14 February 2012 On the construction of topology-preserving deformations
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Abstract
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.
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Dominique Apprato, Dominique Apprato, Christian Gout, Christian Gout, Carole Le Guyader, Carole Le Guyader, } "On the construction of topology-preserving deformations", Proc. SPIE 8314, Medical Imaging 2012: Image Processing, 83141Q (14 February 2012); doi: 10.1117/12.905567; https://doi.org/10.1117/12.905567
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