Most approaches to 3D x-ray imaging geometry calibration use some well-defined calibration phantom containing point markers. The calibration aims at minimizing the so-called re-projection error, i.e., the error between the detected marker locations in the acquired projection image and the projected marker locations based on the phantom model and the current estimate of the imaging geometry. The phantoms that are being employed consist usually of spherical markers arranged in some spatial pattern. One widely used phantom type consists of spherical markers in a helical arrangement.
We present a framework that establishes a good intuitive understanding of the calibration problem, and allows to evaluate the performance of different phantom designs. It is based on a linear approximation of the error propagation between parameters of the imaging geometry, a projection alignment error (which is not identical to the re-projection error), and the "backprojection misalignment", which ultimately dictates 3D image quality. This methodology enables us to characterize the statistics of the parameters describing the imaging geometry, based on simple assumptions on "measurement noise", i.e., phantom and pre-processing accuracy. We also characterize the 3D misalignment in the backprojection (which is used in the 3D reconstruction), which directly impacts 3D image quality. In a comparison of different phantom designs -using backprojection misalignment as a metric- a "candy cane" phantom was found to give superior performance. The presented approach gives many useful intuitive insights into the calibration problem and its key properties. It can also be leveraged, e.g., for an easy implementation of a fast and robust calibration algorithm.