Nicolas Barnafi, Gabriel N. Gatica, Daniel E. Hurtado:
Primal and mixed finite element methods for deformable image registration problems
Deformable image registration (DIR) represent a powerful computational method for image analysis, with promising applications in the diagnosis of human disease. Despite being widely used in the medical imaging community, the mathematical and numerical analysis of DIR methods remain understudied. Further, recent applications of DIR include the quantification of mechanical quantities apart from the aligning transformation, which justifies the development of novel DIR formulations where the accuracy and convergence of fields other than the aligning transformation can be studied. In this work we propose and analyze a primal, mixed and augmented formulations for the DIR problem, together with their finite-element discretization schemes for their numerical solution. The DIR variational problem is equivalent to the linear elasticity problem with a nonlinear source term that depends on the unknown field. Fixed point arguments and small data assumptions are employed to derive the well-posedness of both the continuous and discrete schemes for the usual primal and mixed variational formulations, as well as for an augmented version of the later. In particular, continuous piecewise linear elements for the displacement in the case of the primal method, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) for the stress together with piecewise constants (resp. continuous piecewise linear) for the displacement when using the mixed approach (resp. its augmented version), constitute feasible choices that guarantee the stability of the associated Galerkin systems. A-priori error estimates derived by using Strang-type Lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of the method.