Measurement of 3D displacement fields from few tomographic projections

The present paper aims at providing 3D volume images of a deformed specimen based on i) a full 3D image describing the reference state as obtained e.g., from conventional computed tomography and ii) the 3D displacement field accounting for its motion. The displacement field, which is described by much fewer degrees of freedom than the specimen volume itself, is here proposed to be determined from very few projections. The reduction in number of needed projections may be larger than two orders of magnitude. In the proposed approach, the displacement field is described over an unstructured mesh composed of tetrahedra with linear shape functions. The mesh is based on the reconstructed reference volume so that it provides a faithful and accurate description of the specimen, including its boundary. Nodal displacements are determined from the minimization of the quadratic difference between the computed projections of the deformed configuration and the acquired projections (radiographs) for the selected orientations. Well-posedness of the problem requires the number of kinematic unknowns to be small. However, in cases where the geometry is complex, the displacement field may call for many parameters. To deal with such conflicting demands it is proposed to use a regularization based on the mechanical modeling of the displacement field using a linear elastic description.