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Convex Shapes (CS) are common priors for optic disc and cup segmentation ineye fundus images. It is important to design proper techniques to representconvex shapes. So far, it is still a problem to guarantee that the outputobjects from a Deep Neural Convolution Networks (DCNN) are convex shapes. Inthis work, we propose a technique which can be easily integrated into thecommonly used DCNNs for image segmentation and guarantee that outputs areconvex shapes. This method is flexible and it can handle multiple objects andallow some of the objects to be convex. Our method is based on the dualrepresentation of the sigmoid activation function in DCNNs. In the dual space,the convex shape prior can be guaranteed by a simple quadratic constraint on abinary representation of the shapes. Moreover, our method can also integratespatial regularization and some other shape prior using a soft thresholdingdynamics (STD) method. The regularization can make the boundary curves of thesegmentation objects to be simultaneously smooth and convex. We design a verystable active set projection algorithm to numerically solve our model. Thisalgorithm can form a new plug-and-play DCNN layer called CS-STD whose outputsmust 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 forwardand backward propagation during training and prediction process. As anapplication example, we apply the convexity prior layer to the retinal fundusimages segmentation by taking the popular DeepLabV3+ as a backbone network.Experimental results on several public datasets show that our method isefficient and outperforms the classical DCNN segmentation methods.

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