Deep Learning Methods for Parallel Magnetic Resonance Image Reconstruction

Florian Knoll; Kerstin Hammernik; Chi Zhang; Steen Moeller; Thomas Pock; Daniel K. Sodickson; Mehmet Akcakaya

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Deep Learning Methods for Parallel Magnetic Resonance Image Reconstruction

Florian Knoll, Kerstin Hammernik, Chi Zhang, Steen Moeller, Thomas Pock, Daniel K. Sodickson, Mehmet Akcakaya

arXiv:1904.01112v1Apr 1, 2019eess.SPcs.CVcs.LGeess.IV

TL;DR

Accelerated MRI reconstruction needs to integrate multi-coil information, but this aspect has been studied less frequently than image-based deep-learning reconstruction. The manuscript reviews classical parallel imaging and neural-network approaches for image-domain regularization and k-space completion, highlighting reported reconstruction improvements alongside open data, training, and hallucination concerns.

Problem
Multi-coil recovery of missing k-space lines is less studied than other deep-learning MRI reconstruction approaches despite being standard in accelerated clinical MRI acquisitions.
Method
The manuscript surveys classical parallel imaging and recent machine-learning methods for image-domain regularization and k-space interpolation.
Results
Learned reconstructions are reported to improve artifact removal and small-feature preservation, while RAKI achieves the highest SSIM among compared subject-specific calibration methods in the cited examples.
Takeaways & Limitations
Machine learning methods offer approaches for improving parallel MRI reconstruction quality, but the field remains under development with open problems and community dataset efforts.
Takeaways & Limitations
Machine-learning reconstruction is constrained by the need for curated training data that generalizes to unseen test data.

Abstract

from arXiv · show

Following the success of deep learning in a wide range of applications, neural network-based machine learning techniques have received interest as a means of accelerating magnetic resonance imaging (MRI). A number of ideas inspired by deep learning techniques from computer vision and image processing have been successfully applied to non-linear image reconstruction in the spirit of compressed sensing for both low dose computed tomography and accelerated MRI. The additional integration of multi-coil information to recover missing k-space lines in the MRI reconstruction process, is still studied less frequently, even though it is the de-facto standard for currently used accelerated MR acquisitions. This manuscript provides an overview of the recent machine learning approaches that have been proposed specifically for improving parallel imaging. A general background introduction to parallel MRI is given that is structured around the classical view of image space and k-space based methods. Both linear and non-linear methods are covered, followed by a discussion of recent efforts to further improve parallel imaging using machine learning, and specifically using artificial neural networks. Image-domain based techniques that introduce improved regularizers are covered as well as k-space based methods, where the focus is on better interpolation strategies using neural networks. Issues and open problems are discussed as well as recent efforts for producing open datasets and benchmarks for the community.

I. INTRODUCTION

MRI acceleration is clinically important because MRI acquisition is inherently slow, and parallel imaging reduces scan time by exploiting multi-coil information. The paper introduces image-space and k-space perspectives on reconstructing undersampled data.

Motivation: MRI scan-time reduction is important for clinical use because MRI acquisition is inherently slower than several other imaging modalities.MRI is described as a leading diagnostic modality, making shorter scans a major driver of routine clinical application.
Multi-coil acquisitions: Parallel imaging reduces MRI scan time by using spatial information from multiple receiver coils alongside gradient encoding.Phased-array coils provide multiple spatially localized measurements that are combined into a complete field of view.
Reconstruction domains: K-space methods recover missing data before inverse Fourier transformation and coil combination, whereas image-space methods transform first and then remove aliasing using coil sensitivities.This distinction is illustrated in Figure 1.
Classical categories: The paper organizes classical parallel imaging into image-domain methods inspired by SENSE and k-space methods inspired by SMASH and GRAPPA.The two categories are closely related but are treated separately for historical reasons.

II. CLASSICAL PARALLEL IMAGING IN IMAGE SPACE

Classical image-space parallel imaging reconstructs an image after inverse Fourier transformation using receive-coil sensitivity information. MRI reconstruction is formulated as an inverse problem, with iterative methods addressing computational demands and solving convex objectives in standard parallel imaging.

Image-space formulation: SENSE removes aliasing artifacts in image space after inverse Fourier transformation and uses precomputed receive-coil sensitivity maps.Sensitivity maps may come from a separate reference scan or a fully sampled central k-space block.
Image-space formulation: MRI reconstruction recovers image u from measurements f through a forward encoding operator E that includes sampling and coil sensitivities, with measurement noise n.In classic parallel imaging, the number of receive elements generally exceeds the acceleration factor, producing an over-determined system.
Computational considerations: Uniform Cartesian sampling enables SENSE to resolve a small number of sharply folded pixels efficiently, whereas alternative sampling strategies increase computational and memory demands.The dimensions of E scale with acquired k-space points times N^2, motivating iterative methods such as CG-SENSE.
Computational considerations: CG-SENSE finds a minimizer of a quadratic cost function, and standard parallel imaging solves the resulting convex problem using methods including conjugate gradients and ADMM.Other listed numerical approaches include gradient descent, Landweber iterations, and primal-dual methods.
Didactic example: The manuscript illustrates reconstruction approaches using a single coronal knee slice acquired on a clinical 3T system with a 15-channel phased-array coil.The example uses a turbo spin echo sequence and 0.5 mm^2 in-plane resolution.

