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k-Space Deep Learning for Accelerated MRI

Yoseob Han, Leonard Sunwoo, Jong Chul Ye

arXiv:1805.03779v3cs.CVcs.LGstat.ML

TL;DR

The paper addresses the challenge of efficiently reconstructing missing k-space data while retaining the concise representations exploited by low-rank Hankel methods. It proposes a data-driven convolutional k-space interpolation network inspired by the connection between Hankel decomposition and convolutional framelets. Across Cartesian and non-Cartesian sampling settings, the method consistently outperforms existing image-domain deep-learning approaches.

  • Problem

    Existing MRI reconstruction approaches often process images or alternate between k-space and image domains, while low-rank Hankel completion has substantial computational complexity and memory requirements.

  • Method

    The paper uses a fully data-driven convolutional neural network to directly interpolate missing k-space data, trained with an image-domain loss and extendable to non-Cartesian trajectories.

  • Results

    The proposed k-space interpolation method consistently outperforms existing image-domain deep-learning methods across sampling trajectories and provides the best quantitative values in the reported comparisons.

  • Takeaways & Limitations

    K-space deep learning provides a fully data-driven interpolation framework for Fourier imaging problems, including non-Cartesian sampling and multi-coil data.

Abstract

from arXiv · show

The annihilating filter-based low-rank Hankel matrix approach (ALOHA) is one of the state-of-the-art compressed sensing approaches that directly interpolates the missing k-space data using low-rank Hankel matrix completion. The success of ALOHA is due to the concise signal representation in the k-space domain thanks to the duality between structured low-rankness in the k-space domain and the image domain sparsity. Inspired by the recent mathematical discovery that links convolutional neural networks to Hankel matrix decomposition using data-driven framelet basis, here we propose a fully data-driven deep learning algorithm for k-space interpolation. Our network can be also easily applied to non-Cartesian k-space trajectories by simply adding an additional regridding layer. Extensive numerical experiments show that the proposed deep learning method consistently outperforms the existing image-domain deep learning approaches.

I. INTRODUCTION

Deep learning MRI reconstruction has mainly used image-domain processing, cascaded k-space/image updates, or end-to-end transforms, while this paper introduces direct k-space interpolation with a convolutional network. The approach is motivated by structured low-rank k-space representations and extends to non-Cartesian sampling and multi-coil data.

  • Prior approaches: Existing MR deep-learning methods include image-domain learning, cascaded k-space/image-domain networks, and AUTOMAP-style end-to-end recovery.AUTOMAP estimates the Fourier transform with fully connected layers.
  • Limitations of prior approaches: AUTOMAP is limited to sufficiently small images because its fully connected layers require substantial memory.This motivates alternatives that avoid fully connected layers.
  • Proposed approach: The proposed method directly interpolates missing k-space data and reconstructs images by applying the Fourier transform to the interpolated data.Its convolutional implementation avoids fully connected layers and has minimal GPU memory requirements.
  • Motivation: The method combines k-space structured low-rankness, image-domain sparsity, and deep convolutional framelet theory to learn concise signal representations.The paper links Hankel matrix decomposition with encoder-decoder convolutional representations.
  • Extensions: The framework supports radial and spiral sampling and enables calibration-free multi-channel interpolation by stacking multi-coil k-space data along the channel direction.These extensions are presented as requiring a general sampling framework and channel stacking, respectively.

C. Low-Rank Hankel Matrix Approaches

Compressed sensing and ALOHA exploit image sparsity or low-rank Hankel structure, but Hankel completion is computationally and memory intensive. The paper connects this structure to convolutional framelet and encoder-decoder representations for efficient learned k-space interpolation.

  • Compressed sensing: Compressed sensing MRI seeks a feasible image-domain solution with minimum nonzero support under a sparsifying transform, often requiring iterative k-space/image-domain updates.The undersampled measurements impose data consistency through the downsampling operator.
  • Hankel approaches: Low-rank Hankel methods convert compressed sensing into structured matrix completion or impose a low-rank penalty directly on a Hankel matrix built from k-space data.The Hankel matrix is constructed from the measured k-space signal.
  • Hankel approaches: Large matrix-factorization complexity and memory requirements for storing Hankel matrices are major technical obstacles to low-rank Hankel completion.The paper identifies deep learning as an efficient way to address this problem.
  • ALOHA formulation: ALOHA jointly optimizes a k-space signal and filter bases subject to data consistency and a low-rank Hankel constraint, while its cost is defined in the image domain.The estimated rank is denoted by s.
  • Framelet representation: Deep convolutional framelets represent low-rank Hankel-constrained signals through encoder-decoder convolutions derived from singular-vector bases and generalized pooling operators.The encoder and decoder filters are obtained by reordering matrix factors.
  • Framelet representation: Recursive encoder-decoder application increases the net convolutional filter length, providing a learned signal representation for k-space interpolation.The resulting complex-valued signal space is determined by the encoder and decoder filters.

