Browsing by Author "Barnafi, Nicolas"

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    Adaptive Mesh Refinement in Deformable Image Registration: A Posteriori Error Estimates for Primal and Mixed Formulations
    (2021) Barnafi, Nicolas; Gatica, Gabriel N.; Hurtado, Daniel E.; Miranda, Willian; Ruiz-Baier, Ricardo
    Deformable image registration (DIR) is a popular technique for the alignment of digital images, with highly relevant applications in medical image analysis. However, the numerical solution of DIR problems can be very challenging in computational terms, as the improvement of the DIR solution typically involves a uniform refinement of the underlying domain discretization that exponentially increases the number of degrees of freedom. In this work, we develop adaptive mesh refinement schemes particularly designed for the finite-element solution of DIR problems. We start by deriving residual-based a posteriori error estimators for the primal and mixed formulations of the DIR problem and show that they are reliable and efficient. Based on these error estimators, we implement adaptive mesh-refinement schemes into a finite-element code to register images. We assess the numerical performance of the proposed adaptive scheme on smooth synthetic images, where numerical convergence is verified. We further show that the adaptive mesh refinement scheme can deliver solutions to DIR problems with significant reductions in the number of degrees of freedom without compromising the accuracy of the solution. We also confirm that the adaptive scheme proposed for the mixed DIR formulation successfully handles volume-constrained registration problems, providing optimal convergence in analytic examples. To demonstrate the applicability of the method, we perform adaptive DIR on medical brain images and binary images and study how image noise affects the proposed refinement schemes.
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    New primal and dual-mixed finite element methods for stable image registration with singular regularization
    (2021) Barnafi, Nicolas; Gatica, Gabriel N.; Hurtado, Daniel E.; Miranda, Willian; Ruiz-Baier, Ricardo
    This work introduces and analyzes new primal and dual-mixed finite element methods for deformable image registration, in which the regularizer has a nontrivial kernel, and constructed under minimal assumptions of the registration model: Lipschitz continuity of the similarity measure and ellipticity of the regularizer on the orthogonal complement of its kernel. The aforementioned singularity of the regularizer suggests to modify the original model by incorporating the additional degrees of freedom arising from its kernel, thus granting ellipticity of the former on the whole solution space. In this way, we are able to prove well-posedness of the resulting extended primal and dual-mixed continuous formulations, as well as of the associated Galerkin schemes. A priori error estimates and corresponding rates of convergence are also established for both discrete methods. Finally, we provide numerical examples confronting our formulations with the standard ones, which prove our finite element methods to be particularly more efficient on the registration of translations and rotations, in addition for the dual-mixed approach to be much more suitable for the quasi-incompressible case, and all the above without losing the flexibility to solve problems arising from more realistic scenarios such as the image registration of the human brain.
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    Primal and Mixed Finite Element Methods for Deformable Image Registration Problems
    (2018) Barnafi, Nicolas; Gatica, Gabriel N.; Hurtado Sepúlveda, Daniel