| Summary: | 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.
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