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LHMCF-Net: A Learned Hyperbolic Mean Curvature Flow Network for Medical Images Segmentation

Shuangshuang Duan, Chunlei He, Shoujun Huang, Dexing Kong

arXiv:2608.20942v1cs.CVmath-ph

TL;DR

Medical image segmentation is difficult under noise, low contrast, and weak boundaries. LHMCF-Net embeds feature-space fidelity and deep structural priors in a learned hyperbolic mean-curvature-flow unfolding network, with ablations reporting best performance for learned damping and improved anatomical structure and boundary detail.

  • Problem

    Noise, artifacts, low contrast, and weak or missing boundaries challenge accurate medical-image segmentation, while classical models lack learned high-dimensional features and require task-specific regularization.

  • Method

    LHMCF-Net integrates feature-space data fidelity and deep structural priors into a second-order dissipative hyperbolic PDE unfolded as learnable, physically interpretable evolution stages.

  • Results

    Learned damping achieves the best performance across all evaluated metrics, while the deep prior improves preservation of global anatomical structure and fine-scale boundary details.

  • Takeaways & Limitations

    The study supports physically inspired hyperbolic geometric evolution as a promising basis for robust medical-image segmentation and interpretable deep unfolding.

  • Takeaways & Limitations

    Well-posedness, convergence, and stability under inhomogeneous initial velocities remain analytically open.

Abstract

from arXiv · show

Motivated by the classical Chan-Vese model and the ability of deep priors to capture complex spatial structures, we develop a segmentation model that leverages learned hyperbolic mean curvature flow (LHMCF) as a mathematical foundation for integrating feature space data fidelity and deep structural priors within a unified high-dimensional framework. The proposed LHMCF model is governed by a second-order dissipative hyperbolic PDE, where the introduction of a velocity field provides inertia and momentum to the evolving interface. This hyperbolic mechanism enables the contour to bypass noise-induced local minima and propagate coherently through low-contrast or ambiguous regions, addressing limitations inherent to first-order parabolic flows. To solve the continuous LHMCF model, we construct a deep unfolding network, named LHMCF-Net, which maps the iterative numerical procedure of the PDE into a sequence of discrete evolution stages. Each stage corresponds to one physically interpretable update of the underlying dynamical system, allowing the network to inherit the stability and geometric consistency of the PDE while supporting end-to-end optimization. Comprehensive experiments on three publicly available medical segmentation datasets demonstrate that LHMCF-Net achieves superior performance, particularly in challenging scenarios with low contrast and unclear boundaries. These results highlight the effectiveness of embedding hyperbolic geometric evolution into deep unfolding architectures and underscore the potential of physically inspired models for robust medical image segmentation.

1 Introduction

The introduction frames medical segmentation as challenging because noise, artifacts, low contrast, and weak boundaries undermine conventional methods. LHMCF-Net addresses these limitations by combining feature-space fidelity, deep structural priors, hyperbolic dynamics, and physically interpretable deep unfolding.

  • Motivation: Medical images often contain noise, artifacts, speckle, shadows, low contrast, and lesion regions with similar color and texture.These conditions complicate reliable boundary identification and segmentation.
  • Limitations of Existing Methods: Variational methods provide mathematical and geometric priors but primarily operate in image scale space and omit high-dimensional learned features.Representative formulations include snake, Chan-Vese, Mumford-Shah, and geodesic active contours.
  • Proposed Model: LHMCF integrates curvature-driven smoothing, feature-space data fidelity, and deep structural priors within a unified hyperbolic mean curvature flow framework.The formulation connects classical geometric evolution with modern semantic representation.
  • Hyperbolic Dynamics: Acceleration supplies inertia for escaping shallow or noise-induced local minima, while dissipative damping stabilizes evolution and prevents oscillatory behavior.Together, these terms distinguish the model from first-order parabolic flows and support robust, noise-resistant segmentation.
  • Deep Unfolding: Deep unfolding discretizes the coupled hyperbolic PDE into learnable stages with physically meaningful updates to velocity, level-set functions, and feature means.The architecture links continuous PDE dynamics with discrete neural operators while supporting numerical stability and transparent parameter interpretation.

2 Related works

Related work spans region-based Chan–Vese segmentation and its level-set implementation, hyperbolic mean curvature flow theory and numerics, and deep unfolding architectures that convert iterative model-based procedures into trainable networks.

