HamVision: Hamiltonian dynamics as inductive bias for medical image analysis

We present HamVision, a network architecture for medical image segmentation and classification organised around a shared, interpretable bottleneck. At every spatial location, the bottleneck produces three intermediate maps as natural outputs of its forward pass: a filtered representation of the input, its spatial derivative, and a non-negative saliency map. By analogy with the dynamics of a damped harmonic oscillator that motivates their construction, we refer to the three maps as position q, momentum p, and energy H=12(|q|²+|p|²). Because these maps are produced by the network rather than by dedicated supervised modules, the same bottleneck is reused by two task-specific heads: HamSeg gates encoder skip connections with the energy map and injects momentum at every decoder resolution, and HamCls uses a Phase-Space Spectral Pooling head that combines frequency-domain features of q and p with per-channel energy attention. We evaluate on 14 medical-imaging benchmarks (five segmentation, nine classification) spanning dermoscopy, ultrasound, MRI, optical coherence tomography, histology, blood-cell microscopy, computed tomography, chest X-ray, and retinal fundus photography. On the five segmentation datasets (ISIC 2018, ISIC 2017, TN3K, MMOTU, ACDC), HamSeg leads on Dice score with 8.57M parameters; on ACDC, the 3-seed result of 93.81 ± 0.10% exceeds the prior state of the art (FreqConvMamba, 89.79%) by 4.02 percentage points. On the nine MedMNIST classification benchmarks, HamCls operates at 2.95M parameters and 1.71GFLOPs at 224² input, a 4× parameter and 7× FLOPs reduction relative to the prior state-of-the-art MedKAFormer-T, and leads or ties it on 8 of 9 datasets at 3-seed precision, with margins up to ＋17.76 percentage points macro F1 on DermaMNIST and ＋8.30 percentage points top-1 on OCTMNIST. On the smallest binary benchmark, BreastMNIST (546 training images), HamCls trails MedKAFormer-T; the gap is consistent with the small-data regime. Diagnostic measurements show that momentum carries an interior > boundary > exterior activity gradient on lesion-segmentation tasks, and that the energy map concentrates on the class-discriminative structure of every classification modality without any localisation supervision. An ablation study confirms that the bottleneck contributes signal that a depth-matched conventional convolutional block cannot replicate. Code, configurations, and trained models will be released on acceptance.