Data-driven deformable image registration methods predominantly rely on operations that process grid-like inputs. However, applying deformable transformations to an image results in a warped space that deviates from a rigid grid structure. Consequently, data-driven approaches with sequential deformations have to apply grid resampling operations between each deformation step. While artifacts caused by resampling are negligible in high-resolution images, the resampling of sparse, high-dimensional feature grids introduces errors that affect the deformation modeling process. Taking inspiration from Lagrangian reference frames of deformation fields, our work introduces a novel paradigm for data-driven deformable image registration that utilizes geometric deep-learning principles to model deformations without grid requirements. Specifically, we model image features as a set of nodes that freely move in Euclidean space, update their coordinates under graph operations, and dynamically readjust their local neighborhoods. We employ this formulation to construct a multi-resolution deformable registration model, where deformation layers iteratively refine the overall transformation at each resolution without intermediate resampling operations on the feature grids. We investigate our method's ability to fully deformably capture large deformations across a number of medical imaging registration tasks. In particular, we apply our approach (GeoReg) to the registration of inter-subject brain MR images and inhale-exhale lung CT images, showing on par performance with the current state-of-the-art methods. We believe our contribution open up avenues of research to reduce the black-box nature of current learned registration paradigms by explicitly modeling the transformation within the architecture.

The prevailing deep learning-based methods of predicting cardiac segmentation involve reconstructed magnetic resonance (MR) images. The heavy dependency of segmentation approaches on image quality significantly limits the acceleration rate in fast MR reconstruction. Moreover, the practice of treating reconstruction and segmentation as separate sequential processes leads to artifact generation and information loss in the intermediate stage. These issues pose a great risk to achieving high-quality outcomes. To leverage the redundant k-space information overlooked in this dual-step pipeline, we introduce a novel approach to directly deriving segmentations from sparse k-space samples using a transformer (DiSK). DiSK operates by globally extracting latent features from 2D+time k-space data with attention blocks and subsequently predicting the segmentation label of query points. We evaluate our model under various acceleration factors (ranging from 4 to 64) and compare against two image-based segmentation baselines. Our model consistently outperforms the baselines in Dice and Hausdorff distances across foreground classes for all presented sampling rates.