On the construction of Topology-preserving Deformation Fields.

Carole Le Guyader - LMI

In this talk, we investigate a new method to enforce topology preservation on two-dimensional 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 (Estimating Topology Preserving and Smooth Displacement Fields, IEEE Transactions on Medical Imaging, vol. 23(7), 2004), proposes a new approach based on interval analysis.
- the second one aims to reconstruct the deformation, given its full set of discrete gradient vector fields. 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. Experimental results demonstrate the efficiency of the method. Comparisons and comments on the major differences between our model and the one proposed by Karaçali and Davatzikos are also provided.