  • Chan–Vese and level-set methods: The Chan–Vese model segments images by partitioning the domain into two approximately homogeneous-intensity regions using contour regularization and region-based fidelity terms.Unlike edge-based active contours, it can handle weak or missing gradients because it relies on region statistics rather than local image gradients.
  • Chan–Vese and level-set methods: Level-set methods represent the contour as the zero level set of a Lipschitz function, enabling topology changes such as region splitting and merging.Regularized Heaviside and Dirac delta approximations provide the level-set formulation of the energy.
  • Chan–Vese and level-set methods: The level-set evolution is a parabolic PDE combining mean-curvature contour regularization with a region-based force that drives the interface toward the separating object boundary.Numerical implementation alternates between updating region statistics and evolving the level-set function.
  • Hyperbolic mean curvature flow: Hyperbolic mean curvature flow replaces velocity-driven parabolic evolution with acceleration-driven geometric evolution, with prior work establishing solution properties, finite-time behavior, dissipation, and numerical approximations.Applications of second-order hyperbolic PDEs have also been explored in image denoising, displacement-error correction, and color-image segmentation.
  • Deep unfolding: Deep unfolding translates variational or PDE-based iterative algorithms into neural networks whose stages correspond to trainable algorithmic updates, combining model-based optimization with data-driven learning.Prior examples include ADMM-CSNet, Learned Primal–Dual, and PottsMGNet for reconstruction and segmentation.

3 The proposed method

The proposed LHMCF model unifies curvature-driven geometric evolution, high-dimensional feature-space region statistics, and a deep structural prior through a second-order dissipative hyperbolic flow. LHMCF-Net unfolds its numerical solution into physically interpretable stages that preserve the discrete dynamics while enabling end-to-end learning.

  • LHMCF formulation: The continuous model is a second-order dissipative hyperbolic mean curvature flow with damping, curvature, anisotropy, and an initial normal velocity.Its level-set representation describes the evolving curve through a function whose zero level set defines the interface.
  • LHMCF formulation: The LHMCF model integrates curvature-driven evolution, feature-space data fidelity, and a data-driven deep prior in a unified high-dimensional segmentation framework.The feature representation is extracted by a backbone, while the deep prior preserves semantic structures and regularizes the evolving level-set function.
  • Numerical discretization: The second-order PDE is reformulated as a coupled position-velocity system and solved with an explicit exponential integration scheme based on Duhamel’s principle and piecewise-constant force approximation.The resulting velocity and level-set updates provide the mathematical foundation for deep unfolding.
  • Stage-wise evolution: Each stage refines foreground and background feature means through momentum updates before applying a deterministic hyperbolic evolution layer that preserves the exact learned-parameter-dependent discrete flow.The momentum coefficient balances historical information with current estimates and suppresses oscillations in the background feature field.
  • LHMCF-Net architecture: Deep unfolding maps the iterative PDE procedure into sequential discrete stages, allowing curvature, feature-space data, and deep-prior terms to be learned within the evolution process.Each stage corresponds to one evolution step of the underlying dynamical system.
  • Initialization: The network initializes high-dimensional features, a predicted level set, feature means, and zero velocity, while latent-variable mappings enforce positivity and conservative step-size initialization for learned physical parameters.Spatially adaptive anisotropic curvature modulation is obtained from the feature representation through a 1 × 1 convolution and Sigmoid activation.

4 Experimental results

Experiments on BUSI, Kvasir-SEG, and ISIC 2018 evaluate LHMCF-Net under challenging low-contrast or ambiguous-boundary conditions using segmentation and boundary metrics. Across quantitative, qualitative, and efficiency comparisons, LHMCF-Net shows strong segmentation quality, coherent contours, and competitive accuracy with low model complexity.

  • Datasets: The evaluation uses BUSI, Kvasir-SEG, and ISIC 2018, three publicly available medical-image datasets featuring challenging anatomical or lesion boundaries.BUSI includes 647 pathological scans, Kvasir-SEG contains 1,000 endoscopy images, and ISIC 2018 provides 2,594 dermoscopic images.
  • Evaluation metrics: Evaluation uses Accuracy, IoU, DSC, and HD95, with lower HD95 indicating more accurate boundary localization.Acc, IoU, and DSC emphasize internal object consistency, whereas HD95 focuses on boundary precision.
  • Quantitative comparison: LHMCF-Net outperforms UNet, UNet++, AttnUNet, Deeplab V3+, and TransUNet on Acc, IoU, and DSC while achieving the lowest HD95 across the evaluated datasets.Compared with the second-best method, LHMCF-Net reduces HD95 by 1.4920 on BUSI, 0.8275 on Kvasir, and 0.7955 on ISIC 2018.
  • Qualitative comparison: LHMCF-Net produces contours that adhere closely to ground-truth boundaries and form smoother, more coherent shapes than competing methods.The reported improvement is attributed to second-order hyperbolic dynamics and curvature-based geometric regularization, which stabilize interface evolution and suppress noise-induced distortions.
  • Model complexity and efficiency: 36.69M parameters make LHMCF-Net the smallest model in the complexity comparison, while its Dice performance remains comparable to TransUNet and its HD95 is lowest.TransUNet has 93.23M parameters; the comparison links the efficiency-accuracy tradeoff to the physically guided hyperbolic formulation.

