Join tensors: On 3D-to-3D alignment of dynamic sets

This paper introduces a family of 4 × 4 × 4 tensors, referred to as "join tensors" or Jtensorsfor short, which perform "3D to 3D" alignment between coordinate systems of sets of dynamic 3D points. 3D Configurations of points are obtained by a 3D measuring device (such as a structured light or laser range sensor, or a stereo rig) at times t₁, t₂, t₃ from different viewing positions in addition to the motion of the sensor the points are also allowed to move in space; each point can move along an arbitrary straight-line path - we refer to this situation as "dynamic". The problem is to recover the motion of the sensor given the 3D correspondences of the points over time. We introduce Jtensors to capture the problem described above. Three observations P, P′, P″ of a point measured at three time instants contribute a linear measurement to the Jtensor, regardless of whether the point has moved in space or has remained stationary while the sensor has changed position.