This paper introduces a multilevel method for image registration based on a variational formulation. The method combines consistent discretization of the optimization problem with a multigrid solution of the linear system that evolves in a Gauss-Newton iteration. The discretization is h-elliptic, independent of parameter choice, allowing for a simple multigrid implementation. A multilevel continuation technique is used to overcome potential large nonlinearities and speed up computation. The method is demonstrated on a realistic, highly nonlinear registration problem. Image registration involves finding a transformation that aligns a template image with a reference image. The problem is ill-posed and requires regularization. The authors use a variational approach to minimize a joint energy function consisting of a distance measure and a regularizer. The distance measure is the sum of squared differences (SSD), and the regularizer is an elastic potential. The registration problem is discretized using staggered grids, which are well-suited for stable discretizations of partial differential equations. The discretization is shown to be h-elliptic, making it amenable to multigrid methods. The authors propose a multilevel inexact Gauss-Newton scheme combined with a multigrid solver for the linearized systems. This approach is efficient and modular, allowing for easy adaptation to different distance measures and regularizers. The method is tested on a three-dimensional MRI scan registration problem. The results show a significant reduction in image distance, with the algorithm being efficient and effective on modest computational hardware. The method is also shown to be robust to different regularization parameters, with the optimal choice left to an expert. The numerical experiments demonstrate the effectiveness of the multigrid solver, with a significant reduction in the residual after a single multigrid cycle. The method is efficient and scalable, making it suitable for large-scale problems.This paper introduces a multilevel method for image registration based on a variational formulation. The method combines consistent discretization of the optimization problem with a multigrid solution of the linear system that evolves in a Gauss-Newton iteration. The discretization is h-elliptic, independent of parameter choice, allowing for a simple multigrid implementation. A multilevel continuation technique is used to overcome potential large nonlinearities and speed up computation. The method is demonstrated on a realistic, highly nonlinear registration problem. Image registration involves finding a transformation that aligns a template image with a reference image. The problem is ill-posed and requires regularization. The authors use a variational approach to minimize a joint energy function consisting of a distance measure and a regularizer. The distance measure is the sum of squared differences (SSD), and the regularizer is an elastic potential. The registration problem is discretized using staggered grids, which are well-suited for stable discretizations of partial differential equations. The discretization is shown to be h-elliptic, making it amenable to multigrid methods. The authors propose a multilevel inexact Gauss-Newton scheme combined with a multigrid solver for the linearized systems. This approach is efficient and modular, allowing for easy adaptation to different distance measures and regularizers. The method is tested on a three-dimensional MRI scan registration problem. The results show a significant reduction in image distance, with the algorithm being efficient and effective on modest computational hardware. The method is also shown to be robust to different regularization parameters, with the optimal choice left to an expert. The numerical experiments demonstrate the effectiveness of the multigrid solver, with a significant reduction in the residual after a single multigrid cycle. The method is efficient and scalable, making it suitable for large-scale problems.