Cone-beam reconstruction from n-sin trajectories with transversely-truncated projections

International audienceIn cone-beam tomography, we define the n-sin source trajectory as having n periods of a sinusoid traced on an imaginary cylinder enclosing the object. A 2-sin is commonly known as a saddle, and it is known that the convex hull of a saddle is the same as the union of all of its chords. The convex hull of a closed trajectory is the Tuy region, where cone-beam reconstructionis possible if there are no truncated projections. However, with truncated projections, the method of differentiated backprojection and Hilbert inversion can be applied along a chord if the chordis visible (not truncated) in the projections. Here, we consider a particular transaxial truncation which prevents chords from always being visible, but we establish that the more powerful method of M-lines can be applied to ensure reconstruction in the reduced field-of-view. The 3-sin, on the other hand, has a Tuy region which is not filled by its chords, and we do not have any cone-beam theory to determine if reconstruction is possible with transverse reconstruction. In our preliminary numerical experiment, the 3-sinseemed to perform equally well as the 2-sin trajectory even though there were no chords passing through the slice we examined. We tentatively suggest that there might be other, yet unknown theory that explains why 3-sin reconstruction is possible with the specified transaxial truncation. We believe that these results on transverse truncation and reconstruction from 2-sin and 3-sin trajectories are new