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ADMM-Net: A Deep Learning Approach for Compressive Sensing MRI
Yan Yang, Jian Sun, Huibin Li, Zongben Xu
TL;DR
CS-MRI aims to accelerate MRI by reconstructing images from undersampled k-space, while selecting effective transforms, regularization, and optimization parameters remains challenging. This paper develops ADMM-derived networks whose parameters are learned from data, and reports state-of-the-art reconstruction accuracy with fast computational speed.
Problem
CS-MRI reconstructs images from a few k-space samples to reduce acquisition time, but choosing effective transforms and regularization remains challenging.
Source: §1 · p. 1Method
The paper unfolds ADMM iterations into deep networks that learn CS-MRI model and algorithm parameters from data, including a complex-valued extension.
Source: §1 · p. 2Results
ADMM-Nets achieve state-of-the-art MR reconstruction accuracy with fast computational speed.
Source: §1 · p. 2, §6 · p. 13Takeaways & Limitations
The authors present ADMM-Nets as combining model-based and deep-learning advantages, with potential application to other inverse problems such as image deconvolution.
Source: §1 · p. 2, §6 · p. 13
Abstract
from arXiv · showhide
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.
1 INTRODUCTION
CS-MRI reconstructs images from undersampled k-space to reduce MRI acquisition time, but selecting effective sparse representations and regularization remains challenging. The paper introduces ADMM-inspired networks with discriminatively learned parameters and reports state-of-the-art reconstruction accuracy with fast computational speed.
1 INTRODUCTION
- MRI speed is a fundamental challenge: CS-MRI reconstructs high-quality images from a few k-space samples, reducing scanning time.Long scanning and waiting times may result in motion artifacts and patient discomfort.Source: §1 · p. 1
- Choosing an optimal transform or subspace and its regularization remains challenging because these choices are often based on experience and may be suboptimal for reconstruction quality and speed.ADMM is efficient and generally applicable, but selecting parameters such as update rates and penalty parameters is also nontrivial.Source: §1 · p. 1
- The networks derive their data-flow graphs from ADMM iterations, with stages mapping to iterations and learned operations reconstructing MR images from undersampled k-space.The paper describes Basic-ADMM-Net and Generic-ADMM-Net, plus a complex-valued extension; network parameters are discriminatively learned from training data.Source: §1 · p. 2
- ADMM-Nets achieve state-of-the-art MR reconstruction accuracy with fast computational speed.The paper presents Basic-ADMM-Net and Generic-ADMM-Net, and extends the approach to complex-valued images.Source: §1 · p. 2
- The authors present ADMM-Nets as combining model-based and deep-learning advantages, with potential applications beyond MRI, including deconvolution and general compressive sensing.The paper also reports comparisons with recently published deep-learning methods for CS-MRI.Source: §1 · p. 2
2 RELATED WORK
Prior CS-MRI methods trade reconstruction speed against image quality, while deep-learning methods offer fast reconstruction but bring architecture and training-data challenges. The paper positions ADMM-Net as a discriminatively learned alternative that retains model-based structure and reports improved reconstruction performance.
2.1 Model-based CS-MRI Methods
- Dictionary-learning and nonlocal CS-MRI methods generally produce higher-quality images but reconstruct more slowly than basic transform-based methods.Traditional transform-based methods run fast but are less accurate; particular sparse transforms may also introduce staircase or blocky artifacts.Source: §2.1 · p. 2
2.2 Deep Learning CS-MRI Methods
- Deep-learning CS-MRI methods reconstruct quickly, but network architecture is difficult to define and general networks require many training samples.Prior examples include mappings from down-sampled reconstructions and cascaded or residual networks.Source: §2.2 · p. 2
2.3 Discriminative Learning Methods
- ADMM-Net learns CS-MRI model and ADMM parameters within a non-conventional deep architecture, and the paper reports that it outperforms traditional CS-MRI models and widely used CNN structures.Prior discriminative-learning work includes unfolded optimization methods and learned shrinkage operators.Source: §2.3 · pp. 2–3
3 ADMM-NET FOR CS-MRI
This section derives a data-flow graph from ADMM procedures for a general CS-MRI model and builds deep ADMM-Nets with learned operation-specific parameters. It also describes a complex-valued version that processes real and imaginary parts separately using shared real-valued convolution filters and a shared piecewise linear transform.
