[Paper Review] k-Space Deep Learning for Accelerated MRI

Yoseob Han, Leonard Sunwoo+1 more|arXiv (Cornell University)|May 10, 2018

Sparse and Compressive Sensing Techniques30 references17 citations

TL;DR

This paper proposes a novel k-space deep learning framework for accelerated MRI that directly interpolates missing k-space data using a convolutional neural network inspired by Hankel matrix completion and convolutional framelets. The method outperforms image-domain deep learning approaches by leveraging data-driven, structured low-rank representations in k-space, enabling generalization across diverse sampling patterns and multi-coil data without retraining.

ABSTRACT

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.

Motivation & Objective

• To develop a fully data-driven deep learning method for k-space interpolation in accelerated MRI that avoids image-domain reconstruction.
• To leverage the duality between structured low-rankness in k-space and sparsity in the image domain for improved signal representation.
• To enable generalization across non-Cartesian k-space trajectories (e.g., radial, spiral) and multi-coil, calibration-free data using a single trained network.
• To theoretically and empirically demonstrate the expressive power of deep CNNs in k-space through framelet-based decomposition and ReLU-based spatially variant filtering.

Proposed method

• The method uses a U-Net-like encoder-decoder CNN to learn k-space interpolation directly, with skip connections to enhance feature propagation.
• The network is grounded in the mathematical equivalence between deep convolutional framelets and Hankel matrix decomposition, enabling structured low-rank representation in k-space.
• Recurrent ReLU activations introduce spatially variant filters, increasing model expressiveness by enabling multiple linear representations per input.
• A regridding layer is added to handle non-Cartesian k-space trajectories, allowing the same network to generalize across sampling patterns.
• Multi-coil k-space data are stacked along the channel dimension and fed into the network, enabling calibration-free, multi-channel reconstruction.
• The network is trained end-to-end using supervised loss on k-space data, avoiding iterative optimization or image-domain post-processing.

Experimental results

Research questions

• RQ1 Can a fully data-driven deep learning model in k-space outperform existing image-domain deep learning methods for accelerated MRI?
• RQ2 How does the expressiveness of a CNN in k-space relate to structured low-rank matrix completion and framelet theory?
• RQ3 Can a single trained k-space network generalize across diverse k-space sampling patterns, including non-Cartesian trajectories?
• RQ4 What is the role of ReLU nonlinearities in enhancing the representational capacity of k-space deep learning models?
• RQ5 How does the inclusion of skip connections affect the number of available linear representations in the k-space network?

Key findings

• The proposed k-space deep learning method consistently outperforms state-of-the-art image-domain deep learning approaches in terms of reconstruction accuracy across various acceleration factors.
• The network generalizes well to unseen k-space sampling patterns, including radial and spiral trajectories, due to its invariance to sampling geometry when combined with a regridding layer.
• Theoretical analysis confirms that the number of available linear representations in the network grows exponentially with depth, especially when skip connections are used.
• The use of ReLU activations enables spatially variant filtering, effectively creating diverse convolutional filters from the same set of parameters, enhancing model expressiveness.
• The method achieves calibration-free multi-coil MRI reconstruction by stacking coil channels in the input, eliminating the need for separate coil sensitivity estimation.
• Empirical results show that the network maintains high performance even with limited training data, due to the inductive bias from Hankel matrix structure and framelet-based learning.