FMS${}^2$: Unified Flow Matching for Segmentation and Synthesis of Thin Structures

Segmenting thin structures such as infrastructure cracks and anatomical vessels is hampered by fragile connectivity, expensive annotations, and poor cross-domain generalization. We propose FMS${}^2$, a flow-matching framework that addresses all three with two modules, SegFlow for image$\to$mask segmentation and SynFlow for mask$\to$image synthesis. SegFlow is an encoder–decoder that casts segmentation as deterministic transport, regressing a velocity field on interpolated image–mask states and integrating from the image itself. Two exact identities describe this sampler: the velocity objective is equivalent to time-weighted endpoint regression on interpolated reference states, and the $K$-step output equals the one-step prediction plus a sum of signed corrections. Together with margin and residual assumptions, these identities characterize possible pixel-label changes and give sufficient conditions for reconnecting two components or recovering the decoded binary mask exactly. SynFlow renders mask-conditioned images using multi-scale mask injection and boundary gating, while class-conditional mask generation and controlled propagation vary sparsity, width, and branching. The same endpoint view applies to SynFlow, where extrapolation drifts by time-weighted trajectory acceleration, giving a conditional stability criterion for a fixed structural image readout already aligned with the conditioning mask. Across five crack and vessel benchmarks, SegFlow improves both overlap and topological metrics over nine baselines: mean IoU $0.601$ versus $0.546$ (+10.1%) and Betti matching error $51.524$ versus $73.856$ (-30.2%), each against the strongest baseline on that metric. Synthetic augmentation improves SegFlow when it receives 25% of the real training labels, recovering most of the full-label performance on two crack subsets and increasing source-only cross-domain IoU by $0.109$ on average. We release code, model checkpoints, and 10k crack and 1k vessel image–mask pairs.