Sensor calibration with unknown correspondence: Solving AX=XB using Euclidean-group invariants

The AX = XB sensor calibration problem must often be solved in image guided therapy systems, such as those used in robotic surgical procedures. In this problem, A, X, and B are homogeneous transformations with A and B acquired from sensor measurements and X being the unknown. It has been known for decades that this problem is solvable for X when a set of exactly measured A's and B's, in a priori correspondence, is given. However, in practical problems, the data streams containing the A' and B's will be asynchronous and may contain gaps (i.e., the correspondence is unknown, or does not exist, for the sensor measurements) and temporal registration is required. For the AX = XB problem, an exact solution can be found when four independent invariant quantities exist between two pairs of A's and B's. We formally define these invariants, reviewing and elaborating results from classical screw theory. We then illustrate how they can be used, with sensor data from multiple sources that contain unknown or missing correspondences, to provide a solution for X.