Abstract

Generative diffusion models have emerged as a class of powerful techniques for various imaging applications, including but not limited to synthesis, reconstruction, and segmen- tation. Despite their success, current generative models pose two key limitations. First, they primarily rely on image intensity and texture information, with limited attention to underlying object geometry. As a result, they do not guarantee geometric or topological consistency during the generation process, which is a crucial requirement for high-stakes domains such as computational anatomy, biology, and robotics, where preserving object structure is critical. Second, existing models fail to explicitly learn or represent shape changes in the generative process. Such deformation dynamics remain occluded within network parameters; hence leaving the transformation process uninterpretable and phys- ically uninformed. To address these challenges, we introduce IGG (Image Generation informed by Geodesic dynamics), a novel framework that integrates topology-preserving geodesic principles into the diffusion-based generative process. In contrast to conventional methods that operate in image intensity space, IGG learns and synthesizes diverse sam- ples within geodesic deformation spaces, where geometric object changes are learned as smooth and invertible smooth mappings from a given template/source image. Specifically, IGG employs a two-stage architecture: (i) a geodesic-informed image registration (GIR) module that directly encodes geodesic paths of image deformations into a compact latent space, and (ii) a latent geodesic diffusion (LGD) model that captures the distribution of these deformation representations, conditioned on a template image and/or text prompts. We validate IGG on the datasets of plant growth and brain MRI scans. Experimental results demonstrate that IGG outperforms state-of-the-art image generation and editing models, producing realistic, high-quality images with preserved topology and fewer arti- facts. Furthermore, when used for data augmentation in downstream segmentation tasks, IGG substantially improves segmentation accuracy, particularly in low-data regimes. Our code is publicly available at https://github.com/nellie689/IGG