[Paper Review] ADMM-Net: A Deep Learning Approach for Compressive Sensing MRI

Yan Yang, Jian Sun+2 more|arXiv (Cornell University)|May 19, 2017

Sparse and Compressive Sensing Techniques41 references91 citations

TL;DR

ADMM-Net introduces deep architectures inspired by ADMM to solve CS-MRI reconstruction, learning all parameters end-to-end for improved accuracy and speed. It includes Basic-ADMM-Net, Generic-ADMM-Net, and Complex-ADMM-Net variants.

ABSTRACT

Compressive sensing (CS) is an effective approach for fast Magnetic Resonance Imaging (MRI). It aims at reconstructing MR images from a small number of under-sampled data in k-space, and accelerating the data acquisition in MRI. To improve the current MRI system in reconstruction accuracy and speed, in this paper, we propose two novel deep architectures, dubbed ADMM-Nets in basic and generalized versions. ADMM-Nets are defined over data flow graphs, which are derived from the iterative procedures in Alternating Direction Method of Multipliers (ADMM) algorithm for optimizing a general CS-based MRI model. They take the sampled k-space data as inputs and output reconstructed MR images. Moreover, we extend our network to cope with complex-valued MR images. In the training phase, all parameters of the nets, e.g., transforms, shrinkage functions, etc., are discriminatively trained end-to-end. In the testing phase, they have computational overhead similar to ADMM algorithm but use optimized parameters learned from the data for CS-based reconstruction task. We investigate different configurations in network structures and conduct extensive experiments on MR image reconstruction under different sampling rates. Due to the combination of the advantages in model-based approach and deep learning approach, the ADMM-Nets achieve state-of-the-art reconstruction accuracies with fast computational speed.

Motivation & Objective

• Motivate fast and accurate MRI reconstruction from undersampled k-space data using compressive sensing principles.
• Bridge model-based CS-MRI with data-driven learning by unfolding ADMM iterations into neural network stages.
• Develop variants (Basic, Generic, and Complex) to handle different ADMM formulations and complex-valued MR images.
• Enable end-to-end discriminative training to learn transforms, shrinkage functions, and penalty parameters directly from data.

Proposed method

• Formulate CS-MRI as a penalized inverse problem with data fidelity and transform-domain regularization.
• Unfold two ADMM solvers into corresponding deep networks by mapping iterations to network stages (Basic-ADMM-Net and Generic-ADMM-Net).
• Extend to Complex-ADMM-Net to handle complex-valued MR images by processing real/imaginary parts coherently.
• Represent each stage with learnable layers: reconstruction, convolution (transform), nonlinear transform (piecewise linear shrinkage), and multiplier update, all trained end-to-end.
• Train parameters via L-BFGS with back-propagation on NMSE loss using pairs of undersampled y and fully sampled x.
• Initialize parameters with model-based or random schemes to guide optimization.

Experimental results

Research questions

• RQ1 Can ADMM-inspired deep networks achieve superior reconstruction quality for CS-MRI compared to traditional CS-MRI methods and generic CNNs?
• RQ2 How does unfolding ADMM iterations into trainable network stages affect reconstruction speed and accuracy across varying undersampling rates?
• RQ3 What is the impact of extending ADMM-Net to complex-valued MR images on reconstruction performance?
• RQ4 Do Basic-ADMM-Net and Generic-ADMM-Net offer complementary benefits in network design and performance for CS-MRI?
• RQ5 Can the learned parameters generalize across sampling patterns and different MRI datasets?

Key findings

• ADMM-Nets achieve state-of-the-art reconstruction accuracy with fast computational speed on CS-MRI tasks.
• Two novel networks (Basic-ADMM-Net and Generic-ADMM-Net) are constructed by unfolding ADMM iterations into trainable stages.
• A Complex-ADMM-Net variant enables handling complex-valued MR images for improved clinical relevance.
• Networks learn transforms, shrinkage functions, and penalty/update parameters end-to-end from data.
• The approach combines model-based advantages with deep learning flexibility for CS-MRI and potentially other inverse problems.