HamVision: Hamiltonian Dynamics as Inductive Bias for Medical Image Analysis

1. Problem Statement

Medical image analysis requires extracting discriminative features from complex, noisy data where critical information often resides in subtle gradients (tissue boundaries), textural patterns, or spatial intensity distributions. Current dominant paradigms rely on specialized architectures (e.g., U-Nets for segmentation, Transformers/ResNets for classification) that learn generic feature transformations purely from data. These approaches often lack structural priors, relying heavily on model capacity rather than embedding domain-specific physical or mathematical inductive biases. Consequently, they may struggle to efficiently capture boundary information, frequency selectivity, and spatial saliency without massive parameter counts.

2. Methodology: The HamVision Framework

The authors propose HamVision, a unified framework that replaces the standard bottleneck in neural networks with a damped harmonic oscillator. This acts as a structured inductive bias, decomposing feature maps into three functionally distinct, physics-derived representations without requiring explicit supervision for these specific properties.

A. The Hamiltonian Bottleneck

The core innovation is a bottleneck layer based on the dynamics of a damped harmonic oscillator. For a spatially varying signal (a row or column of a feature map), the system models the evolution of:

1. Position ($q$): The filtered signal content.
2. Momentum ($p = \dot{q}$): The rate of spatial change, acting as a frequency-selective spatial derivative.
3. Energy ($H = \frac{1}{2}(q^2 + p^2)$): A scalar measure of local excitation, integrating both content and gradients.

Mathematical Formulation:
The system is governed by the state-space equation $\dot{x} = Ax + Bu$, where $x = [q, p]^\top$. The matrix $A$ is structurally constrained to represent a damped oscillator with natural frequency $\omega$ and damping $\nu$.

• Complex Diagonalization: The coupled system is diagonalized into a complex scalar $z = q + ip$, satisfying $\dot{z} = (-\nu + i\omega)z + u$.
• Discretization: The system is discretized using an exponential integrator (exact for the homogeneous part), ensuring BIBO (Bounded-Input Bounded-Output) stability.
• Implementation: To handle 2D images, the oscillator bank scans rows and columns in four directions (left-right, right-left, top-bottom, bottom-up). The outputs are merged via learned projections.

B. Task-Specific Heads

The framework uses a shared encoder and Hamiltonian bottleneck, with different heads for segmentation and classification:

1. HamSeg (Segmentation Head):

• Architecture: A U-Net decoder with skip connections.
• Mechanism:
  • Energy Gating: The energy map ($H_{map}$) is interpolated to each decoder level and used to gate skip connections from the encoder. This highlights salient regions (boundaries/structures) and suppresses homogeneous backgrounds.
  • Momentum Injection: The momentum tensor ($p$) is projected and concatenated at every decoder level, injecting explicit boundary and gradient information to refine spatial details.
• Key Design: Centering the energy map before gating is crucial to prevent sigmoid saturation, allowing the gate to operate dynamically around 0.5.

2. HamCls (Classification Head):

• Architecture: Replaces the U-Net decoder with a global pooling head.
• Mechanism:
  • Phase-Space Pooling: The three representations are globally averaged:
    • $\bar{f}$ (from $q$): Encodes content.
    • $\bar{p}$ (from $|p|$): Encodes texture complexity and boundary density (spatial activity).
    • $\bar{h}$ (from $H$): Encodes overall excitation levels.
  • These are concatenated into a phase-space vector and passed through an MLP for classification.

3. Key Contributions

1. Structured Inductive Bias: Introduces a Hamiltonian bottleneck that naturally produces position, momentum, and energy as intermediate representations, grounded in physical laws rather than learned heuristics.
2. Dual-Task Architecture: Demonstrates that a single shared oscillator bottleneck can serve both segmentation (via spatial gating and injection) and classification (via statistical pooling) without modification to the core dynamics.
3. State-of-the-Art Efficiency: Achieves top-tier performance with only 7–9 million parameters, significantly fewer than competing methods (e.g., U-Nets, Transformers, Mamba variants) which often use 16–105M parameters.
4. Interpretability: Provides diagnostic evidence that the oscillator's outputs carry intrinsic physical meaning (e.g., momentum consistently encodes boundary/interior gradients) without post-hoc explanation tools.

4. Experimental Results

The framework was evaluated on 10 benchmarks across 5 imaging modalities (dermoscopy, ultrasound, MRI, CT, microscopy).

• Segmentation (HamSeg):

  • ISIC 2018: 89.38% Dice (SOTA).
  • ISIC 2017: 88.40% Dice.
  • TN3K (Thyroid): 87.05% Dice.
  • ACDC (Cardiac MRI): 92.40% Dice (outperforming FreqConvMamba by +2.61%).
  • Note: The multi-scale momentum injection was particularly effective for ACDC, where cardiac structures vary greatly in size.

• Classification (HamCls):

  • BloodMNIST: 98.85% Accuracy (SOTA, +0.45% over MedMamba-S).
  • PathMNIST: 96.65% Accuracy (SOTA, +0.25% over MedMamba-B).
  • DermaMNIST: 77.96% Accuracy (competitive with MedViT-S).
  • BreastMNIST: 89.60% Accuracy (handling severe class imbalance).

5. Significance and Analysis

• Physical Interpretability: Unlike attention mechanisms which are often "black boxes," the oscillator's outputs have a priori physical meanings. Diagnostics confirmed that momentum consistently follows an "Interior > Boundary > Exterior" gradient (due to resonance accumulation), and energy maps correlate with discriminative regions.
• Parameter Efficiency: By leveraging the physics of the oscillator to generate three distinct functional signals (content, gradient, saliency) from a single process, HamVision eliminates the need for separate, parameter-heavy modules for edge detection or saliency mapping.
• Generalization: The success across diverse modalities (from smooth MRI to noisy ultrasound) suggests that the oscillator captures fundamental properties of visual analysis (frequency selectivity, boundary detection) rather than modality-specific artifacts.
• Theoretical Connection: The work bridges the gap between State-Space Models (like Mamba) and physical systems. While Mamba-3 validated the algebraic necessity of complex transitions, HamVision demonstrates the practical utility of the phase-space decomposition (position/momentum/energy) for vision tasks.

In conclusion, HamVision establishes that embedding Hamiltonian dynamics as an inductive bias provides a principled, interpretable, and highly efficient foundation for medical image analysis, outperforming larger, less structured models across segmentation and classification tasks.