## 1 Introduction  
Given the rapid advancements in medical imaging technology, magnetic resonance imaging (MRI) has emerged as the preferred method for brain diagnosis and treatment planning. The swift and precise automatic segmentation of MRI images through computer-aided diagnosis technology is crucial for the treatment and prognosis of brain tumors. Researchers are progressively turning to computer-assisted methods for brain-tumor segmentation, leveraging a variety of machine-learning and deep-learning algorithms to achieve optimal results. In 2012, the authors of [1] designed AlexNet, a deep convolutional neural network, and were the winners of ImageNet LSVRC in the same year. Subsequently, other researchers then introduced AlexNet into the fields of image segmentation and object detection. AlexNet introduced the ReLU activation function and Dropout technique: The ReLU activation function addressed the gradient vanishing problem of sigmoid in deep networks, while the Dropout technique effectively reduced overfitting. However, in AlexNet, information propagation primarily relied on local convolution operations, which meant that information interactions mainly occurred between adjacent pixels or feature maps. This locality constraint limited the network's ability to perceive global information, making it difficult for the network to fully utilize contextual information across the entire image. In 2014, ref. [2] proposed the deep convolutional structure VGGNet, which uses pre-training to train a shallow network and then reuses the weight of the previous layer to train the subsequent layer of the network. Through repeated training, problems, such as the weight initialization of the model, are solved. It also allows the model to converge faster during training. VGGNet consists of five convolutional layers, three fully connected layers, and a softmax output layer. Max-pooling is used to separate layers, and multiple smaller convolutional layers (3 \(\times\) 3) are used instead of larger convolutional layers to reduce the number of parameters. However, this approach also limits the network's ability to perceive global information. In 2015, ref. [3] proposed the UNet network, which adopts an encoder-decoder architecture, combining both lightweight and high performance and intensively integrating shallow features and deep features, making this model outstanding in the field of medical image segmentation. However, traditional UNet encoders and decoders mainly perform local operations, which limits the model's ability to integrate global information. In biomedical image segmentation, the integration of global information is crucial for improving segmentation accuracy. In 2018, ref. [4] proposed an improved UNet++ network based on the UNET network. The model not only uses the structure of UNet for reference but also adopts the dense connection mode of the DenseNet network and introduces deep supervision. It not only preserves and reconstructs global information and local information but also makes the model more efficient. Due to the introduction of more nested structures and skip connections in UNet++, this results in an increase in the model's parameter count and computational complexity. Later, ref. [5] proposed the vision transformer and applied it in the field of brain-tumor segmentation. Compared with convolutional neural networks, the Vision Transformer enables the model to better obtain global semantic information. However, since ViT mainly captures global information through self-attention mechanisms, it may not be as effective as convolutional neural networks (CNNs) in handling local details and texture features. CNNs can naturally extract local features of images through convolution operations, while ViT requires additional design or integration with other techniques to enhance its ability to capture local context information.