Cone-beam calibration using small balls: Centroids or ellipse centers?

When small dense balls are used in geometric calibration of x-ray cone-beam (CB) scanners, it is vital to accurately identify the CB projections of the centers of the balls, called the “center projections.” The detector location of the center projection is usually estimated either from the center of the elliptical projection, or from the centroid of the intensity values inside the ellipse. The ellipse center is easily seen to differ from the center projection, and probably for this reason the centroid has been the preferred estimate. We have demonstrated mathematically that the centroid also differs from the center projection, and we have shown using numerical methods that the centroid lies much closer to the ellipse center than to the center projection. We have established that the centroid is only 20% more accurate than the ellipse center for a very wide range of CB geometries, including all situations that would be encountered in practice. Since both estimates present a systematic bias, it is important to evaluate the magnitude of this geometric error, especially for the purposes of CB calibration. We have proved that this geometric error cannot be ascertained from the elliptical projection alone, and we have selected the source-to-detector distance f(also called the SID) as the easiest parameter to estimate independently. We demonstrate that the absolute ellipse-center error can be estimated by eA/πf where e and A are the ellipse eccentricity and ellipse area of the CB projection of the ball. The centroid error is 4/5 of the ellipse error. Finally we point out that physical effects such as beam hardening and Compton scatter, which typically generate cupping or capping artifacts, can be treated as undistorted projections of balls of non-uniform density, and we extend our analysis to such cases. The centroid error remains well within 70% to 90% of the ellipse-center error for density variations under 50%. In conclusion, the preprocessing r- quired to obtain good centroid estimates in the presence of noisy data is considerably more than for finding the ellipse center, whereas the geometric benefit is relatively small.