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

arXiv: https://arxiv.org/abs/2605.19210 · HTML: https://mlsanity.com/paper/2605.19210/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 41 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 16 of 16)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1802.02611/) (2018)
- [Swin-Unet: Unet-like Pure Transformer for Medical Image Segmentation](https://mlsanity.com/paper/2105.05537/) (2021)
- [REFUGE Challenge: A Unified Framework for Evaluating Automated Methods for Glaucoma Assessment from Fundus Photographs](https://mlsanity.com/paper/1910.03667/) (2019)
- [CE-Net: Context Encoder Network for 2D Medical Image Segmentation](https://mlsanity.com/paper/1903.02740/) (2019)
- [DCAN: Deep Contour-Aware Networks for Accurate Gland Segmentation](https://mlsanity.com/paper/1604.02677/) (2016)
- [Joint Optic Disc and Cup Segmentation Based on Multi-label Deep Network and Polar Transformation](https://mlsanity.com/paper/1801.00926/) (2018)
- [Active Boundary Loss for Semantic Segmentation](https://mlsanity.com/paper/2102.02696/) (2021)
- [Deep Retinal Image Understanding](https://mlsanity.com/paper/1609.01103/) (2016)
- [Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation](https://mlsanity.com/paper/2002.03989/) (2020)
- [Convexity Shape Prior for Level Set based Image Segmentation Method](https://mlsanity.com/paper/1805.08676/) (2018)
- [Convex Shape Prior for Deep Neural Convolution Network based Eye Fundus Images Segmentation](https://mlsanity.com/paper/2005.07476/) (2020)
- [Convexity Shape Constraints for Image Segmentation](https://mlsanity.com/paper/1509.02122/) (2015)
- [A Binary Characterization Method for Shape Convexity and Applications](https://mlsanity.com/paper/2203.11395/) (2022)
- [Conv-MCD: A Plug-and-Play Multi-task Module for Medical Image Segmentation](https://mlsanity.com/paper/1908.05311/) (2019)
- [Contour Flow Constraint: Preserving Global Shape Similarity for Deep Learning based Image Segmentation](https://mlsanity.com/paper/2504.09384/) (2025)

## Abstract (arXiv)

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.