Estimating the location of a tracked tool’s tip relative to its Dynamic Reference Frame (DRF) and localizing a
specific point in a tracking system’s coordinate frame are fundamental tasks in image-guided procedures. The
most common approach to estimating these values is by pivoting a tool around a fixed point. The transformations
from the tracking system’s frame to the tool's DRF are the input. The output is the translation from the DRF
to the tool’s tip and the translation from the tracker’s frame to the pivoting point. While the input and output
are unique, there are multiple mathematical formulations for performing this estimation task. The question is,
are these formulations equivalent in terms of precision and accuracy? In this work we empirically evaluate three
common formulations, a geometry based sphere fitting formulation and two algebraic formulations. In addition
we evaluate robust variants of these formulations using the RANSAC framework. Our evaluation shows that the
algebraic formulations yield estimates that are more precise and accurate than the sphere fitting formulation.
Using the Vicra optical tracking system from Northern Digital Inc., we observed that the algebraic approaches
have a mean(std) precision of 0.25(0.11)mm localizing the pivoting point relative to the tracked DRF, and yield a
fiducial registration error with a mean(std) 0.15(0.08)mm when registering a precisely constructed divot phantom
to the localized points in the tracking system's frame. The sphere fitting formulation yielded less precise and
accurate results with a mean(std) of 0.35(0.21)mm for precision and 0.25(0.14)mm for accuracy. The robust
versions of these formulations yield similar results even when the data is contaminated with 30% outliers.