A new algorithm for determining 3D biplane imaging geometry: Theory and implementation

Biplane imaging is a primary method for visual and quantitative assessment of the vasculature. A key problem (called Imaging Geometry Determination problem or IGD for short) in this method is to determine the rotation matrix (R) and the translation vector (t) which relate the two coordinate systems. In this paper, we propose a new approach, called IG-Sieving, to calculate R and t using corresponding points in the two images. Our technique first generates an initial estimate of R and t from the gantry angles of the imaging system, and then optimizes them by solving an optimal-cell-search problem in a 6-D parametric space (three variables defining R plus the three variables of t). To efficiently find the optimal imaging geometry (IG) in 6-D, our approach divides the high dimensional search domain into a set of lower-dimensional regions (holding two variables constant at each optimization step), thereby reducing the optimal-cell-search problem to a set of optimization problems in 3D sub-spaces (one other variable is correlated). For each such sub-space, our approach first applies efficient computational geometry techniques to identify "possibly-feasible" IG's, and then uses a criterion we call fall-in-number to sieve out good IGs. We show that in a bounded number of optimization steps, a (possibly infinite) set of near optimal IGs (which are equally good) can be determined. Simulation results indicate that our method can reconstruct 3D points with average 3D center-of-mass errors of about 0.8cm for input image-data errors as high as 0.1cm, which is comparable to existing techniques. More importantly, our algorithm provides a novel insight into the geometric structure of the solution space, which could be exploited to significantly improve the accuracy of other biplane algorithms.