R2Net: Efficient and flexible diffeomorphic image registration using Lipschitz continuous residual networks

Classical diffeomorphic image registration methods, while being accurate, face the challenges of high computational costs. Deep learning based approaches provide a fast alternative to address these issues; however, most existing deep solutions either lose the good property of diffeomorphism or have limited flexibility to capture large deformations, under the assumption that deformations are driven by stationary velocity fields (SVFs). Also, the adopted squaring and scaling technique for integrating SVFs is time- and memory-consuming, hindering deep methods from handling large image volumes. In this paper, we present an unsupervised diffeomorphic image registration framework, which uses deep residual networks (ResNets) as numerical approximations of the underlying continuous diffeomorphic setting governed by ordinary differential equations, which is parameterized by either SVFs or time-varying (non-stationary) velocity fields. This flexible parameterization in our Residual Registration Network (R2Net) not only provides the model's ability to capture large deformation but also reduces the time and memory cost when integrating velocity fields for deformation generation. Also, we introduce a Lipschitz continuity constraint into the ResNet block to help achieve diffeomorphic deformations. To enhance the ability of our model for handling images with large volume sizes, we employ a hierarchical extension with a multi-phase learning strategy to solve the image registration task in a coarse-to-fine fashion. We demonstrate our models on four 3D image registration tasks with a wide range of anatomies, including brain MRIs, cine cardiac MRIs, and lung CT scans. Compared to classical methods SyN and diffeomorphic VoxelMorph, our models achieve comparable or better registration accuracy with much smoother deformations. Our source code is available online at https://github.com/ankitajoshi15/R2Net.