  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.
