We introduce a fluid-based image augmentation method for medical image analysis. In contrast to existing methods, our framework generates anatomically meaningful images via interpolation from the geodesic subspace underlying given samples. Our approach consists of three steps: 1) given a source image and a set of target images, we construct a geodesic subspace using the Large Deformation Diffeomorphic Metric Mapping (LDDMM) model; 2) we sample transformations from the resulting geodesic subspace; 3) we obtain deformed images and segmentations via interpolation. Experiments on brain (LPBA) and knee (OAI) data illustrate the performance of our approach on two tasks: 1) data augmentation during training and testing for image segmentation; 2) one-shot learning for single atlas image segmentation. We demonstrate that our approach generates anatomically meaningful data and improves performance on these tasks over competing approaches.