On the Construction of Topology-Preserving Deformation Fields

In this paper, we investigate a new method to enforce topology preservation on deformation fields. The method is composed of two steps. The first one consists in correcting the gradient vector fields 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 Karaçali and Davatzikos, proposes a new approach based on interval analysis. The second one aims to reconstruct the deformation, given its full set of discrete gradient vectors. The problem is phrased as a functional minimization problem on the convex subset
K
of the Hilbert space
V
. The 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
. Experimental results demonstrate the efficiency of the method.