  We propose a minimal solution for the similarity registration (rigid pose and
scale) between two sets of 3D lines, and also between a set of co-planar points
and a set of 3D lines. The first problem is solved up to 8 discrete solutions
with a minimum of 2 line-line correspondences, while the second is solved up to
4 discrete solutions using 4 point-line correspondences. We use these
algorithms to perform the extrinsic calibration between a pose tracking sensor
and a 2D/3D ultrasound (US) curvilinear probe using a tracked needle as
calibration target. The needle is tracked as a 3D line, and is scanned by the
ultrasound as either a 3D line (3D US) or as a 2D point (2D US). Since the
scale factor that converts US scan units to metric coordinates is unknown, the
calibration is formulated as a similarity registration problem. We present
results with both synthetic and real data and show that the minimum solutions
outperform the correspondent non-minimal linear formulations.
