Deformation Field Correction to Preserve Topology for Image Registration

In this paper, the author addresses the issue of designing a theoretically well-motivated and computationally efficient method ensuring topology preservation on image-registration-related deformation fields. The model is motivated by a mathematical characterization of topology preservation for a deformation field mapping two subsets of Z2, namely, positivity of the four approximations to the Jacobian determinant of the deformation on a square patch. The first step of the proposed algorithm thus consists in correcting the gradient vector field of the deformation at the discrete level in order to fulfill this positivity condition. Once this step is achieved, it thus remains to reconstruct the deformation field, given its full set of discrete gradient vectors. The author propose to decompose the reconstruction problem into independent problems of smaller dimensions, yielding a natural parallelization of the computations and enabling us to reduce drastically the computational time (up to 80 in some applications). For each subdomain, a functional minimization problem under Lagrange interpolation constraints is introduced and its well-posedness is studied: existence/uniqueness of the solution, characterization of the solution, convergence of the method when the number of data increases to infinity, discretization with the Finite Element Method and discussion on the properties of the matrix involved in the linear system. Numerical simulations based on OpenMP parallelization and MKL multi-threading demonstrating the ability of the model to handle large deformations (contrary to classical methods) and the interest of having decomposed the problem into smaller ones are provided.