2D Gum-Net: Unsupervised Dense Elastic Image Registration via Symmetric Correlation and Spectral Pooling

Dense non-rigid registration, the recovery of a pixel-wise deformation field that warps one image into geometric agreement with another, is fundamental across medical imaging, structural biology, and correspondence estimation, yet learning-based approaches face a severe bottleneck: ground-truth deformation fields are notoriously difficult to obtain for real data. We present 2D Gum-Net, an end-to-end architecture that predicts dense elastic deformation fields fully unsupervised. We transfer the Geometric Unsupervised Matching Network (Gum-Net), originally developed for 3D subtomogram alignment in cryo-electron tomography, to 2D elastic registration: all three modules are redesigned in 2D, and the rigid transformation head is replaced by a Dense Spatial Transformer that regresses a sparse control-point grid, lifted to a full-resolution field $\mathbf{\Phi}$ by bicubic interpolation. Downsampling uses DCT spectral pooling, implemented as frozen-basis matrix multiplications ($\approx 50\times$ GPU speedup), preserving the fine, quasi-periodic structure that max and average pooling destroy. We evaluate on contact-based fingerprints, a challenging testbed for this task: locally self-similar ridge patterns make correspondence maximally ambiguous, skin pressed against a sensor generates authentic large-displacement elastic distortion, and minutiae act as precise landmarks through which an independent matcher externally verifies the topological correctness of predicted fields. Trained purely on synthetic impressions and evaluated zero-shot on real benchmarks, our method improves structural correlation by up to 15.7% on FVC2004 and raises genuine match scores under an independent matcher while leaving imposter scores essentially unchanged. This suggests that the predicted deformations restore true structure without fabricating correspondences.