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The pose of a rigid object is usually regarded as a rigid transformation,described by a translation and a rotation. However, equating the pose spacewith the space of rigid transformations is in general abusive, as it does notaccount for objects with proper symmetries -- which are common among man-madeobjects.In this article, we define pose as a distinguishable static state of anobject, and equate a pose with a set of rigid transformations. Based solely ongeometric considerations, we propose a frame-invariant metric on the space ofpossible poses, valid for any physical rigid object, and requiring no arbitrarytuning. This distance can be evaluated efficiently using a representation ofposes within an Euclidean space of at most 12 dimensions depending on theobject's symmetries. This makes it possible to efficiently perform neighborhoodqueries such as radius searches or k-nearest neighbor searches within a largeset of poses using off-the-shelf methods. Pose averaging considering thismetric can similarly be performed easily, using a projection function from theEuclidean space onto the pose space. The practical value of those theoreticaldevelopments is illustrated with an application of pose estimation of instancesof a 3D rigid object given an input depth map, via a Mean Shift procedure.

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