Most of the norms used in the field of digital image (and volume) correlation to register two images (or volumes) lead to ill-posed problems. One of the frequent solutions is to enforce a restricted kinematics requiring a compromise between the richness of the solution (i.e., the spatial resolution) and the measurement uncertainty. An alternative route is to use a displacement norm that permits to alleviate this compromise by the means of a mechanical regularization used when the gray levels do not give enough information. It is then possible to compute a displacement vector for each pixel or voxel, inducing lower residuals (in terms of experimental data) while decreasing the noise sensitivity. The resolution performance of these different approaches is discussed, and compared for the analysis of a tensile test on a cast iron specimen based on a pair of tomographic images. As representative reconstructed volumes lead to a large number of degrees of freedom, a dedicated GPU computational strategy has been developed and implemented.