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.

中文翻译：

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定义任何 3D 刚性对象的姿势和相关距离

刚性物体的姿态通常被视为刚性变换，由平移和旋转来描述。然而，将姿势空间等同于刚性变换空间通常是滥用的，因为它没有考虑具有适当对称性的物体——这在人造物体中很常见。在本文中，我们将姿势定义为物体可区分的静态状态，并将姿势等同于一组刚性变换。仅基于几何考虑，我们提出了一个关于可能姿势空间的框架不变度量，对任何物理刚性对象都有效，并且不需要任意调整。根据对象的对称性，可以使用最多 12 维的欧几里得空间中的姿态表示来有效地评估该距离。这使得可以使用现成的方法在大量姿势中有效地执行邻域查询，例如半径搜索或 k-最近邻搜索。使用从欧几里得空间到姿势空间的投影函数，可以类似地轻松执行考虑到该度量的姿势平均。这些理论发展的实际价值通过对给定输入深度图的 3D 刚性对象实例的姿态估计的应用来说明，通过 Mean Shift 过程。