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

A pose of a rigid object is usually regarded as a rigid transformation, described by a translation and a rotation. In this article, we define a pose as a distinguishable static state of the considered object, and show that the usual identification of the pose space with the space of rigid transformations is abusive, as it is not adapted to objects with proper symmetries. Based solely on geometric considerations, we propose a frame-invariant metric on the pose space, valid for any physical object, and requiring no arbitrary tuning. This distance can be evaluated efficiently thanks to a representation of poses within a low dimension Euclidean space, and enables to perform efficient neighborhood queries such as radius searches or k-nearest neighbor searches within a large set of poses using off-the-shelf methods. We lastly solve the problems of projection from the Euclidean space onto the pose space, and of pose averaging for this metric. 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.