International Journal of Computer Vision

, Volume 126, Issue 6, pp 571–596 | Cite as

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

  • Romain Brégier
  • Frédéric Devernay
  • Laetitia Leyrit
  • James L. Crowley


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 to 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 a 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.


Pose 3D rigid object Symmetry Distance Metric Average Rotation \(\textit{SE}(3)\) \(\textit{SO}(3)\) Object recognition 



We would like to thank the anonymous reviewers for their insightful comments and suggestions that greatly helped to improve this article. Some of our illustrations are based on the following mesh models: “Stanford bunny”, from the Stanford University Computer Graphics Laboratory; “Eiffel Tower” created by Pranav Panchal; and “Şamdan 2” (candlestick), from Metin N. Those were respectively available online at, and the GrabCAD and 3D Warehouse plateforms on May 2016.


Funding is provided by Association Nationale de la Recherche et de la Technologie (CIFRE 2014/0173).


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Authors and Affiliations

  1. 1.SiléaneSaint-ÉtienneFrance
  2. 2.Inria Grenoble Rhône-AlpesUniversité Grenoble Alpes (UGA)GrenobleFrance

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