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Deep Residual Learning for Accelerated MRI using Magnitude and Phase Networks

Dongwook Lee, Jaejun Yoo, Sungho Tak, Jong Chul Ye

arXiv:1804.00432v1cs.CVcs.AIcs.LGstat.ML

TL;DR

Accelerated MRI reconstruction must recover images from undersampled k-space, while conventional CS and parallel-imaging methods can be computationally costly. This paper proposes separately trained magnitude and phase residual networks, using image-domain post-processing for magnitude-only data and iterative k-space interpolation when phase is available. The approach removed strong coherent aliasing artifacts and produced good single- and multi-coil reconstructions with substantially faster computation than existing CS methods.

  • Problem

    Existing accelerated MRI reconstruction methods can have high computational complexity, and k-space-dependent methods cannot improve images when only magnitude data are available.

  • Method

    The paper separately trains deep residual magnitude and phase networks for image restoration and, when phase is available, iterative k-space interpolation using framelet representation.

  • Results

    The proposed network removed strong coherent aliasing artifacts and provided good reconstruction results for single- and multiple-coil data with much faster computation than existing compressed-sensing methods.

  • Takeaways & Limitations

    The framework supports both magnitude-only image-domain processing and complex-data iterative reconstruction, indicating potential for accelerated MRI reconstruction.

Abstract

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Accelerated magnetic resonance (MR) scan acquisition with compressed sensing (CS) and parallel imaging is a powerful method to reduce MR imaging scan time. However, many reconstruction algorithms have high computational costs. To address this, we investigate deep residual learning networks to remove aliasing artifacts from artifact corrupted images. The proposed deep residual learning networks are composed of magnitude and phase networks that are separately trained. If both phase and magnitude information are available, the proposed algorithm can work as an iterative k-space interpolation algorithm using framelet representation. When only magnitude data is available, the proposed approach works as an image domain post-processing algorithm. Even with strong coherent aliasing artifacts, the proposed network successfully learned and removed the aliasing artifacts, whereas current parallel and CS reconstruction methods were unable to remove these artifacts. Comparisons using single and multiple coil show that the proposed residual network provides good reconstruction results with orders of magnitude faster computational time than existing compressed sensing methods. The proposed deep learning framework may have a great potential for accelerated MR reconstruction by generating accurate results immediately.

I. INTRODUCTION

Accelerated MRI relies on undersampled k-space reconstruction, but established CS and parallel-imaging methods can be computationally expensive. The paper therefore investigates deep CNN and residual-learning approaches for removing aliasing artifacts, including complex-valued MR handling through separate magnitude and phase processing.

  • Motivation: Accelerated MRI reduces scan time by reconstructing images from insufficient k-space samples acquired through undersampling.Parallel MRI and compressed sensing are established approaches for accurate reconstruction from incomplete measurements.
  • Limitations of existing methods: Low-rank k-space methods such as SAKE and ALOHA achieve strong reconstruction performance but are limited by computational complexity.CS methods also require k-space access, preventing image-domain post-processing when only reconstructed images are available.
  • Deep-learning motivation: Deep convolutional networks are used to capture image-structure features for medical image reconstruction.CNNs learn convolutional filters from training data and have succeeded in several image regression tasks.
  • Proposed approach: The proposed residual approach separately trains phase and magnitude networks to reconstruct complex-valued MR data and correct globalized undersampling artifacts.Residual learning uses identity mappings to support deeper networks, while the proposed scheme handles magnitude-only and complex-image settings.
  • Reconstruction formulation: The reconstruction formulation models coil sensitivity-weighted images and undersampled measurements within a penalized least-squares framework.The formulation includes regularization and can incorporate GRAPPA-type constraints for parallel imaging.

A. Magnitude and Phase Networks

The method learns an image-domain mapping that can be embedded into iterative k-space interpolation when complex data are available, while magnitude-only data are processed directly in the image domain. Separate magnitude and phase networks support these two operating modes.

  • Frame-based interpolation: The method seeks accelerated MR reconstruction through k-space interpolation inspired by classical frame-based inpainting and perfect-reconstruction mappings.The fully sampled multi-coil k-space is represented as the target of the mapping.
  • Iterative reconstruction: Measured k-space samples are retained while estimated values from the learned mapping update the unmeasured locations during iterative reconstruction.The formulation uses a shrinkage operator to avoid the trivial solution.
  • Magnitude and phase networks: Because MRI data are complex-valued, the image-domain mapping is decomposed into separate magnitude and phase mappings.QM denotes magnitude mapping and QP denotes phase mapping.
  • Magnitude-only reconstruction: Magnitude-only images can be improved by image-domain post-processing without access to raw k-space data.Magnitude images from public datasets such as HCP can also be used to pre-train the magnitude network.
  • Complex-data reconstruction: When phase information is available, combining the phase network enables iterative k-space interpolation using a relaxed Krasnoselskii–Mann procedure.Algorithm 1 summarizes this iterative process for improving frame-based interpolation convergence.

