A mathematical explanation of UNet

The UNet architecture has transformed image segmentation. UNet's versatility and accuracy have driven its widespread adoption, significantly advancing fields reliant on machine learning problems with images. In this work, we gave a clear and concise mathematical explanation of UNet. We explained what is the meaning and function of each of the components of UNet. We showed that UNet was solving a control problem. We decomposed the control variables using multigrid methods. Then, operator-splitting techniques were used to solve the problem, whose architecture exactly recovered the UNet architecture. Our result showed that UNet was a one-step operator-splitting algorithm for the control problem.