3.1 General CS-MRI Model and ADMM Algorithm
- The general CS-MRI model reconstructs an image from undersampled k-space using Fourier measurements and transform-domain regularization, and can be solved with ADMM.The x-update can be computed efficiently by FFT; the paper notes that choosing transforms, nonlinear functions, and parameters across sampling rates is challenging.Source: §3.1 · pp. 3–4
3.2 Data Flow Graph for ADMM Algorithm
- The ADMM data-flow graph repeats reconstruction, denoising, and multiplier-update operations across stages, processing undersampled k-space data into a reconstructed image.The graph maps each ADMM iteration to a stage and provides the structure over which the deep networks are defined and trained.Source: §3.2 · p. 4, Figure 1 · p. 4
3.3 Deep ADMM-Nets
- Deep ADMM-Nets retain the ADMM graph structure while generalizing its operations into learnable network layers.The generalized operations include reconstruction, convolution, nonlinear transform, and multiplier-update layers.Source: §3.3 · p. 4
3.3.1 Basic-ADMM-Net
- Basic-ADMM-Net maps ADMM solver I into reconstruction, convolution, nonlinear-transform, and multiplier-update layers, using learnable piecewise-linear transforms beyond fixed hard or soft thresholds.Its network parameters include reconstruction filters, convolution filters, nonlinear-function control values, and multiplier-update parameters.Source: §3.3.1 · pp. 4–5, Figure 2 · p. 5
3.3.2 Generic-ADMM-Net
- Generic-ADMM-Net maps ADMM solver II into stages with distinct reconstruction and denoising layers, decomposing denoising into sublayers that implement gradient descent.Each stage corresponds to an ADMM iteration, and each denoising substage corresponds to a gradient-descent iteration.Source: §3.3.2 · pp. 5–6
3.3.3 Complex-ADMM-Net
- Complex-ADMM-Net extends Generic-ADMM-Net to complex-valued MR images and generalizes the operation in its nonlinear-transform layer.The paper motivates this extension by the importance of phase information in medical applications.Source: §3.3.3 · pp. 6–7
- Real-valued filters convolve complex-valued inputs, equivalently applying shared filters separately to the real and imaginary parts.The two parts use the same convolution filters.Source: §3.3.3 · p. 7
- The nonlinear transform processes real and imaginary parts separately, and both parts share the same piecewise linear transform function.Source: §3.3.3 · p. 7
4 NETWORK TRAINING
ADMM-Net parameters are trained end-to-end by minimizing normalized mean square error (NMSE) between network outputs and fully sampled ground truth. The section describes two initialization strategies and back-propagation through the network layers.
4 NETWORK TRAINING
- Network parameters are optimized against NMSE loss on pairs of undersampled k-space inputs and fully sampled ground-truth images.The loss compares the network output with the ground truth; L-BFGS is used for gradient-based parameter optimization.Source: §4 · p. 7
4.1 Initialization
- Model-based initialization sets DCT filters and soft-thresholding parameters from an ADMM solver, making the initialized network an exact realization of its optimization process.The baseline model uses an l1 regularizer in the DCT transform space.Source: §4.1 · p. 7
- Random initialization samples filters from a zero-mean Gaussian and initializes nonlinear transform values to implement ReLU.The Gaussian variance is described as well chosen; the nonlinear layer uses max(pi, 0).Source: §4.1 · p. 7
4.2 Gradient Computation by Back-propagation
- Back-propagation reverses the forward layer order: each stage processes X, Z, then M, while gradients are computed in the inverse order.The paper describes gradient computation for the layers of Generic-ADMM-Net and says the Basic-ADMM-Net method is similar.Source: §4.2 · p. 7
- For Complex-ADMM-Net, each parameter gradient sums contributions from the real and imaginary parts of layer outputs.The passage contrasts this with gradient computation in Generic-ADMM-Net.Source: §4.2 · p. 8
5 EXPERIMENTS
Experiments evaluate ADMM-Net configurations and compare reconstruction quality and speed across sampling rates, datasets, and data conditions. Generic-ADMM-Net is selected as the default, and the reported comparisons show strong reconstruction performance alongside computational efficiency.
