Rigorous Geometric Self-Calibrating Bundle Adjustment for a Dual Fluoroscopic Imaging System

High-speed dual fluoroscopy is a noninvasive imaging technology for three-dimensional skeletal kinematics analysis that finds numerous biomechanical applications. Accurate reconstruction of bone translations and rotations from dual-fluoroscopic data requires accurate calibration of the imaging geometry and the many imaging distortions that corrupt the data. Direct linear transformation methods are commonly applied for performing calibration using a two-step process that suffers from a number of potential shortcomings including that each X-ray source and corresponding camera must be calibrated separately. Consequently, the true imaging set-up and the constraints it presents are not incorporated during calibration. A method to overcome such drawbacks is the single-step self-calibrating bundle adjustment method. This procedure, based on the collinearity principle augmented with imaging distortion models and geometric constraints, has been developed and is reported herein. Its efficacy is shown with a carefully controlled experiment comprising 300 image pairs with 48 507 image points. Application of all geometric constraints and a 31 parameter distortion model resulted in up to 91% improvement in terms of precision (model fit) and up to 71% improvement in terms of 3-D point reconstruction accuracy (0.3–0.4 mm). The accuracy of distance reconstruction was improved from
$0.3\pm 2.0$
mm to
$0.2 \pm 1.1$
mm and angle reconstruction accuracy was improved from
$-0.03\pm 0.55^{\circ}$
to
$0.01\pm 0.06^{\circ}$
. Such positioning accuracy will allow for the accurate quantification of in vivo arthrokinematics crucial for skeletal biomechanics investigations.