Deep-ADMM-Net

Introduction of Deep ADMM-Net

I am currently working in Math AI lab, and Deep ADMM Net is one of our main targets recently depends on papers:

  • Yang, Y., Sun, J., Li, H., & Xu, Z. (2016). Deep ADMM-Net for compressive sensing MRI
  • Yang, Y., Sun, J., Li, H., & Xu, Z. (2017). ADMM-Net: A Deep Learning Approach for Compressive Sensing MRI

A deep dive into Deep ADMM-Net

Deep ADMM-Net for FastMRI

Compressive Sensing MRI Model and ADMM Algorithm

General CS-MRI Model problem

where $A=P F \in \mathbb{R}^{N^{\prime} \times N}$ is a measurement matrix, $P \in \mathbb{R}^{N^{\prime} \times N}$ is a under-sampling matrix and $F$ is a Fourier transform.

ADMM solver
By introducing $ z=\left\{z_{1}, z_{2}, \cdots, z_{L}\right\} $,
the problem equals to

and its augmented Lagrangian function is

where $α = {α_l}$ are Lagrangian multipliers and $ρ = {ρ_l}$ are penalty parameters. ADMM alternatively optimizes ${x, z, α}$ by solving the following three subproblems:

Deep ADMM-Net

  • Reconstruction layer
  • Convolution layer
  • Nonlinear Transform layer
  • Multiplier update layer

Network Training

Given pairs of training data, the loss between the network output and ground truth is defined as

Initialization

Gradient Computation by Backpropagation over Data Flow Graph

  • Multiplier update layer
  • Nonlinear Transform layer
  • Convolution layer
  • Reconstruction layer