Nicolas Barnafi, Gabriel N. Gatica, Daniel E. Hurtado, Willian Miranda, Ricardo Ruiz-Baier:
A posteriori error estimates for primal and mixed finite element approximations of the deformable image registration problem
In this paper we consider primal and mixed variational formulations that have been recently proposed for the deformable image registration (DIR) problem, and derive reliable and efficient residual-based a posteriori error estimators suitable for adaptive mesh-refinement methods. Our theoretical results, being based on the a posteriori error analysis for the linear elasticity problem with Neumann boundary conditions, make use of the standard tools for that purpose. In particular, these include global inf-sup conditions, Helmholtz decompositions, and the approximation properties of the Raviart-Thomas and Clement interpolants for proving reliability. Localization techniques using bubble functions and inverse inequalities are employed to prove the corresponding efficiency estimates. The adaptive mesh-refinement schemes for the primal and mixed DIR formulations are implemented and tested using synthetic images as well as brain images, and the corresponding numerical results confirm the theoretical properties of the estimators.