# Convex Shape Prior for Deep Neural Convolution Network based Eye Fundus Images Segmentation

arXiv: https://arxiv.org/abs/2005.07476 · HTML: https://mlsanity.com/paper/2005.07476/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2020.

## Built on by (most cited first, 10 of 10)

- [Medical Image Segmentation with Limited Supervision: A Review of Deep Network Models](https://mlsanity.com/paper/2103.00429/) (2021)
- [Deep Convolutional Neural Networks Meet Variational Shape Compactness Priors for Image Segmentation](https://mlsanity.com/paper/2406.19400/) (2024)
- [Two-headed eye-segmentation approach for biometric identification](https://mlsanity.com/paper/2209.15471/) (2022)
- [Detection-Guided Deep Learning-Based Model with Spatial Regularization for Lung Nodule Segmentation](https://mlsanity.com/paper/2410.20154/) (2024)
- [A Mathematical Explanation of Transformers](https://mlsanity.com/paper/2510.03989/) (2025)
- [JOINEDTrans: Prior Guided Multi-task Transformer for Joint Optic Disc/Cup Segmentation and Fovea Detection](https://mlsanity.com/paper/2305.11504/) (2023)
- [Inverse Evolution Layers: Physics-informed Regularizers for Deep Neural Networks](https://mlsanity.com/paper/2307.07344/) (2023)
- [D-Convexity: A Unified Differentiable Convex Shape Prior via Quasi-Concavity for Data-driven Image Segmentation](https://mlsanity.com/paper/2605.19210/) (2026)
- [HadBalance: A Plug-and-Play Unified Global Geometric Prior Framework for Generalizable Biomedical Segmentation](https://mlsanity.com/paper/2606.15976/) (2026)
- [Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers](https://mlsanity.com/paper/2609.35080/) (2026)

## Abstract (arXiv)

Convex Shapes (CS) are common priors for optic disc and cup segmentation in eye fundus images. It is important to design proper techniques to represent convex shapes. So far, it is still a problem to guarantee that the output objects from a Deep Neural Convolution Networks (DCNN) are convex shapes. In this work, we propose a technique which can be easily integrated into the commonly used DCNNs for image segmentation and guarantee that outputs are convex shapes. This method is flexible and it can handle multiple objects and allow some of the objects to be convex. Our method is based on the dual representation of the sigmoid activation function in DCNNs. In the dual space, the convex shape prior can be guaranteed by a simple quadratic constraint on a binary representation of the shapes. Moreover, our method can also integrate spatial regularization and some other shape prior using a soft thresholding dynamics (STD) method. The regularization can make the boundary curves of the segmentation objects to be simultaneously smooth and convex. We design a very stable active set projection algorithm to numerically solve our model. This algorithm can form a new plug-and-play DCNN layer called CS-STD whose outputs must be a nearly binary segmentation of convex objects. In the CS-STD block, the convexity information can be propagated to guide the DCNN in both forward and backward propagation during training and prediction process. As an application example, we apply the convexity prior layer to the retinal fundus images segmentation by taking the popular DeepLabV3+ as a backbone network. Experimental results on several public datasets show that our method is efficient and outperforms the classical DCNN segmentation methods.