## 1 Introduction  
Image segmentation is a fundamental task in computer vision. It plays an important role in medical imaging technology, material science and other fields. Typical approaches for image segmentation include intensity thresholds [1], region growing [2], deformable models [3] and some methods based on machine learning. However, these methods over rely on predefined characteristics, which makes it difficult to automated medical image segmentation tasks. Recently, many deep learning methods have been proposed to automatically segment medical images [4, 5]. For instance, [PERSON] et al. [6] proposed a residual learning framework to improve accuracy by 28% on common objects in context (COCO) dataset, which compared with previous methods. Moreover, [PERSON] et al. [7] constructed an end-to-end fully convolutional network, which could input an image of any size and could output the corresponding segmentation results. This method improved performance on the visual object classes (VOC) dataset [8] by 20%. Generally, deep learning methods can be classified into three categories based on the dimension of input data: (1) 2D models: Cascaded-FCNs [9], and VGG based on fully convolutional network (FCN) [10]; (2) 3D models: ConVNet based densely connected convolutional networks [11], 3D FCN [12] and 3D U-Net [13]; and (3) 2.5D models: U-Net based residual networks [14] and recursive neural networks (RNNs) based on intra-slice features [15]. In general, a 2D model inputs a preprocessed image and outputs a probabilistic map. The 3D model can be regarded as an enhancement of the 2D model, which inputs a series of adjacent images with related information and outputs a corresponding set of probabilistic maps. Different from 2D and 3D models, the 2.5D model inputs several adjacent images with related information and outputs the probability map in the middle of these images. Although the current deep learning methods have improved the performance of automated segmentation of 3D medical images, they still have some problems and need to be improved. For example, the previous 2D FCNs [9; 16; 17] ignore the context information between the slices in the _z-axis_ direction, which results in low segmentation accuracy. On the other hand, the effect of the 2.5D network on the fuzzy boundary segmentation of the medical images was not sufficient [14]. In addition, the training of 3D networks requires higher hardware configurations and more computational resources. With the same computational resources, the larger number of parameters and computational consumption from 3D networks limit the design of deeper and more complex network structures. For instance, [PERSON] et al. [18] used 3D network structures to segment 3D medical images, and used 24G video random access memory (VRAM) to train and test their networks. Moreover, the loss function is also key for the optimization of the deep learning model. For the medical segmentation tasks, cross-entropy [19], similarity coefficient [20] and contour [21] are usually used as loss functions. These loss functions tend to focus on feature extraction in a specific region and lack the ability to learn features with fuzzy boundary information since the boundary includes very important feature information for image processing. With deep learning models, the learning of boundary information can improve the performance of image segmentation. To learn more about the boundary information for medical image segmentation, a boundary loss function is introduced into the proposed 2.5D deep learning model. Here, we perform segmentation of the liver and tumors in contrast-enhanced computed tomography (CT) as experimental cases. The liver is one of the most important organs in the human body and it can assist in the digestion of food and the breakdown of toxic substances [22; 23]. Liver cancer is known as the sixth most common cancer in the world and is one of the most common cancers [24].