Functional Kolmogorov-Arnold Network: A Hilbert-Space Perspective on Spatial Representation Learning for Medical Image Segmentation

Medical image segmentation requires spatial transformations that remain effective across heterogeneous image statistics and boundary conditions. We introduce FunKAN, a KAN-inspired operator that acts directly on spatial feature maps: analytical Hermite basis maps are evaluated on an input-conditioned deformed grid, scored against learned channel-wise templates, mixed by a softmax over modes, and aggregated across channels. A Hilbert-space functional approximation result motivates the function-space viewpoint, while the exact finite FunKAN operator is treated separately and is not claimed to inherit a universal-approximation theorem. We embed FunKAN in a U-shaped architecture (U-FunKAN) and evaluate it under the validation-only selection protocol on BUSI, GlaS, and CVC-ClinicDB. The frozen model is tested once per seed after three-seed training. U-FunKAN improves mean test IoU over a same-protocol U-Net by 6.35 percentage points on BUSI and 1.85 points on CVC-ClinicDB, while GlaS performance is comparable, with U-FunKAN 0.50 points lower. U-FunKAN uses fewer parameters than the retrained U-Net (15.95 M vs. 31.03 M) but is substantially slower in measured GPU latency, so we do not claim universal computational efficiency. The results support dataset-dependent gains from functional spatial parameterization rather than uniform dominance.