In this paper, we investigate a new method to enforce topology preservation on two/three-dimensional deformation
fields for non-parametric registration problems involving large-magnitude deformations. The method is
composed of two steps. The first one consists in correcting the gradient vector field of the deformation at the
discrete level, in order to fulfill a set of conditions ensuring topology preservation in the continuous domain
after bilinear interpolation. This part, although related to prior works by Kara¸cali and Davatzikos (Estimating
Topology Preserving and Smooth Displacement Fields, B. Kara¸cali and C. Davatzikos, IEEE Transactions on
Medical Imaging, vol. 23(7), 2004), proposes a new approach based on interval analysis and provides, unlike
their method, uniqueness of the correction parameter α at each node of the grid, which is more consistent with
the continuous setting. The second one aims to reconstruct the deformation, given its full set of discrete gradient
vector field. The problem is phrased as a functional minimization problem on a convex subset K of an Hilbert
space V . Existence and uniqueness of the solution of the problem are established, and the use of Lagrange's
multipliers allows to obtain the variational formulation of the problem on the Hilbert space V . The discretization
of the problem by the finite element method does not require the use of numerical schemes to approximate the
partial derivatives of the deformation components and leads to solve two/three uncoupled sparse linear subsystems.
Experimental results in brain mapping and comparisons with existing methods demonstrate the efficiency
and the competitiveness of the method.