  Correct fusion of data from two sensors is not possible without an accurate
estimate of their relative pose, which can be determined through the process of
extrinsic calibration. When two or more sensors are capable of producing their
own egomotion estimates (i.e., measurements of their trajectories through an
environment), the 'hand-eye' formulation of extrinsic calibration can be
employed. In this paper, we extend our recent work on a convex optimization
approach for hand-eye calibration to the case where one of the sensors cannot
observe the scale of its translational motion (e.g., a monocular camera
observing an unmapped environment). We prove that our technique is able to
provide a certifiably globally optimal solution to both the known- and
unknown-scale variants of hand-eye calibration, provided that the measurement
noise is bounded. Herein, we focus on the theoretical aspects of the problem,
show the tightness and stability of our solution, and demonstrate the
optimality and speed of our algorithm through experiments with synthetic data.
