Centers and Centroids of the Projection of a Sphere
Dr. Rolf Clackdoyle
Directeur de Recherche
Laboratoire Hubert Curien, Universite Jean Monnet
Tuesday, April 17, 2012

Consider the elliptical shadow on a screen of a small solid sphere
illuminated by an ideal point light source. The question is to find
the point inside the ellipse corresponding to the (projection of the)
center of the sphere, when the position of the light source is unknown.
The center of the ellipse is not the right answer. This problem arises
in geometric calibration of tomographic scanners where the objective
is to establish the mapping between points in space and their projected
detector locations by imaging small dense ball bearings (BBs). In this
case, an ideal X-ray source is assumed so the BB is not opaque and the
elliptical shadow exhibits intensity variations. A popular approach
is to use the centroid of these intensities to estimate the projected
center of the BB. It has recently been established that the centroid
is not the right answer either. A method is presented for quantifying
the errors incurred when the ellipse center estimate or the centroid
estimate is used, with an example from Dr. Tong Xu's X-ray laboratory.