Estimation of a set of parameters that describe the geometry of the cone-beam computed tomography system plays an important role in the geometrical calibration. In the calibration process, the helical phantom consisting of spherical markers arranged on a helical trajectory has been widely applied. To directly determine the complete nine geometric calibration parameters using the helical phantom, we propose a novel calibration method using explicit mathematical formulae. In the method, the geometric characteristics of the helix are utilized by converting the helix to delicately designed parallelograms. Then, the projections of the intersection points of the diagonals of parallelograms are obtained and used to identify the projections of the phantom coordinate axes, which are integrated into the calibration algorithm to calculate the geometric parameters. Our method makes full use of the markers, and has the property that flexible selection of the phantom coordinate system, which can deal with degenerate cases. To validate this method, simulation studies with various system geometries, different number of markers and different noise types are performed. A comparison of our proposed method with projection matrix method is also presented. The results show that our method can provide comparable accuracy of parameter estimation with the projection matrix method. The estimation of piercing points is even better using our method, which shows a factor of 8 × error reduction. The small animal studies also verify the accuracy and robustness of the proposed method.

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