Cylindrical Flow Matching for Complex-Valued Medical Image Synthesis

While deep generative models have achieved remarkable success in MRI reconstruction, most operate strictly on magnitude images, discarding critical phase information. This magnitude-only paradigm hinders the development and validation of complex-valued neural networks for phase-sensitive diagnostic imaging. Current methods attempting to synthesize fully complex data suffer from a fundamental topological mismatch: they naively embed the complex signal into a flat Euclidean space ($\mathbb{R}^2$). This ignores the intrinsic circular topology of the phase ($S^1$), leading to destructive interference, magnitude attenuation, and non-physical artifacts. To address this, we propose Cylindrical Flow Matching, a geometry-aware generative framework that explicitly models amplitude on $\mathbb{R}^+$ and phase on $S^1$. By constructing probability paths along geodesics on a cylindrical manifold, our approach natively resolves phase discontinuities while preserving the computational efficiency of Optimal Transport. We demonstrate that Cylindrical Flow Matching ensures topological validity and physical signal plausibility, providing a reliable synthetic baseline for safety-critical medical downstream tasks.