B. Network Architecture

The proposed MR reconstruction uses a dual frame residual U-net with separately trained magnitude and phase networks. Magnitude-only inference supports image-domain restoration, while phase availability enables iterative k-space interpolation.

  • Network Architecture: The dual frame U-net replaces standard U-net processing to avoid duplicated low-frequency content.Its low-resolution residual is up-sampled and subtracted after unpooling.
  • Network Architecture: The residual network combines convolution, batch normalization, ReLU, pooling, unpooling, and contracting-path concatenation.Each stage uses sequential 3 × 3 convolutional, normalization, and ReLU layers, with a final 1 × 1 convolution.
  • Magnitude and Phase Networks: Magnitude and phase networks share the same architecture but are trained separately on aliased inputs and alias-free targets.The inputs and targets are formed from inverse-Fourier-transformed down-sampled k-space data.
  • Magnitude and Phase Networks: Phase masking removes random outside-brain phase fluctuations before phase-network training and inference.Masks are generated by thresholding reconstructed magnitude images, and phase outside the region of interest is set to zero.
  • Iterative Reconstruction: When phase is available, recursive training augments measured k-space with estimated samples for further reconstruction refinement.The iterative procedure is motivated by framelet representations and uses improved images as additional training data.

IV. METHOD

The method evaluates accelerated brain MRI reconstruction using HCP structural data and in vivo 3T and 7T data under multiple undersampling patterns. Images are normalized per scan, augmented in the complex domain, and acquired with a fixed acceleration factor.

  • Datasets: Three brain-image datasets were used, including HCP structural MRI for magnitude-network pre-training and in vivo k-space data from 3T and 7T scanners.The HCP set contributed 200 selected T2-weighted images from 1200 images for pre-training.
  • Datasets: The 3T in vivo dataset contained 81 axial brain slices from nine subjects, with separate training, validation, and testing subjects.Seven subjects supplied 61 training images, one supplied 8 validation images, and one supplied 12 test images.
  • Down-Sampled Data Generation: Five undersampling patterns varied low-frequency sampling, Gaussian or uniform random sampling, and ACS-line counts.The patterns included 12 ACS lines at 5% of phase encoding and 52 ACS lines at 20%.
  • Down-Sampled Data Generation: The acceleration factor was fixed at ×3.5 across all sampling patterns.Additional ACS lines were used to make aliasing primarily high-frequency and were also required for GRAPPA comparisons.
  • Preprocessing: Each image was normalized separately, and complex-domain flipping generated four additional in vivo training samples.Magnitude inputs were scaled to [0,256], while phase values were modified from [−π,π] to [0,2π].

C. Network Training

The networks were trained with stochastic gradient descent, early stopping, staged pre-training and recursive training, and evaluation against ALOHA and GRAPPA. Reconstruction quality used NRMSE and structural similarity with coil-combined references.

  • Training Procedure: Convolution weights used Xavier Gaussian initialization, and training minimized l2 loss with momentum-based stochastic gradient descent.Early stopping was applied as regularization against overfitting.
  • Training Procedure: Training on HCP data used 90 pre-training epochs, followed by 1410 in vivo epochs and 500 recursive-training epochs.The learning rate was reduced logarithmically from 10^-1 to 10^-2 each epoch.
  • Training Procedure: Phase training lasted 1000 epochs, with mini-batches of 10, because phase structures were considered simpler than magnitude structures.Training each network took approximately 14 hours for 3T data and 20 hours for 7T data.
  • Comparative Studies: ALOHA was the representative compressed-sensing baseline, while GRAPPA was additionally used for multi-channel uniform-sampling comparisons.GRAPPA used the same ACS lines and acceleration factor, with a 5×5 kernel.
  • Evaluation: Final magnitude reconstructions were compared using SSOS images, with SSOS of alias-free reconstructions as ground truth.Phase ground truth came from phase images reconstructed from fully sampled k-space.
  • Evaluation: Reconstruction performance was measured using normalized root mean square error and structural similarity index.NRMSE compares estimated and ground-truth images after normalization by the ground-truth mean.

V. EXPERIMENTAL RESULTS

Experiments evaluated sampling choices, network design, reconstruction quality, iterative refinement, phase-mask thresholds, and computational time. The proposed method reconstructed single- and multi-channel images accurately while substantially reducing reconstruction time.

