###### Abstract  
The segmentation of optic disc (OD) and optic cup (OC) are used in the automatic diagnosis of glaucoma. However, the spatially ambiguous boundary and semantically uncertain region-of-interest area in pictures may lead to the degradation of the performance of precise segmentation of the OC and OD. Unlike most existing methods, including the variants of CNNs (Convolutional Neural Networks) and U-Net, which limit the contributions of rich global features, we instead propose a hybrid CNN-transformer and polar transformation network, dubbed as Polarformer, which aims to extract discriminative and semantic features for robust OD and OC segmentation. Our Polarformer typically exploits contextualized features among all input units and models the correlation of structural relationships under the paradigm of the transformer backbone. More specifically, our learnable polar transformer module optimizes the polar transformations by sampling images in the Cartesian space and then mapping them back to the polar coordinate system for masked-image reconstruction. Extensive experimental results present that our Polarformer achieves superior performance in comparison to most state-of-the-art methods on three publicly available datasets. deep learning; multi-model learning; medical segmentation; transformer; attention +
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performance. However, these deep-learning methods only used original fundus images. According to our observations, a valid geometrical constraint is that the closed nested structures of OD and OC are both of approximately elliptical shape, and there is variability of the object (size, position, etc.). However, representing this information and these constraints is difficult to implement within the network. With these in mind, M-Net [6] converts images to polar coordinates to learn representations for improving the performance of segmentation. Under the polar coordinate system, this geometric constraint can be easily converted into spatial relationships and can present an ordered layer structure. However, these similar methods only treat the center of the image as the polar origin in the transformation and coordinate system, while we formulate a Polar Origin Predictor capable of automatically detecting the centroid of the OC as the polar origin. The previous methods take polar transformation as a preprocessing process with no parameters learned. Since the hyperparameter is not optimal for segmentation evaluation and effective training, they cannot exploit all supervisory signals incorporating the extracted features. To solve this problem, we formulate a polar converter, which conducts the original image, and its polar origin performs a polar transform. Consequently, a learnable polar transformer module (LPTM) is proposed, which consists of a polar origin detector and a polar converter. It is trained for transforming the original images into the polar coordinate system. Thus, the network enables learning the polar representation. Another challenge is that the blurred edges, shape and size of the OC and OD vary among patients. Features of the background region (such as blood vessels and exudates) easily interfere with the foreground (the OD and OC regions). Accordingly, it is necessary to introduce sufficient contextual information and global information under different receptive fields to search for relationships between features. The image features extracted by the methods [6; 7; 8; 9; 10; 11] mentioned before mostly focus on local information, which cannot accurately capture the global contextual information, particularly from the OD and OC regions, thus, limiting the model's generalizability. The underlying reason is that convolutions have a limited field of perception. In other words, the network focuses more on local features when extracting features and cannot effectively consider the large-range contextual information.