B. Nonlinear regularization and compressed sensing

Classical MRI reconstruction can incorporate prior knowledge through penalty terms that balance data fidelity against regularization. Compressed sensing commonly uses nonlinear ℓ1-based regularization, and the manuscript demonstrates a combined parallel-imaging and compressed-sensing reconstruction with Total Generalized Variation.

Nonlinear regularization: Additional penalty terms extend the quadratic reconstruction objective into a constrained optimization problem, forming the basis of modern MRI reconstruction methods.The penalties encode a priori knowledge beyond the measurement model.
Nonlinear regularization: Regularization terms Ψi are weighted by parameters λi that balance data fidelity against the incorporated prior.Since compressed sensing was introduced and adopted for MRI, nonlinear ℓ1-norm-based penalties have become common in parallel imaging.
Demonstration: A Total Generalized Variation constraint demonstrates nonlinear combined parallel imaging and compressed-sensing reconstruction using λ=2.5 · 10^-5 and 1000 primal-dual iterations.The example uses equidistant sampling, which the passage notes is not optimal for compressed-sensing incoherence.

III. CLASSICAL PARALLEL IMAGING IN K-SPACE

K-space parallel imaging treats reconstruction as interpolation of unacquired lines from acquired multi-coil data. Unlike image-domain methods that typically output one combined image, k-space methods output data across all coils.

K-space formulation: Parallel imaging can be formulated in k-space as an interpolation procedure connected to the image-domain SENSE inverse problem.The acquired data across coils are represented through a linear operator Gacq.
K-space formulation: Acquired k-space lines across all coils are modeled with a linear operator, while unacquired lines are expressed through the corresponding reconstruction formulation.The two formulations can be combined to relate missing lines to acquired measurements.
Interpolation: Unacquired k-space lines can be interpolated from acquired lines across all coils when the pseudo-inverse G†acq of Gacq exists.This provides the algebraic basis for classical k-space parallel-imaging interpolation.
Output representation: K-space methods produce reconstructed data across all coils, whereas image-domain methods typically produce a single image combining coil information.The distinction concerns the output representation of the reconstruction.

A. Linear k-space interpolation in GRAPPA

GRAPPA reconstructs missing uniformly undersampled k-space lines through calibrated linear convolutional kernels that combine neighboring multi-coil data. Its clinical utility is supported by favorable noise and motion characteristics relative to SENSE.

GRAPPA formulation: GRAPPA uses linear shift-invariant convolutional kernels to interpolate missing k-space lines from uniformly spaced acquired lines.The kernels estimate missing locations for each target coil using neighboring data across coils.
Calibration: The kernels are calibrated for each subject using either a separate reference scan or a fully sampled central autocalibration signal region.Sliding windows form calibration-matrix rows from acquired multi-coil data and target entries from missing locations.
Properties and use: GRAPPA has favorable properties compared with SENSE, including lower or locally sub-unity g-factors, smoother g-factor maps, and reduced motion sensitivity.These properties have supported its use in large-scale projects such as the Human Connectome Project.

B. Advances in k-space interpolation methods

Parallel MRI reconstruction has expanded beyond linear interpolation toward self-consistency, nonlinear modeling, low-rank completion, and learned image-domain networks. These approaches address noise and modeling limitations through alternative k-space constraints or trainable regularization and data-consistency schemes, with learned reconstruction showing improved example quality.

Motivation: GRAPPA is widely used but can amplify noise as a function of coil geometry and acceleration rate, motivating alternative reconstruction strategies.The limitation is attributed to GRAPPA’s linear formulation.
Alternative k-space methods: SPIRiT enforces self-consistency across receiver-coil k-space data by exploiting correlations between neighboring k-space points.Like GRAPPA, it estimates a linear shift-invariant convolutional kernel from ACS data, and subspace analysis connects its kernels with coil sensitivities.
Alternative k-space methods: Nonlinear GRAPPA uses a kernel approach because measurement noise in calibration data creates errors in both regressors and regressands.The paper presents this as a nonlinear interpolation strategy for uniformly undersampled parallel imaging acquisitions.
Alternative k-space methods: SAKE reformulates k-space reconstruction as low-rank matrix completion by arranging local multi-coil neighborhoods into a block-Hankel matrix.It is described as an early matrix-completion approach to parallel imaging reconstruction.
Learned image-domain reconstruction: Learned image-domain reconstructions unroll iterative algorithms into T network stages, combining trainable regularization with data consistency.Different architectures vary in parameter sharing and in whether data consistency uses gradient-based or unrolled conjugate-gradient updates.
Training: Supervised learned reconstruction retrospectively undersamples fully sampled multi-coil k-space, compares the network output with a reference, and optimizes trainable parameters using a selected error metric.Common metrics include MSE, the ℓ1 norm, and SSIM; training uses non-convex optimization with standard numerical optimizers and backpropagation.
Example result: In a retrospectively accelerated coronal knee example, the learned reconstruction removed artifacts and preserved small features better than CG-SENSE and constrained reconstructions, achieving the highest SSIM.The comparison was made against a fully sampled reference.

