HadBalance: A Plug-and-Play Unified Global Geometric Prior Framework for Generalizable Biomedical Segmentation

Abstract: Precise biomedical image segmentation is crucial for clinical diagnosis. Geometric cues (e.g., boundary, shape, and topology) can improve structural consistency, yet most are task-specific and lack a unified geometric foundation that generalizes across organs and modalities. We are motivated by the observation that several medical segmentation targets can be approximated as globally near-convex shapes. A convex region is one in which any two interior points can be connected by a line segment entirely contained within the region. In practice, medical targets may exhibit small local concavities or boundary irregularities; we refer to such globally convex-like shapes as near-convex. Motivated by this, we derive Hadwiger Shape Priors from Hadwiger's theorem as an interpretable global regularizer using three 2D measures: area A, perimeter P, and Euler characteristic chi, enabling transfer across organs and modalities. However, because medical datasets are shape-heterogeneous, enforcing near-convex priors uniformly can over-regularize non-convex anatomy with significant concavities, washing out concavities and fine details and degrading segmentation accuracy. To address this challenge, we propose Conflict-Aware Objective Balancing (CAOB), which integrates shape priors with segmentation in a gradient-aware manner. For each prior, CAOB removes only the gradient component that conflicts with segmentation while preserving the remaining aligned component, and adaptively regulates objective influences to prevent prior dominance. This enables stable use of shape priors on shape-heterogeneous data without erasing genuine concavities or fine structural details. We call this plug-and-play framework HadBalance.

• The paper introduces HadBalance, a plug-and-play framework that combines differentiable area, perimeter, and Euler-characteristic priors with conflict-aware gradient optimization for generalizable biomedical segmentation.
• HadBalance improves the strongest baseline on CVC-ClinicDB from 93.38% to 94.41% DSC and achieves a 75.07% BFScore, while delivering smaller gains on CUBS and DRIONS-DB.
• The ablations show that geometric priors alone can harm non-convex structures, whereas Primary Gradient Projection plus Adaptive Gradient Balancing preserves useful shape information and reduces optimization conflicts.

Motivation and problem setting

Geometric priors—boundary, shape, and topology constraints—are widely used to improve structural consistency in biomedical segmentation, but existing instances are largely task-specific: topology terms target vessel connectivity, boundary losses target polyp delineation, and so on. The paper observes that many medical targets (polyps, carotid lumen in ultrasound, optic disc) are globally near-convex despite local concavities, and leverages Hadwiger's characterization theorem from integral geometry as a principled basis for a unified prior. In $\mathbb{R}^2$, any motion-invariant, additive, continuous valuation on convex bodies is a linear combination of the three intrinsic volumes—area $A$, perimeter $P$, and Euler characteristic $\chi$—so constraining these three quantities yields an interpretable global regularizer that transfers across organs and modalities.

The central difficulty is data heterogeneity: near-convex and non-convex anatomies coexist within the same dataset, so uniformly enforcing near-convexity over-regularizes non-convex structures, erasing genuine concavities and fine boundary detail. Heuristic gating or auxiliary networks for distinguishing convex cases introduce hyperparameters and unstable decisions on ambiguous samples. The authors instead address the problem at the optimization level.

Method

Hadwiger Shape Priors. Each functional is computed differentiably on soft predictions via multi-scale local maps with pooling windows $\mathcal{D}=\{7,15,31\}$:

• Area: $\Delta\times\Delta$ average pooling of the prediction and ground truth, with MSE between local area maps.
• Perimeter: Gaussian smoothing followed by gradient magnitude (a differentiable total-variation proxy), then pooled and matched by MSE.
• Euler characteristic: the Gray-code bit-quad decomposition expresses $\chi$ as a sum over $2\times2$ binary neighborhoods; fixed convolutional kernels count bit-quad configurations, and pooled maps are combined linearly before MSE matching.

The prior loss is $$\mathcal{L}_H = \mathcal{L}_A + \mathcal{L}_P + \mathcal{L}_\chi$$, added to a standard segmentation loss $\mathcal{L}_S$.