B. Generalization and Depth

The training formulation separates learning filter bases from inference, while ReLU nonlinearities provide input-dependent linear representations that support generalization. The reconstruction flow also accommodates skipped connections, weighting, and multi-coil inputs.

  • Generalization: Training is separated into learning filter bases from training data and estimating the interpolated k-space signal for new inputs during inference.The paper identifies generalization from multiple training examples as a key issue in translating the signal-representation theory to neural-network training.
  • Generalization: Figure 2 illustrates input-space partitioning for a two-channel, three-layer ReLU network and the associated linear representation selected for each input.The example is defined over R2.
  • Generalization: ReLU nonlinearities partition the input space into regions, with each region sharing a linear signal representation selected according to the input k-space data.This provides an input-adaptive mechanism for a fixed trained filter set.
  • Reconstruction flow: The proposed reconstruction flow uses a skipped connection, optionally combined with weighting and unweighting layers, before inverse Fourier transformation.The weighting-layer variant is shown as a separate reconstruction flow.
  • Reconstruction flow: For parallel imaging, multi-coil k-space data are stacked along the channel direction as both network input and output.This describes the multi-channel reconstruction configuration.

C. Extension to parallel imaging

Parallel imaging extends the low-rank Hankel formulation across receiver coils by stacking multi-coil k-space data along the channel direction. The resulting network retains the single-channel structure apart from input and output channel counts.

  • The extended Hankel structured matrix formed from the k-space measurements of multiple receiver coils is low-ranked.
  • Multi-coil k-space data are incorporated by stacking each coil's data along the channel direction.
  • The parallel-imaging network is identical to the single-channel k-space-learning network except for its input and output channel numbers.

D. Sparsification

The proposed network sparsifies k-space learning through residual or transform-based representations that yield low-rank Hankel structure. It uses a U-Net-style complex-valued k-space architecture, with trajectory-specific regridding for non-Cartesian data.

  • Residual sparsification: Residual k-space data become mostly high-frequency and sparse when low-frequency measurements, including the DC component, are acquired.
  • Residual sparsification: The skipped connection before the network exploits the residual's low-rank Hankel structure, while inner skipped connections make the frame basis more expressive.
  • Transform sparsification: Finite-difference or wavelet whitening transforms produce innovation signals whose weighted k-space data have low-rank Hankel matrices.
  • Trajectory handling: Non-Cartesian radial and spiral trajectories use NUFFT regridding to Cartesian coordinates, whereas Cartesian data use zero-filling for unacquired regions.
  • Network implementation: The network uses complex-valued k-space input and output with convolution, ReLU, batch normalization, pooling, unpooling, and concatenation operations.
  • Training: Training places a Fourier transform after k-space interpolation and computes an l2 loss in the image domain using SGD.

V. MATERIAL AND METHODS

The study evaluates k-space deep learning across Cartesian, radial, and spiral sampling in single- and multi-coil MRI, using quantitative and visual comparisons against compressed-sensing and CNN baselines. The proposed method directly interpolates k-space and generally provides strong reconstruction quality, while its convergence behavior depends on the sampling trajectory and architecture.

  • Data and sampling: Evaluations covered single- and multi-coil k-space data with Cartesian, radial, and spiral trajectories.Cartesian data used R = 3 sampling; radial and spiral experiments used R = 6 and R = 4, respectively.
  • Data and sampling: Cartesian experiments used 18 knee cases for training, one for validation, and one for testing after Gaussian undersampling with 10% ACS lines.The net acceleration factor was about 3 (R = 3).
  • Comparative methods: The comparison included TV-penalized CS, ALOHA, variational and cascade CNNs, KIKI, and an image-domain CNN trained on the same data.The cascade and cross-domain networks were modified for parallel imaging, and TV-based k-space weighting was used for ALOHA and the proposed method.
  • Training and architecture: In single-coil training, the selected architecture was Fig. 3(b) for Cartesian data and the skipped-connection architecture in Fig. 3(a) for radial and spiral data.Fig. 3(b) had the lowest Cartesian validation objective, whereas Fig. 3(a) generalized better for non-Cartesian trajectories.
  • Results: The proposed method produced the best quantitative values across the reported single-coil Cartesian, multi-coil Cartesian, radial, and spiral comparisons.For single-coil Cartesian reconstruction it preserved high-frequency edges and textures; computation was about 3.5 times faster than TV-penalized CS, while ALOHA was more than 100 times slower in the eight-coil comparison.

VII. DISCUSSION

The discussion attributes the proposed method’s performance to combining efficient k-space signal representation with deep-network expressivity. It reports that fully data-driven k-space interpolation outperformed image-domain deep learning across sampling trajectories.

  • Large matrix-pencil sizes needed by ALOHA are limited by memory and computational burdens.
  • The proposed network uses cascaded convolutions to obtain longer filters while retaining adaptation to different input images through ReLU nonlinearity.
  • The method combines concise k-space signal representation with neural-network expressivity to improve performance over image-domain learning.
  • Fully data-driven k-space interpolation is feasible using k-space deep learning with an image-domain loss function.
  • The proposed k-space interpolation network outperformed existing image-domain deep learning methods for various sampling trajectories.

