https://mlsanity.com/paper/2103.16317/) (2021)
- [ES6D: A Computation Efficient and Symmetry-Aware 6D Pose Regression Framework](https://mlsanity.com/paper/2204.01080/) (2022)
- [Chordal Averaging on Flag Manifolds and Its Applications](https://mlsanity.com/paper/2303.13501/) (2023)
- [Transferring Experience from Simulation to the Real World for Precise Pick-And-Place Tasks in Highly Cluttered Scenes](https://mlsanity.com/paper/2101.04781/) (2021)
- [Investigations on Output Parameterizations of Neural Networks for Single Shot 6D Object Pose Estimation](https://mlsanity.com/paper/2104.07528/) (2021)
- [A Representation Separation Perspective to Correspondences-free Unsupervised 3D Point Cloud Registration](https://mlsanity.com/paper/2203.13239/) (2022)
- [SD-Net: Symmetric-Aware Keypoint Prediction and Domain Adaptation for 6D Pose Estimation In Bin-picking Scenarios](https://mlsanity.com/paper/2403.09317/) (2024)
- [BOP-Distrib: Revisiting 6D Pose Estimation Benchmarks for Better Evaluation under Visual Ambiguities](https://mlsanity.com/paper/2408.17297/) (2024)
- [Learning Implicit Probability Distribution Functions for Symmetric Orientation Estimation from RGB Images Without Pose Labels](https://mlsanity.com/paper/2211.11394/) (2022)
- [Alignist: CAD-Informed Orientation Distribution Estimation by Fusing Shape and Correspondences](https://mlsanity.com/paper/2409.06683/) (2024)
- [Extended to Reality: Prompt Injection in 3D Environments](https://mlsanity.com/paper/2602.07104/) (2026)
- [Towards Symmetry-sensitive Pose Estimation: A Rotation Representation for Symmetric Object Classes](https://mlsanity.com/paper/2604.18208/) (2026)
- [Multistream ValidNet: Improving 6D Object Pose Estimation by Automatic Multistream Validation](https://mlsanity.com/paper/2106.06684/) (2021)
- [Precise Object Placement with Pose Distance Estimations for Different Objects and Grippers](https://mlsanity.com/paper/2110.00992/) (2021)
- [Dynamic multi feature-class Gaussian process models](https://mlsanity.com/paper/2112.04495/) (2021)
- [Pseudo-Trilateral Adversarial Training for Domain Adaptive Traversability Prediction](https://mlsanity.com/paper/2306.14370/) (2023)
- [NormNet: Scale Normalization for 6D Pose Estimation in Stacked Scenarios](https://mlsanity.com/paper/2311.09269/) (2023)
- [AnchorPose for Geometry-Aware MOF Assembly through Meso-Grained Pose Generation](https://mlsanity.com/paper/2610.05843/) (2026)

## Abstract (arXiv)

The pose of a rigid object is usually regarded as a rigid transformation, described by a translation and a rotation. However, equating the pose space with the space of rigid transformations is in general abusive, as it does not account for objects with proper symmetries -- which are common among man-made objects.In this article, we define pose as a distinguishable static state of an object, and equate a pose with a set of rigid transformations. Based solely on geometric considerations, we propose a frame-invariant metric on the space of possible poses, valid for any physical rigid object, and requiring no arbitrary tuning. This distance can be evaluated efficiently using a representation of poses within an Euclidean space of at most 12 dimensions depending on the object's symmetries. This makes it possible to efficiently perform neighborhood queries such as radius searches or k-nearest neighbor searches within a large set of poses using off-the-shelf methods. Pose averaging considering this metric can similarly be performed easily, using a projection function from the Euclidean space onto the pose space. The practical value of those theoretical developments is illustrated with an application of pose estimation of instances of a 3D rigid object given an input depth map, via a Mean Shift procedure.