HyperGen: Controllable Pathological Anomaly Synthesis via Hyperbolic Semantic Manifold Learning

Medical anomaly synthesis has emerged as a promising strategy for alleviating the scarcity of annotated pathological data in clinical image analysis. Recent diffusion-based methods can generate anatomically realistic lesions, yet their latent representations remain largely unstructured, making it difficult to systematically control pathological characteristics and generate diverse abnormalities. To address this challenge, we propose **HyperGen**, a novel framework that reformulates anomaly synthesis as geometry-guided semantic navigation on a learned hyperbolic manifold. HyperGen first learns a lesion-centric hyperbolic embedding, where the radial coordinate is aligned with Hounsfield Unit (HU) density while the angular coordinates capture complementary morphological and textural variations. By manipulating the manifold in polar coordinates and projecting the navigated embedding back to the Euclidean space, HyperGen provides semantic conditioning for a diffusion model to generate diverse and anatomically consistent pathological images. Extensive experiments on lung CT demonstrate that HyperGen produces more realistic and structurally coherent lesions than strong diffusion-based baselines. Furthermore, the synthesized anomalies consistently improve downstream lesion segmentation, validating the effectiveness of geometry-guided semantic manifold learning for controllable medical anomaly generation.