Conflict-Aware Objective Balancing (CAOB). Rather than linear scalarization (LS), which is weight-sensitive and requires dataset-specific tuning, CAOB computes per-objective gradients $A$0 for $A$1 each iteration, treating segmentation as the anchor objective. Two stages follow:

1. Primary Gradient Projection (PGP): if a prior gradient is negatively aligned with $A$2 ($A$3), only its opposing component is removed, guaranteeing $A$4 while preserving aligned information.
2. Adaptive Gradient Balancing (AGB): because scale heterogeneity can still let priors dominate even after projection, AGB seeks weights $A$5 such that the joint direction $A$6 yields positive, balanced per-objective utilities $A$7. Minimizing the average inverse utility under feasibility constraints ($A$8, $A$9) and applying KKT stationarity yields a fixed-point equation $P$0, solved online via constrained nonlinear least squares and normalized to remove scale ambiguity.

The framework is plug-and-play: it wraps any backbone without architectural modification.

Experimental results

Evaluation uses DSC, IoU, and Boundary F-score (BFScore) on three public datasets spanning organs and modalities: CUBS (carotid ultrasound), CVC-ClinicDB (colonoscopy polyps), and DRIONS-DB (optic disc fundus images). HadBalance is applied to the strongest backbone per dataset and compared against six backbones plus two generic gradient-balancing methods (Nash-MTL, FairGrad).

Dataset	Backbone	Baseline DSC	HadBalance DSC	HadBalance IoU	HadBalance BFScore
CUBS	TransUNet	86.02	86.35	76.24	71.66
CVC-ClinicDB	nnUNet	93.38	94.41	89.85	75.07
DRIONS-DB	nnUNet	95.79	95.90	92.25	71.18

On CVC-ClinicDB—the most shape-heterogeneous dataset—HadBalance delivers the clearest gains over LS (+1.04 DSC, +1.50 IoU, +1.64 BFScore) and dramatically outperforms Nash-MTL and FairGrad on boundary quality (75.07 vs. 65.86 and 65.05 BFScore respectively), indicating that generic gradient balancing alone does not improve contour accuracy. On CUBS, LS already performs strongly (86.19 DSC) and HadBalance adds only marginal gains, which the authors attribute to limited headroom on this largely near-convex dataset. On DRIONS-DB, improvements over nnUNet are modest (+0.11 DSC, +0.21 IoU).

Ablation analysis

The ablation reveals a notable asymmetry that motivates the full framework. Adding Hadwiger priors alone improves CUBS (+0.17 DSC, +1.09 BFScore) but slightly hurts CVC-ClinicDB (−0.01 DSC, −0.20 IoU), confirming that the priors are sample-dependent and do not transfer naively to non-convex-heavy data. PGP alone also reduces performance relative to plain priors on both datasets, showing that conflict removal is insufficient when remaining priors retain scale dominance—without PGP, the $P$1 gradient in particular can dominate the joint update and steer optimization away from segmentation. Only the combination of PGP and AGB produces consistent gains across both datasets, with training-time diagnostics showing reduced gradient-conflict rates after PGP and increasing cosine alignment between the update direction and $P$2 over training.

Limitations and open questions

The paper concedes several limitations. First, CAOB requires separate backward passes for each loss term, adding training cost that the authors identify as future work but do not quantify precisely. Second, the empirical scope covers three 2D datasets; generalization to volumetric segmentation (where Hadwiger's theorem involves four intrinsic volumes including mean breadth) is untested. Third, the near-convexity motivation itself bounds applicability: for anatomy with substantial inherent concavity, the value of the priors depends entirely on CAOB's ability to suppress conflicting components, and the mechanism by which AGB handles persistently conflicting samples remains implicit. Finally, the fixed-point solve introduces a preset step-scale constant $P$3, whose sensitivity is not analyzed.

Conclusion

HadBalance couples Hadwiger-derived intrinsic-volume priors—a minimal, interpretable, modality-agnostic geometric basis—with a conflict-aware optimizer that projects out gradient components opposing segmentation and adaptively balances residual contributions. The results support the paper's central claim that unified global shape priors are viable on heterogeneous medical data only when paired with conflict-aware optimization, with the largest benefits appearing precisely where naive weighting fails. Open questions include computational overhead reduction, extension to 3D intrinsic volumes, and behavior on predominantly non-convex anatomies.