Join Tensors: on 3D-to-3D alignment of Dynamic Sets

Lior Wolf Amnon Shashua and Yoni Wexler

School of Computer Science and Engineering, Center for Automation Research

The Hebrew University, University of Maryland

Jerusalem 91904, Israel College park MD

This paper introduces a family of 4x4x4 tensors,

referred to as ``join tensors'' or Jtensors for 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 $t_1,t_2,t_3$ 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.