5 Ablation analysis and physical interpretability · 5.1 Ablation on Hyperbolic vs. Parabolic Dynamics · 5.2 Ablation study on damping coefficient

The ablation analysis shows that second-order hyperbolic dynamics outperform a first-order parabolic reduction, while learnable damping provides the most effective balance between inertia and dissipation. These physically interpretable components improve stability, interface propagation, and segmentation precision on BUSI and Kvasir.

  • 5.1 Ablation on Hyperbolic vs. Parabolic Dynamics: LHMCF-Net is compared with the first-order parabolic counterpart LMCF-Net on the BUSI and Kvasir datasets.Table 3 evaluates the proposed second-order hyperbolic model against its first-order parabolic counterpart.
  • 5.1 Ablation on Hyperbolic vs. Parabolic Dynamics: Removing the inertial term ϕ_tt and fixing β = 1 degenerates LHMCF-Net into a classical gradient-descent-type parabolic evolution.The reduction isolates the contribution of the second-order term ϕ_tt by eliminating inertia and setting the damping coefficient to β = 1.
  • 5.1 Ablation on Hyperbolic vs. Parabolic Dynamics: LHMCF-Net achieves consistently superior performance across the evaluated BUSI and Kvasir comparisons.The supplied passage reports consistent superiority in Table 3, while the table caption identifies the datasets and model comparison.
  • 5.2 Ablation study on damping coefficient: The damping coefficient β critically shapes hyperbolic level-set evolution, with configurations ranging from β = 0 to β = 10.The ablation compares an undamped purely hyperbolic system, under-damped β = 0.1, and strictly over-damped settings including β = 10.
  • 5.2 Ablation study on damping coefficient: When β = 0, the purely hyperbolic system lacks dissipation and exhibits noticeable instability in boundary evolution.This configuration is used to assess the role of dissipative dynamics in stabilizing the evolving interface.
  • 5.2 Ablation study on damping coefficient: The learned β consistently achieves the best performance across all metrics on the BUSI and Kvasir datasets.Treating β as learnable adaptively balances inertia and dissipation, producing more coherent interface propagation and enhancing segmentation precision.

5.3 Ablation on deep prior term

The ablation shows that adding the deep prior term consistently improves segmentation across BUSI and Kvasir, while providing semantic and geometric guidance that stabilizes hyperbolic evolution.

  • Ablation setup: Removing αR(ϕ) leaves the hyperbolic evolution driven solely by mean curvature and feature space data fidelity.The ablation compares LHMCF with and without the deep prior term.
  • Quantitative results: On BUSI, DSC increases from 0.8513 to 0.8664 when the deep prior is incorporated.This comparison is reported in Table 5.
  • Quantitative results: On BUSI, HD95 decreases from 17.9611 to 14.6110 with the deep prior.The lower HD95 indicates the reported improvement for this metric.
  • Mechanism: The learned deep prior provides implicit geometric regularization and high-level semantic guidance, preserving global anatomy while capturing fine-scale boundary details.It steers the flow toward a plausible semantic manifold, stabilizes evolution, and prevents anatomically unrealistic interface drift.

5.4 Ablation on Feature-Mean Evolution

This ablation examines momentum-based feature evolution (MFE) and EMA for dynamically updating foreground and background feature means. It compares this temporally evolving approach with a memory-less baseline using instantaneous coarse estimates on BUSI and Kvasir.

  • Feature-Mean Evolution: MFE and EMA dynamically evolve the foreground and background feature means used to compute the total force.The feature evolution operator is momentum-based, providing temporal evolution of the feature means.
  • Feature-Mean Evolution: The baseline removes temporal evolution and obtains feature means directly from instantaneous coarse estimates through Equation (18).This produces a memory-less update scheme whose feature means rely solely on potentially noisy current estimates.
  • Feature-Mean Evolution: The ablation evaluates performance with and without MFE on the BUSI and Kvasir datasets.Table 6 reports the comparative ablation study, with best results highlighted in bold.

5.5 Impact of the number of stages K

An ablation study varying unfolding depth K from 1 to 4 found that LHMCF-Net performs best with K = 2 across BUSI, Kvasir, and ISIC 2018. A single stage provides insufficient geometric evolution for low-contrast regions and complex anatomical boundaries.

  • Impact of the number of stages K: The ablation study varied the number of unfolding stages K ∈ {1, 2, 3, 4}, with each stage representing one discrete integration step of the second-order hyperbolic system.This experiment examined how unfolding depth affects the learned hyperbolic mean curvature flow.
  • Impact of the number of stages K: LHMCF-Net achieves its best performance at K = 2 across the BUSI, Kvasir, and ISIC 2018 datasets.The comparison is summarized in Figure 8.
  • Impact of the number of stages K: With K = 1, minimal geometric evolution makes it difficult for the level-set function ϕ to propagate through low-contrast regions or align accurately with complex anatomical boundaries.The limitation arises because a single unfolding stage provides too little evolution of the interface.

