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.

翻译：摘要：精确的生物医学图像分割对临床诊断至关重要。几何线索（如边界、形状和拓扑）能够提升结构一致性，然而大多数方法依赖于特定任务，缺乏可跨器官和模态泛化的统一几何基础。我们的研究动机源于观察：多个医学分割目标可近似为全局近凸形状。凸区域定义为其中任意两点间的线段完全位于该区域内的区域。在实际应用中，医学目标可能呈现微小局部凹陷或边界不规则性；我们将此类全局近似凸的形状称为“近凸”。受此启发，我们从Hadwiger定理推导出Hadwiger形状先验，作为一种可解释的全局正则化器，使用三个二维测度量（面积A、周长P和欧拉示性数χ），从而实现了跨器官和模态的迁移。然而，由于医学数据集具有形状异质性，对非凸解剖结构（具有显著凹陷）统一施加近凸先验可能导致过度正则化，抹平凹陷和细节信息，降低分割精度。为应对这一挑战，我们提出冲突感知目标平衡（CAOB）方法，以梯度感知方式将形状先验与分割任务结合。对于每个先验，CAOB仅移除与分割任务冲突的梯度分量，保留其余对齐分量，并自适应调节目标影响力以防止先验主导。这使得在形状异质性数据上稳定使用形状先验成为可能，而不会消除真实凹陷或精细结构细节。我们将此即插即用框架命名为HadBalance。