Yoseob Han^{1} and Jong Chul Ye^{1}

The annihilating filter-based
low-rank Hankel matrix approach (ALOHA) [1] is one of the most recent
compressed sensing (CS) approaches that directly interpolates the missing *k*-space data using low-rank Hankel matrix completion.
Inspired by the recent low-rank Hankel matrix decomposition using data-driven
framelet basis [2], we propose a completely data-driven deep learning algorithm
for *k*-space interpolation. In particular, our method can be
applied directly by simply adding an additional re-gridding layer to
non-Cartesian *k*-space trajectories such as radial trajectories.

Introduction

Recently, inspired by the tremendous success of deep learning in low-level computer vision problems, many researchers have investigated deep learning approaches for various biomedical image reconstruction problems and successfully demonstrated significant performance gain.

In particular, Wang et al [1] used the deep learning reconstruction either as an initialization or a regularization term. Deep network architecture using unfolded iterative compressed sensing (CS) algorithm was also proposed. The authors [2] tried to learn a set of regularizers under a variational framework. Multilayer perceptron [3] was introduced in for accelerated parallel MRI. An extreme form of the neural network called AUtomated TransfOrm by Manifold APproximation (AUTOMAP) [4] even attempts to estimate the Fourier transform itself using fully connected layers. All these pioneering works have consistently demonstrated superior reconstruction performances over the CS at significantly lower run-time computational complexity.

However, although the end-to-end recovery approach like AUTOMAP may directly recover the image without ever interpolating the missing k-space samples, it works only for the sufficiently small size images due to its huge memory requirement for fully connected layers. Accordingly, most of the popular deep learning MR reconstruction algorithms are either in the form of image domain post-processing, or iterative updates between the k-space and the image-domain using a cascaded network.

Therefore, The proposed deep learning approach directly interpolates the missing k-space data so that accurate reconstruction can be obtained by simply taking the Fourier transform of the interpolated k-space data. Specifically, the recent theory of deep convolutional framelets [5] showed that an encoder-decoder network emerges from the data-driven low-rank Hankel matrix decomposition [6], whose rank structure is controlled by the number of filter channels. This discovery gives us important clues to develop a successful deep learning approach for k-space interpolation. We further show that our deep learning approach for Non-Cartesian k-space interpolation such as radial, spiral. Moreover, all the network are implemented in the form of convolutional neural network (CNN) without requiring fully connected layer, so the GPU memory requirement is minimal.

Methods

As shown in Fig. 1, most of the existing deep learning approaches were applied directly to an image domain, whereas our deep neural network is directly applied to[1] Wang, Shanshan, et al. "Accelerating magnetic resonance imaging via deep learning." Biomedical Imaging (ISBI), 2016 IEEE 13th International Symposium on. IEEE, 2016.

[2] Hammernik, Kerstin, et al. "Learning a variational network for reconstruction of accelerated MRI data." Magnetic resonance in medicine 79.6 (2018): 3055-3071.

[3] Kwon, Kinam, Dongchan Kim, and HyunWook Park. "A parallel MR imaging method using multilayer perceptron." Medical physics 44.12 (2017): 6209-6224.

[4] Zhu, Bo, et al. "Image reconstruction by domain-transform manifold learning." Nature 555.7697 (2018): 487.APA

[5] Ye, Jong Chul, Yoseob Han, and Eunju Cha. "Deep convolutional framelets: A general deep learning framework for inverse problems." SIAM Journal on Imaging Sciences 11.2 (2018): 991-1048.

[6] Jin, Kyong Hwan, Dongwook Lee, and Jong Chul Ye. "A general framework for compressed sensing and parallel MRI using annihilating filter based low-rank Hankel matrix." IEEE Transactions on Computational Imaging 2.4 (2016): 480-495.

Fig. 1. Network backbone of the proposed method.

Fig. 2. Overall reconstruction flowcharts of the proposed method with (a) weighting layers and (b) skipped connection.

Fig. 3. Reconstruction results for (a) radial trajectory at R = 6 and (b) spiral trajectory at R = 4. The normalized mean square error (NMSE) values are written at the corner.