Glaucoma is a progressive eye disease that results in permanent vision loss, and the vertical cup to disc ratio (vCDR) in colour fundus images is essential in glaucoma screening and assessment. Previous fully supervised convolution neural networks segment the optic disc (OD) and optic cup (OC) from color fundus images and then calculate the vCDR offline. However, they rely on a large set of labeled masks for training, which is expensive and time-consuming to acquire. To address this, we propose a weakly and semi-supervised graph-based network that investigates geometric associations and domain knowledge between segmentation probability maps (PM), modified signed distance function representations (mSDF), and boundary region of interest characteristics (B-ROI) in three aspects. Firstly, we propose a novel Dual Adaptive Graph Convolutional Network (DAGCN) to reason the long-range features of the PM and the mSDF w.r.t. the regional uniformity. Secondly, we propose a dual consistency regularization-based semi-supervised learning paradigm. The regional consistency between the PM and the mSDF, and the marginal consistency between the derived B-ROI from each of them boost the proposed model's performance due to the inherent geometric associations. Thirdly, we exploit the task-specific domain knowledge via the oval shapes of OD & OC, where a differentiable vCDR estimating layer is proposed. Furthermore, without additional annotations, the supervision on vCDR serves as weakly-supervisions for segmentation tasks. Experiments on six large-scale datasets demonstrate our model's superior performance on OD & OC segmentation and vCDR estimation. The implementation code has been made available 1.