I. HANKEL MATRIX CONSTRUCTION

The supplement represents convolution operations as structured matrix products involving wrap-around and extended Hankel matrices. This construction extends from single-channel to multi-channel convolutions through block-structured filter matrices.

  • For periodic one-dimensional signals, SISO convolution is represented using a wrap-around Hankel matrix.
  • The matrix pencil parameter d determines the column structure of the wrap-around Hankel matrix.
  • MIMO convolution maps a P-channel input to a Q-channel output through channel-specific filters of length d.
  • The corresponding MIMO convolution has a block-structured matrix representation defined by the multi-channel filter kernel.
  • The multi-channel Hankel construction uses an extended Hankel matrix formed by stacking P Hankel matrices side by side.

II. REAL AND IMAGINARY CHANNEL SPLITTING

The supplement uses an orthonormal matrix representation and a surrogate rank constraint to simplify Hankel matrix decomposition. Experiments compare reconstruction quality across radial and spiral single-coil settings.

  • The matrix T is orthonormal in the proposed decomposition.
  • Equation (30) is replaced by equation (31) to avoid implementation complications from interactions between real and imaginary channels.
  • 0.9809 versus 0.9741 was obtained for brain-region SSIM by the proposed method and image-domain network, respectively, in radial sampling at R = 6.
  • The radial and spiral experiments report better image quality and quantitative metrics for the proposed method, except for overall SSIM.
  • 0.9872 versus 0.9747 was obtained for brain-region SSIM by the proposed method and image-domain network, respectively, in spiral sampling at R = 4.

B. Radiological evaluation

Radiological and robustness evaluations examine non-Cartesian parallel imaging, noise sensitivity, weighting, acceleration factors, and unseen sampling patterns. The reported results favor k-space learning for structural detail and robustness, with a noise-dependent limitation.

  • B. Radiological evaluation: Radiologist evaluation found that k-space deep learning preserved sophisticated structure and detailed texture better than comparative methods in non-Cartesian parallel imaging.
  • B. Radiological evaluation: The proposed method clearly displayed separate internal cerebral veins and small T2 hyperintensity lesions that comparative methods blurred or failed to show.
  • C. Sensitivity with respect to noisy data: 0% to 3% noise scales favored k-space learning over image-domain learning, while higher noise produced more PSNR and NMSE errors for k-space learning.
  • C. Sensitivity with respect to noisy data: Training with noisy data was suggested as a way to learn noisy-to-noiseless k-space mapping and potentially improve robustness.
  • Weighted networks produced sharper images and better quantitative performance than non-weighted networks.
  • Fourier-domain methods were reported as more robust than image-domain methods across acceleration and sampling patterns not used during training.

IV. UNDERSTANDING GEOMETRY OF ENCODER-DECODER CNNS

This section extends the ALOHA–deep-learning relationship from a single image to multiple images, where generalization across differing sparsity patterns remains unclear. It motivates understanding how encoder-decoder CNN geometry and data-adaptive representations address this challenge.

  • Multiple-image training raises whether large-dimensional subspaces are needed to approximate all images and how learning generalizes across image sparsity patterns.
  • Dictionary learning offers a related perspective because sparsity priors select basis functions adapted to each specific image.
  • The companion theoretical framework relates encoder-decoder CNNs to nonlinear basis representations using combinatorial convolution frames.

A. Expressivity

The section characterizes symmetric encoder-decoder CNNs through input-dependent ReLU masks and combinatorial basis selection. Network depth and skipped connections increase the number of available linear representations, partitioning the input space into regions with shared representations.

  • Architecture: A symmetric encoder-decoder maps input x through an encoder feature space and a decoder to output y, with equal encoder and decoder depth κ.The input and output dimensions are both d0.
  • ReLU-based representation: ReLU masks enable combinatorial basis selection, allowing representations to adapt to different inputs rather than relying on one fixed frame basis.Without nonlinearity, the estimated frame and dual are optimal for training data but are described as less expressive for different inputs.
  • ReLU-based representation: ReLU determines input-dependent diagonal masks with 0 or 1 entries, yielding multiple matrix representations from masked basis factors.The diagonal entries are determined by the ReLU output.
  • Expressivity: The maximum number of distinct configurations represented by the masked matrix is 2^d1.The configurations arise because each diagonal element of Λ(x) is either 0 or 1.
  • Expressivity: The number of linear representations increases exponentially with network depth, while skipped connections further increase encoder-decoder CNN expressivity.The input space is partitioned into Nrep non-overlapping regions whose inputs share the same linear representation.
  • ReLU-based representation: With ReLU, cascaded convolutional interpretation remains valid as spatially varying masks produce spatially variant cascaded convolutions.
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