V. MACHINE LEARNING METHODS FOR PARALLEL IMAGING IN K-SPACE

Machine learning improves k-space parallel imaging through scan-specific and database-trained neural-network interpolation, with RAKI-family methods reducing noise or artifacts in several acquisitions while retaining application-specific limitations.

Training strategies: Neural k-space interpolation methods divide into scan-specific approaches trained on ACS lines and database-trained approaches using prior datasets.Both groups adapt the interpolation strategy beyond conventional linear calibration methods.
RAKI: RAKI trains multi-layer CNNs on ACS data to interpolate missing k-space points for each scan without requiring a training database.Its CNN parameters are learned using an MSE loss, analogous to GRAPPA but with a nonlinear interpolation function.
RAKI: In 7T brain imaging at 0.6 mm isotropic resolution, RAKI visibly reduced noise amplification compared with linear GRAPPA under prospective acceleration.The scan-specific design is useful when fully sampled reference datasets are difficult to acquire.
RAKI: For retrospectively accelerated knee imaging, RAKI reduced residual aliasing and noise amplification, although residual artifacts remained and RAKI had the highest SSIM among calibrated methods.The artifacts were attributed to the small ACS region relative to the acceleration rate.
Residual RAKI: Residual RAKI extends the method to arbitrary sampling and learns a residual CNN based on GRAPPA interpolation to reduce associated noise amplification and artifacts.The residual output is given a physical interpretation as a correction to the GRAPPA reconstruction.
Other neural k-space methods: CNNs have also replaced ALOHA’s weighted k-space completion step, improving computational time and reconstruction quality by exploiting structures beyond Hankel-matrix low-rankness.This approach uses an MSE-trained CNN for completion of the weighted k-space.
DeepSPIRiT: DeepSPIRiT uses distinct CNNs for different k-space regions and successively refines reconstructed resolution, while database training removes the need for scan-specific calibration data.The method was reported to reduce aliasing artifacts but had difficulty with high-resolution content.

A. Issues and open problems

Machine-learning reconstruction can reduce inference time and tailor regularization, but its clinical use remains constrained by training-data requirements, computational demands, uncertain generalization, and risks from loss-function choices.

Training and inference have different computational profiles: a reported experiment required 40 hours of training, while one-slice reconstruction took 200ms.The same comparison reports 10ms for zero filling, 150ms for CG-SENSE, and 10000ms for PI-CS TGV reconstruction.
Some end-to-end k-space-to-image models avoid acquisition information in the forward operator, but their parameter counts and memory needs challenge clinical matrix sizes.The passage also notes that k-space machine-learning methods are less numerous and difficult to generalize because k-space intensities differ between central and outer regions.
Machine-learning models require representative curated training data, which remains difficult for applications lacking fully sampled reference labels.This particularly affects moving-organ imaging and diffusion MRI, where spatio-temporal constraints can prevent fully sampled acquisitions.
Training requires computing resources, although the paper reports experiments using hardware costing less than 10,000 USD.The cited experiments used resources described as generally available in academic institutions, with increasing access to cloud-based solutions.
Non-convex models have less-understood failure modes and generalization than conventional convex iterative methods.Reported generalization across image contrast does not eliminate susceptibility to systematic SNR deviations, and model specificity across exam and scanner settings remains open.
GAN-based losses can produce visually realistic images containing features unsupported by acquired data, so hallucinations must be avoided in medical imaging.Combining GANs with conventional error metrics such as MSE is described as one mitigation strategy.

B. Availability of training databases and community challenges

Public multi-channel raw k-space databases are scarce, limiting participation and making published reconstruction methods difficult to compare when training data are not shared.

The lack of publicly available multi-channel raw k-space databases restricts participation largely to researchers at academic medical centers with access to such data.This excludes much of the broader machine-learning community despite its relevant theoretical and algorithmic expertise.
Unshared training data make it nearly impossible to compare proposed reconstruction approaches fairly across publications.Training data are described as an essential part of a model’s performance, while public raw k-space initiatives are growing but datasets remain scarce.

VII. CONCLUSION

Machine learning has been proposed to improve parallel-imaging MRI reconstruction through image-domain regularization and k-space completion, while the field retains open problems and high-impact applications.

Machine-learning methods target improved parallel-imaging MRI reconstruction through image-domain regularization and k-space completion.The conclusion identifies both approaches as recent developments in a field that remains under development.
Open problems and high-impact applications remain relevant to the broader signal-processing community.The conclusion characterizes the field as still developing rather than settled.