D-Convexity：面向数据驱动图像分割的、基于拟凹性的统一可微凸形状先验
D-Convexity: A Unified Differentiable Convex Shape Prior via Quasi-Concavity for Data-driven Image Segmentation
把凸性先验转成无需阈值的可微卷积损失（基于拟凹性与超水平集凸性），可直接嵌入任意分割网络输出端，用于视网膜血管等管状/凸结构分割的形状正则化，替代离散线约束或图割凸性建模。
原文依据 The first and second-order formulations produce a compact convolutional loss that can be densely applied across the image without thresholding.
摘要译文
由模型翻译 · 术语统一 · 请对照原文核对关键结论凸性是一种基本几何先验，广泛存在于自然与人造结构中，但在端到端可训练的分割网络中仍难以有效施加。本文从泛函视角重新审视凸性，提出一种统一的、无需阈值的凸性先验，其基础是网络输出掩码函数 u 的拟凹性。我们不再约束单个二值分割结果，而是要求 u 的所有超水平集均为凸集，从而将全局形状约束转化为关于 u 及其导数的局部、可微不等式。由此原理出发，我们推导出零阶、一阶与二阶刻画，分别得到局部中点凸化算法、与支撑超平面相关联的梯度条件，以及以切平面上的二次型表示的充分二阶不等式。一阶与二阶形式给出一个紧凑的卷积损失，可在整幅图像上稠密施加而无需阈值化。我们的拟凹性损失通过所提出的凸梯度投影模块（CGPM）与现代分割网络无缝集成。它们在多个数据集上一致地施加凸性并改善形状规整性，优于为视网膜分割定制的网络，并超越以往的形状感知方法。值得注意的是，我们的分析将以往一大类凸形状模型——从离散 1-0-1 直线约束、图割凸性形式，到基于曲率或符号距离拉普拉斯算子的水平集先验——统一在单一连续可微框架之内。
英文原文摘要 1630 字符
Convexity is a fundamental geometric prior that underlies many natural and man-made structures, yet remains challenging to impose effectively in end-to-end trainable segmentation networks. We revisit convexity from a functional perspective and propose a unified, threshold-free convexity prior based on the quasi-concavity of the network's output mask function u. Instead of constraining a single binary segmentation, we require all super-level sets of u to be convex, transforming global shape constraints into local, differentiable inequalities on u and its derivatives. From this principle, we derive zero, first, and second-order characterizations, yielding respectively a local midpoint convexification algorithm, a gradient-based condition linked to supporting hyperplanes, and a sufficient second-order inequality expressed as a quadratic form on the tangent plane. The first and second-order formulations produce a compact convolutional loss that can be densely applied across the image without thresholding. Our quasi-concavity losses integrate seamlessly with modern segmentation networks via the proposed convex gradient projection module (CGPM). They consistently enforce convexity and improve shape regularity across multiple datasets, outperforming networks tailored for retinal segmentation and surpassing previous shape-aware methods. Remarkably, our analysis unifies a wide spectrum of previous convex shape models, from discrete 1-0-1 line constraints and graph-cuts convexity formulations to curvature or signed distance Laplacian based level-set priors, within a single continuous and differentiable framework.