A cheap and easy method for 3D C-arm reconstruction using elliptic curves

For quantitative C-arm fluoroscopy, we had earlier proposed a unified mathematical framework to tackle the issues of pose estimation, correspondence and reconstruction, without the use of external trackers. The method used randomly distributed unknown points in the imaging volume, either naturally present or induced by placing beads on the patient. These points were then inputted to an algorithm that computed the 3D reconstruction. The algorithm had an 8° region of convergence, which in general could be considered sufficient for most applications. Here, we extend the earlier algorithm to make it more robust and clinically acceptable. We propose the use of a circle/ellipse, naturally found in many images. We show that the projection of elliptic curves constrain 5 out of the 6 degrees of freedom of the C-arm pose. To completely recover the true C-arm pose, we use constraints in the form of point correspondences between the images. We provide an algorithm to easily obtain a virtual correspondence across all the images and show that two correspondences can recover the true pose 95% of the time when the seeds employed are separated by a distance of 40 mm. or greater. Phantom experiments across three images indicate a pose estimation accuracy of 1.7° using an ellipse and two sufficiently separated point correspondences. Average execution time in this case is 130 seconds. The method appears to be sufficiently accurate for clinical applications and does not require any significant modification of clinical protocol.