Auto-calibration of cone beam geometries from arbitrary rotating markers using a vector geometry formulation of projection matrices.

An efficient method for the determination of the projection geometry of cone beam micro computed tomography systems based on two or more fiducial markers of unknown position within the field of view is derived. By employing the projection matrix formalism commonly used in computer graphics, a very clear presentation of the resulting self consistent calibration problem can be given relating the sought-for matrix to observable parameters of the markers' projections. Both an easy to implement solution procedure for both the unknown projection matrix and the marker assembly as well as the mapping from projection matrices to real space positions and orientations of source and detector relative to the rotational axis are provided. The separate treatment of the calibration problem in terms of projection matrices on the one hand and the independent transformation to a more intuitive geometry representation on the other hand proves to be very helpful with respect to the discussion of the ambiguities occurring in reference-free calibration. In particular, a link between methods based on knowledge on the sample and those based on knowledge solely on the detector geometry can be drawn. This further provides another intuitive view on the often reported difficulty in the estimation of the detector tilt towards the rotational axis. A simulation study considering 10randomly generated cone beam imaging configurations and fiducial marker distributions within a range of typical scenarios is performed in order to assess the stability of the proposed technique. An experimental example supports the simulation results.