Two-Parameter Flow Map Learning for Continuous-Time Diffeomorphic Image Registration

Diffeomorphic image registration is central to medical im-age analysis, enabling anatomically consistent alignment across subjects.Most learning-based diffeomorphic methods model autonomous ODEs(ordinary differential equations) by parameterizing a stationary velocityfield and recovering deformations via scaling-and-squaring. While non-autonomous ODEs with time-dependent velocities increase expressive-ness, existing approaches rely on numerical integration to implicitly en-force flow structure that entangles model expressiveness with discretiza-tion accuracy. We propose a framework to directly learn the continuous-time solution of a non-autonomous ODE formulated as a two-parameterflow map. By enforcing cocycle consistency, a fundamental structuralproperty of time-varying flows, we learn the flow maps without timediscretization and velocity integration during training. The frameworkrecovers diffeomorphic mappings at inference using a small number ofcompositions. Our proposed framework seamlessly incorporates stan-dard registration backbones and improves alignment accuracy consis-tently across nine datasets while preserving diffeomorphic structure. No-tably, the proposed method achieves an average Dice improvement of2.1% on brain MRI benchmarks, a 12% TRE reduction on lung CT,and a 2.6% Dice gain on cardiac MRI and ultrasound datasets (https://mattkia.github.io/TPFMDIR/).