Double-well Net for Image Segmentation
Hao Liu Jun Liu Raymond Chan Xue-Cheng Tai
In this study, our goal is to integrate classical mathematical models with
deep neural networks by introducing two novel deep neural network models for
image segmentation known as Double-well Nets. Drawing inspiration from the
Potts model, our models leverage neural networks to represent a region force
functional. We extend the well-know MBO (Merriman-Bence-Osher) scheme to solve
the Potts model. The widely recognized Potts model is approximated using a
double-well potential and then solved by an operator-splitting method, which
turns out to be an extension of the well-known MBO scheme. Subsequently, we
replace the region force functional in the Potts model with a UNet-type
network, which is data-driven, and also introduce control variables to enhance
effectiveness. The resulting algorithm is a neural network activated by a
function that minimizes the double-well potential. What sets our proposed
Double-well Nets apart from many existing deep learning methods for image
segmentation is their strong mathematical foundation. They are derived from the
network approximation theory and employ the MBO scheme to approximately solve
the Potts model. By incorporating mathematical principles, Double-well Nets
bridge the MBO scheme and neural networks, and offer an alternative perspective
for designing networks with mathematical backgrounds. Through comprehensive
experiments, we demonstrate the performance of Double-well Nets, showcasing
their superior accuracy and robustness compared to state-of-the-art neural
networks. Overall, our work represents a valuable contribution to the field of
image segmentation by combining the strengths of classical variational models
and deep neural networks. The Double-well Nets introduce an innovative approach
that leverages mathematical foundations to enhance segmentation performance.