D-Convexity: A Unified Differentiable Convex Shape Prior via Quasi-Concavity for Data-driven Image Segmentation

2026-05-18

AI Summary

This work addresses the challenge of effectively incorporating convexity—a crucial geometric prior—into end-to-end image segmentation networks. By adopting a functional perspective, the authors reformulate the global convexity constraint of masks into local, differentiable inequalities involving a function and its derivatives. They propose, for the first time, a unified differentiable convexity prior based on quasi-concavity that operates without thresholding and encompasses diverse convex shape models across discrete and continuous domains, as well as both classical and deep learning frameworks. Leveraging zeroth-, first-, and second-order characterizations of quasi-concavity, they design convolutional dense loss functions and introduce a Convex Gradient Projection Module (CGPM) to enable end-to-end training. Experiments demonstrate that the method significantly enhances shape regularity in segmentation outputs and outperforms specialized retinal segmentation networks and existing shape-aware approaches across multiple datasets.

Abstract

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.

Problem

• convexity
• image segmentation
• shape prior
• quasi-concavity
• differentiable constraint

Innovation

• quasi-concavity
• differentiable convexity prior
• convex gradient projection module
• super-level set convexity
• shape regularization