Optimal calibration marker mesh for 2D X-ray sensors in 3D reconstruction

Image intensifiers suffer from distortions due to magnetic fields. In order to use this X-ray projections images for computer-assisted medical interventions, image intensifiers need to be calibrated. Opaque markers are often used for the correction of the image distortion and the estimation of the acquisition geometry parameters. Information under the markers is then lost. In this work, we consider the calibration of image intensifiers in the framework of 3D reconstruction from several 2D X-ray projections. In this context, new schemes of marker distributions are proposed for 2D X-ray sensor calibration. They are based on efficient sampling conditions of the parallel-beam X-ray transform when the detector and source trajectory is restricted to a circle around the measured object. Efficient sampling are essentially subset of standard sampling in this situation. The idea is simply to exploit the data redundancy of standard sampling and to replace some holes of efficient schemes by markers. Optimal location of markers in the sparse efficient sampling geometry can thus be found. In this case, the markers can stay on the sensor during the measurement with--theoretically--no loss of information (when the signal-to-noise ratio is large). Even if the theory is based on the parallel-beam X-ray transform, numerical experiments on both simulated and real data are shown in the case of weakly divergent beam geometry. We show that the 3D reconstruction from simulated data with interlaced markers is essentially the same as those obtained from data with no marker. We show that efficient Fourier interpolation formulas based on optimal sparse sampling schemes can be used to recover the information hidden by the markers.