CLAIRE-MG: Efficient Diffeomorphic Image Registration Using Multigrid-Preconditioned Newton–Krylov Methods

Abstract

In this thesis, I describe significant enhancements to CLAIRE (Constrained Large Deformation Diffeomorphic Image Registration), a framework formulating diffeomorphic image registration as a PDE-constrained optimization problem. Image registration aims to find a spatial transformation mapping points between images. Diffeomorphic image registration specifically restricts these transformations to diffeomorphisms, i.e., transformations that are smooth maps that possess a smooth inverse. In principle, image registration is an infinite-dimensional, nonlinear, ill-posed inverse problem, leading to ill-conditioned, large-scale inversion operators. This makes its effective solution a significant mathematical and computational challenge. CLAIRE parameterizes the sought-after diffeomorphic map via a stationary velocity field. Within CLAIRE, governing equations consist of transport equations for image intensities. The regularization operator for the inversion variable is a biharmonic differential operator. As a result, the reduced space optimality condition for the inversion variable is described by a biharmonic equation. Optimization in CLAIRE employs a Newton–Krylov method. The central contribution of my thesis is the design and analysis of a sophisticated multigrid preconditioning strategy for the reduced space Hessian. Explicitly forming and storing the Hessian matrix is computationally intractable for problems of this scale, necessitating matrix-free numerical schemes. This imperative drove my extensive exploration and meticulous implementation of various multigrid techniques, including the basic two-grid method, and the more advanced V-cycle and W-cycle multigrid algorithms. A prototype implementation was realized in a two-dimensional setting (ambient space). I conducted a comprehensive analysis of the proposed numerical scheme's convergence behavior, evaluating algorithms on both synthetic (smooth) and real-world (non-smooth) imaging data. I compared their convergence rates, time to solution, and computational complexity across varying mesh sizes and regularization parameter choices. The results indicate the proposed scheme achieves mesh-independent convergence rates, a significant accomplishment as method efficiency is maintained even as image resolution increases. While this independence did not extend to the regularization parameter, its practical benefits are substantial. Furthermore, I meticulously assessed my designed multigrid scheme's efficacy against various pre-existing CLAIRE preconditioning schemes. My results conclusively demonstrate the proposed multigrid preconditioner is highly effective and strongly competitive, often superior to, established methods, thereby substantially elevating CLAIRE's performance for challenging diffeomorphic image registration