5.3 Performances of Different Network Architecture
- The experiments vary filter count and size, iterations within stages, and stage count to examine accuracy and speed trade-offs.The authors describe wider and deeper networks as generally more accurate but computationally costlier.Source: §5 · p. 9, §5.3 · p. 9
5.1 Results of Two Versions of ADMM-Nets
- Generic-ADMM-Net converges faster than Basic-ADMM-Net, and its 38.03 dB result is 1.02 dB higher than Basic-ADMM-Net with 15 stages.The authors adopt Generic-ADMM-Net as the default for subsequent experiments.Source: Table 1 · p. 9, §5.1 · p. 9
5.2 Effect of Different Network Initialization
- Model-based initialization yields marginally higher reconstruction accuracy than random initialization at the same stage count, but DCT initialization limits the available filter counts.For example, the stated limits include 8 filters of size 3 × 3, 24 of size 5 × 5, or 48 of size 7 × 7.Source: §5.2 · p. 9, Table 2 · p. 9
5.3.1 Network Width
- With 3 × 3 filters, 128 filters achieve 0.28 dB higher accuracy than 64 filters but take around twice as long; increasing filter size improves results but slows reconstruction.The 7 × 7 network is only 0.08 dB more accurate than the 5 × 5 network and takes longer to test.Source: §5.3.1 · pp. 9–10, Table 3 · p. 10, Table 4 · p. 10
5.3.2 Network Depth
- Increasing sub-stage iterations improves accuracy, but adding stages is more effective: errors fall quickly through 10 stages, then marginally, while testing time grows linearly.With other parameters fixed, the authors select Nt = 1 and Ns = 10 as the default settings.Source: §5.3.2 · p. 10, Figure 9 · p. 10
5.4 Comparison
- Across brain-data sampling rates, ADMM-Net produces the best quality with comparable speed to conventional methods and outperforms state-of-the-art methods; at 30%, it beats PANO and FDLCP by 1.82 dB and runs around 13 times faster.At that rate, it is also 0.62 dB more accurate and around 10 times faster than BM3D-MRI.Source: Table 6 · p. 11, §5.4 · p. 11
- Brain-trained ADMM-Net remains competitive on chest data, while the chest-trained network performs best; on brain data, ADMM-Net also outperforms the compared deep-learning methods in reconstruction quality and training time.Visual comparisons at 20% and 10% sampling report preserved fine details without obvious artifacts; at 10%, ADMM-Net reconstructs image structure where other methods fail.Source: §5.4 · pp. 12–13, Table 7 · p. 12
5.5 Complex-valued and Noisy Data Reconstruction
- On complex-valued brain data at 20% sampling, Complex-ADMM-Net10 beats FDLCP by 1.96 dB; with noise, its PSNR falls from 35.40 dB to 35.37 dB while FDLCP drops 1.24 dB at noise level 0.015.BM3D-MRI cannot directly reconstruct complex-valued MR images.Source: Table 9 · p. 13, §5.5 · p. 13
6 CONCLUSION
The conclusion presents Basic-ADMM-Net and Generic-ADMM-Net as deep architectures derived from ADMM data-flow graphs for CS-MRI. It reports higher reconstruction accuracy with ADMM-like computational efficiency and notes that the reformulation idea may extend to other applications.
6 CONCLUSION
- The proposed ADMM-Nets combine a model-based approach with data-learned parameters, achieving higher reconstruction accuracy while keeping ADMM's computational efficiency.The authors state that reformulating ADMM as a deep network may also apply to image deconvolution and super-resolution.Source: §6 · p. 13