  For problems in image processing and many other fields, a large class of
effective neural networks has encoder-decoder-based architectures. Although
these networks have made impressive performances, mathematical explanations of
their architectures are still underdeveloped. In this paper, we study the
encoder-decoder-based network architecture from the algorithmic perspective and
provide a mathematical explanation. We use the two-phase Potts model for image
segmentation as an example for our explanations. We associate the segmentation
problem with a control problem in the continuous setting. Then, multigrid
method and operator splitting scheme, the PottsMGNet, are used to discretize
the continuous control model. We show that the resulting discrete PottsMGNet is
equivalent to an encoder-decoder-based network. With minor modifications, it is
shown that a number of the popular encoder-decoder-based neural networks are
just instances of the proposed PottsMGNet. By incorporating the
Soft-Threshold-Dynamics into the PottsMGNet as a regularizer, the PottsMGNet
has shown to be robust with the network parameters such as network width and
depth and achieved remarkable performance on datasets with very large noise. In
nearly all our experiments, the new network always performs better or as good
on accuracy and dice score than existing networks for image segmentation.
