The main objective of image segmentation is to divide an image into homogeneous regions for further analysis. This is a significant and crucial task in many applications such as medical imaging. Deep learning (DL) methods have been proposed and widely used for image segmentation. However, these methods usually require a large amount of manually segmented data as training data and suffer from poor interpretability (known as the black box problem). The classical Mumford-Shah (MS) model is effective for segmentation and provides a piece-wise smooth approximation of the original image. In this paper, we replace the hand-crafted regularity term in the MS model with a data adaptive generalized learnable regularity term and use a multi-grid framework to unroll the MS model and obtain a variational model-based segmentation network with better generalizability and interpretability. This approach allows for the incorporation of learnable prior information into the network structure design. Moreover, the multi-grid framework enables multi-scale feature extraction and offers a mathematical explanation for the effectiveness of the U-shaped network structure in producing good image segmentation results. Due to the proposed network originates from a variational model, it can also handle small training sizes. Our experiments on the REFUGE dataset, the White Blood Cell image dataset, and 3D thigh muscle magnetic resonance (MR) images demonstrate that even with smaller training datasets, our method yields better segmentation results compared to related state of the art segmentation methods.

翻译：图像分割的主要目标是将图像划分为同质区域以进行进一步分析，这是医学成像等众多应用中一项重要且关键的任务。深度学习方法已被提出并广泛应用于图像分割。然而，这些方法通常需要大量人工标注数据作为训练数据，且存在可解释性差（即黑箱问题）的缺陷。经典的Mumford-Shah（MS）模型能有效实现分割，并提供原图像的分片光滑近似。本文用数据自适应的广义可学习正则项替换MS模型中手工设计的正则项，并采用多网格框架展开MS模型，从而得到泛化能力与可解释性更强的基于变分模型的分割网络。该方法可将可学习的先验信息融入网络结构设计中。此外，多网格框架能实现多尺度特征提取，并为U形网络结构在获得良好图像分割结果方面的有效性提供了数学解释。由于所提出的网络源自变分模型，它还能处理小训练样本量。我们在REFUGE数据集、白细胞图像数据集以及三维大腿肌肉磁共振图像上的实验表明，即使使用较小的训练数据集，我们的方法相较于相关的最先进分割方法仍能取得更优的分割结果。