A method to determine the detector locations of the cone-beam projection of the balls’ centers

Abstract

In geometric calibration of cone-beam computed tomography (CBCT), sphere-like objects such as balls are widely imaged, the positioning information of which is obtained to determine the unknown geometric parameters. In this process, the accuracy of the detector location of CB projection of the center of the ball, which we call the center projection, is very important, since geometric calibration is sensitive to errors in the positioning information. Currently in almost all the geometric calibration using balls, the center projection is invariably estimated by the center of the support of the projection or the centroid of the intensity values inside the support approximately. Clackdoyle's work indicates that the center projection is not always at the center of the support or the centroid of the intensity values inside, and has given a quantitative analysis of the maximum errors in evaluating the center projection by the centroid. In this paper, an exact method is proposed to calculate the center projection, utilizing both the detector location of the ellipse center and the two axis lengths of the ellipse. Numerical simulation results have demonstrated the precision and the robustness of the proposed method. Finally there are some comments on this work with non-uniform density balls, as well as the effect by the error occurred in the evaluation for the location of the orthogonal projection of the cone vertex onto the detector.

Measurement of the X-ray computed tomography instrument geometry by minimization of reprojection errors—Implementation on simulated data

“…Correctly determining the location of the center projection from the imaged sphere in the radiograph is not trivial and has been an ongoing topic of research [10][11][12]. In the absence of other objects in the field of view, a sphere is projected onto the detector as an elliptical disk [11].…”

“…The longest path through a perfect sphere is the path that contains the sphere center; therefore, the projection point of the sphere center P is theoretically the position on the projected elliptical disk with a minimum intensity, i.e. highest X-ray attenuation [12].…”

“…Deng et al [12] propose an enhanced estimate of the sphere center projection coordinates P by applying a correction to the fit ellipse center along the ellipse major axis in the direction of the principal point. The correction is calculated from the lengths of the major and minor axes of the fit ellipse major and minor , respectively, as shown in equation (10).…”

Measurement of the X-ray computed tomography instrument geometry by minimization of reprojection errors—Implementation on experimental data

“…Within each acquired radiographic image, the coordinates of each projected sphere center ( obs , obs ) are estimated from the center of an ellipse fitted to the edge of the projected sphere. An enhanced method for estimating the center projection coordinates is presented in [6] and consists of applying a correction to the ellipse center using the lengths of the major and minor axes of the fit ellipse.…”

Enhanced dimensional measurement by fast determination and compensation of geometrical misalignments of X-ray computed tomography instruments

“…The CT instrument geometry is determined by least-squares minimization of reprojection errors [6]. For each radiograph of the reference object = 1,2, … , N at its respective rotation position of the stage, the observed center projection coordinates ( obs, , obs, ) of spheres = 1,2, … , M are determined by way of a dedicated image analysis procedure [7]. A separate ray-tracing algorithm provides a corresponding set of modelled sphere center projection coordinates ( mod , mod ) given an initial set of the seven instrument geometrical parameters and six "nuisance" parameters describing the position and orientation of the reference object in the instrument.…”