  This work is concerned with fan- and cone-beam computed tomography with
circular source trajectory, where the reconstruction inverse problem requires
an accurate knowledge of source, detector and rotational axis relative
positions and orientations. We address this additional inverse problem as a
preceding step of the reconstruction process directly from the acquired
projections. In the cone-beam case, we present a method that estimates both the
detector shift (orthogonal to both focal and rotational axes) and the in-plane
detector rotation (over the focal axis) based on the variable projection
optimization approach. In addition and for the fan-beam case, two new
strategies with low computational cost are presented to estimate the detector
shift based on a fan-beam symmetry condition. The methods are validated with
simulated and experimental industrial tomographic data with code examples
available for both fan- and cone-beam geometries.
