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

TL;DR

This paper introduces a differentiable convexity prior based on quasi-concavity for image segmentation, enabling shape regularization within neural networks and unifying various previous convex shape models.

Contribution

It proposes a novel, unified, threshold-free convexity prior using quasi-concavity, with local inequalities and a convolutional loss integrated into segmentation networks.

Findings

Improves shape regularity in segmentation results across multiple datasets.

Outperforms previous shape-aware methods and tailored networks for retinal segmentation.

Unifies diverse convex shape models within a continuous differentiable framework.

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.