5.6 Interpretability of learned parameters

LHMCF-Net learns physically interpretable parameters that alter dissipation, integration, prior weighting, feature homogeneity, and curvature across unfolding stages. These parameters rapidly stabilize and promote anatomically plausible, coherent segmentations in challenging medical images.

  • Damping and integration: The learned damping coefficient β decreases from 0.6931 to approximately 0.394 (BUSI), 0.353 (Kvasir), and 0.347 (ISIC 2018), while ∆t increases from 0.094 to roughly 0.150–0.162.The reduced β indicates weaker dissipation and stronger physical inertia.
  • Deep-prior weighting: The deep-prior coefficient α increases to 1.084 (BUSI), 1.164 (Kvasir), and 0.9055 (ISIC 2018), indicating greater emphasis on geometric and semantic regularization.The strengthened prior guides hyperbolic flow toward anatomically plausible shapes, improving boundary coherence and reducing noise sensitivity.
  • Homogeneity weighting: Across unfolding stages, λ1 consistently exceeds λ2, emphasizing foreground homogeneity and tighter alignment between the evolving interface and the representative foreground feature mean c1.The asymmetric weighting reflects deliberate emphasis on the foreground manifold.
  • Spatially varying curvature: The curvature coefficient µ evolves from a nearly uniform initialization into a highly anisotropic spatially varying field µ(x, y) generated from feature space Z.The evolution is illustrated through heatmaps across initialization and refined unfolding stages.
  • Stage-wise stabilization: Parameter values remain highly consistent between Stage 1 and Stage 2, indicating rapid stabilization of the physical regime and explaining performance saturation at K = 2.Once the physical parameters settle into a stable configuration, additional unfolding stages yield limited benefit.

5.7 Evolution of the level-set function

LHMCF-Net evolves the level-set function from a semantically informed initialization toward a precise anatomical boundary across unfolding stages. The learned initial contour is already well localized, unlike manual or random initializations used in traditional variational models.

  • Initialization and boundary evolution: The level-set interface evolves across unfolding stages from a semantically informed initialization toward the precise anatomical boundary.Figure 11 visualizes this evolution on BUSI, Kvasir, and ISIC 2018.
  • Initialization and boundary evolution: A convolutional mapping from feature space Z generates an initial contour ϕ0 that already provides a well-localized estimate of the target structure.This initialization is derived from the feature space rather than manually or randomly specified.
  • Initialization and boundary evolution: Figure 11 uses blue contours for groundtruth boundaries and red contours for the evolving level-set interfaces.The visualization spans the BUSI, Kvasir, and ISIC 2018 datasets.

5.8 Impact of initial velocity

The section shows that initial velocity materially affects second-order hyperbolic flow, unlike first-order parabolic evolution, which is insensitive to initial momentum. Experiments compare four velocity configurations and report improved segmentation accuracy for all inhomogeneous initializations, with curvature-adaptive velocity performing best.

  • Impact of initial velocity: Proper initial velocity can make second-order flow behave significantly differently from homogeneous initial velocity.This distinguishes hyperbolic dynamics from first-order parabolic flows, whose evolution depends only on the instantaneous gradient-descent direction.
  • Impact of initial velocity: The study evaluates homogeneous v0 = 0, constant v0 = 1, oscillatory v0 = sinx · siny, and curvature-adaptive v0 = −γκ0 initial velocities.Here, γ is learned and κ0 denotes the curvature of the initial level-set function ϕ0.
  • Impact of initial velocity: All inhomogeneous initial velocities improve segmentation accuracy over the standard zero-velocity initialization, with curvature-adaptive velocity achieving the best results.The comparison is reported on the BUSI datasets in Table 7.

6 Conclusion

The paper proposes LHMCF, a hyperbolic mean-curvature-flow segmentation model integrating feature-space data fidelity and deep structural priors, and develops the lightweight LHMCF-Net deep unfolding network. It also identifies open well-posedness, convergence, and stability questions for hyperbolic geometric flows with inhomogeneous initial velocities.

  • LHMCF integrates feature-space data fidelity and deep structural priors within a second-order dissipative hyperbolic PDE framework.
  • LHMCF-Net is a lightweight deep unfolding network that discretizes the coupled hyperbolic formulation.
  • For inhomogeneous initial velocities, fundamental well-posedness problems remain analytically open, motivating studies of convergence and stability under general conditions.These questions are identified as important directions for future research in PDE and geometric analysis.
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