  • Sampling patterns: 20% ACS lines with uniform random sampling were selected for subsequent experiments because they outperformed 5% ACS lines and slightly exceeded Gaussian random sampling.The comparison used multiple sampling patterns and ACS-line distributions.
  • Network comparison: The proposed separate magnitude-and-phase networks performed similarly to the real-and-imaginary baseline, with slightly better SSIM.This comparison supported using separate magnitude and phase training in later experiments.
  • Reconstruction quality: In single-channel reconstruction, the proposed method removed aliasing artifacts more accurately than ALOHA and achieved minimum NRMSE.ALOHA improved the input but retained some aliasing artifacts.
  • Reconstruction quality: For four-channel data, GRAPPA showed many reconstruction errors, ALOHA removed many but not all artifacts, and the proposed method provided excellent reconstruction.The comparison used magnitude images with SSOS for multi-channel MR images.
  • Iterative refinement: KM iteration produced minimum NRMSE and maximum SSIM, while pre-training on large HCP datasets improved network reconstruction performance.These results were summarized through Table II and Fig. 8.
  • Computational time: The proposed network required approximately 30 ms for single-channel and less than 41 ms for multi-channel reconstruction, versus approximately 30 s for GRAPPA multi-channel reconstruction.Phase reconstruction took approximately 61 ms per coil, and 15 KM iterations required approximately 9 s.

VI. DISCUSSION

The proposed residual network addresses artifact-pattern variability, phase-related concerns, and reconstruction quality while enabling very short post-training reconstruction times. It handled single- and multi-channel data and removed strong coherent aliasing artifacts that compared methods could not remove.

  • Robustness: The image-domain network’s performance depends on artifact patterns, so varied acquisition parameters in training improved robustness.Training included spin echo and GRE images with varied TEs and TRs.
  • Phase handling: Phase wrapping could potentially complicate training, but wrapped-phase training caused no problems in these experiments.The final complex images were not affected by phase wrapping.
  • Image quality: A different loss function may improve visual quality because the l2 loss reduces MSE but can produce blur through output averaging.The authors identify adversarial or perceptual losses as future directions.
  • Computational cost: The network required several hours of training but reconstructed images quickly after training, a major advantage over iterative CS methods.The method’s short reconstruction time is emphasized as a practical benefit.
  • Reconstruction results: The method worked on single- and multi-channel data and removed strong coherent aliasing artifacts that current parallel and CS methods could not remove.The reported advantages included both reconstruction quality and computational time.

APPENDIX

The appendix presents deep convolutional framelets as a signal-representation foundation for encoder-decoder CNNs. It explains how non-local and local bases, frame conditions, and learned filters connect framelet expansions to magnitude and phase networks.

  • Framelet representation: The framelet theory uses non-local and local bases to represent multichannel signals while learning shrinkage behavior from training data.The signal is modeled as a p-channel signal of length n, with non-local basis Φ and local basis Ψ.
  • Framelet representation: When the frame condition holds, the signal can be expressed through framelet coefficients.The frame condition uses the identity matrix I_n×n.
  • CNN connection: The convolutional operations in the framelet expansion correspond to standard multichannel CNN convolutions.The local basis acts as convolutional filters, while the non-local basis corresponds to generalized pooling.
  • CNN connection: A deep encoder-decoder architecture emerges recursively from the convolutional framelet expansion, with learned local bases controlling the network’s low-dimensional structure.The local bases are learned from training data, and filter-channel counts control the dimensionality.

B. Dual frame U-Net Derivation

The dual-frame U-Net is introduced to address frame-condition defects in average-pooling U-Nets. Its augmented framelet representation preserves the intended signal structure through a low-resolution residual bypass and unpooling design.

  • Limitation of U-Net: Average-pooling U-Net bases violate the frame condition and cause high-frequency signal loss.A conventional bypass does not fully solve the issue because it duplicates low-frequency content.
  • Dual-frame construction: The dual-frame U-Net replaces bypass-derived convolutional framelet coefficients with an augmented framelet coefficient matrix.This design is derived to satisfy the frame condition.
  • Dual-frame construction: The augmented dual frame is constructed using a matrix identity, the matrix inversion lemma, and average-pooling orthogonality.The resulting coefficients define the reconstructed signal through the dual frame.
  • Architecture: Unlike U-Net, the dual-frame design upsamples the low-resolution residual through unpooling and subtracts duplicated low-frequency content.This is implemented with an additional low-resolution bypass connection.

C. Application to 7T Data

The 7T application evaluated the method on high-resolution axial brain images acquired with a 32-channel coil. Under 20% ACS-line sampling, the algorithm significantly improved image quality and reduced aliasing artifacts.

  • Dataset: 567 images from nine subjects were acquired on a 7T scanner using a 292×292 matrix, 0.75 mm slices, and 32 head coils.Seven subjects provided 441 training images, one provided 63 validation images, and one provided 63 test images.
  • Results: Under uniform random sampling with 20% ACS lines, the proposed algorithm significantly improved image quality and reduced aliasing artifacts.The application used the 7T dataset to demonstrate performance on high-resolution images.
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