We present a new method for computing the internal displacement fields associated with permanent deformations of 3D composite objects with complex internal structure for fields satisfying the small displacement gradient approximation of continuum mechanics. We compute the displacement fields from a sequence of 3D X-ray computed tomography (CT) images. By assuming that the intensity of the tomographic images represents a conserved property which is incompressible, we develop a constrained nonlinear regression model for estimation of the displacement field. Successive linear approximation is then employed and each linear subsidiary problem is solved using variational calculus. We approximate the resulting Euler-Lagrange equations using a finite set of linear equations using finite differencing methods. We solve these equations using a conjugate gradient algorithm in a multiresolution framework. We validate our method using pairs of synthetic images of plane shear flow. Finally, we determine the 3D displacement field in the interior of a cylindrical asphalt/aggregate core loaded to a state of permanent deformation.