Hyungseob Shin^{1}, Taejoon Eo^{1}, Yohan Jun^{1}, Taeseong Kim^{1}, and Dosik Hwang^{1}

In this study, we compare two deep learning approaches to reconstruct multi-channel magnetic resonance (MR) images subsampled along phase-encoding (PE) direction. They are both based on the Fully-Connected (FC) layers but are performed in two different domains : Image domain, and k-x domain which is 1D inverse Fourier transformed (IFT) k-space. We demonstrate that the latter method shows superior performance to the former one in terms of removing the aliasing artifacts and recovering the details of MR images. The performance of the proposed method to the conventional MR reconstruction on image domain was qualitatively and quantitatively evaluated.

MR reconstruction has
been a very actively studied research area and parallel imaging (PI) has been
proposed as a technique to accelerate the MR acquisition time since it utilizes
the spatial sensitivity of each multi receiver coil ^{1,2}. Today, deep
learning is replacing the conventional MR reconstruction methods and this is
also the case in the fields of PI ^{3}. Most of the deep learning-based
MR reconstruction methods are performed on either image domain or k-space
domain. Recently, a manifold learning framework which directly reconstructs
the MR images from the raw data has also emerged ^{4}.

In this work, we propose a deep learning architecture based on FC layers to specifically reconstruct multi-channel MR data under-sampled along PE direction using cartesian sampling pattern. The input to the proposed network is multi-channel k-space data that is initially 1D inverse Fourier transformed along frequency-encoding (FE) direction. This causes no aliasing since the subsampling was performed in only PE direction. Then, the FC network processes the multi-channel column of the input line-by-line by interpolating the missing data using the true-sampled points. The FC network outputs the root-sum-of-squares line but with filled points that were initially missed from under-sampling. Lastly, 1D IFT along PE direction was performed to form the reconstructed images.

The advantage of the proposed method is that
it utilizes the true-sampled data which is free of aliasing artifacts induced
by inverse transforming the under-sampled data. The initial 1D FT along the FE
direction enables line-by-line processing which dramatically reduces the number
of parameters required for FC network from N^{2} to N. In addition, the
complexity of the learning is much less compared to recently emerging domain
transform learning^{4} since there is no need for the network to learn
domain transform.

Standford Fullysampled 3D FSE Knee datasets 5 from 10 subjects were used for this study. The imaging parameters were as follows : FOV = 160mm x 160mm x 153.6mm, Matrix size = 320 x 320 x 1, Number of channels = 8, Number of slices = 256, TR/TE = 1550ms/25ms, Flip angle = 90’. Matrix size of each slice was resized to 256 x 256.

Figure.1. shows the structures of the two networks designed for this study. Figure.1-(a) shows the deep learning-based MR reconstruction performed on image domain, which maps multi-channel aliased input images into alias-free root-sum-of-squares images through line-by-line processing. This can be done since the subsampling was performed only in one direction. Figure.1-(b) shows the proposed method. First, the under-sampled input k-space is initially 1D inverse Fourier transformed along FE direction since this causes no aliasing due to the characteristic of the cartesian sampling pattern. Second, each multi-channel column line under-sampled along PE direction passes the FC network which is identically designed as the one in Figure.1-(a) for fair comparison. The FC network predicts the missing points by interpolating the true-sampled points and outputs the root-sum-of-squares line. Then, the output lines of the FC network are 1D inverse Fourier transformed along PE direction to form knee MR images. The mean square error was computed between the reconstructed MR images and full-sampled Ground Truth (GT) MR images.

The input and label data to the networks in Fig.1 were divided into real and imaginary components since the deep learning library used cannot handle complex data. The FC network comprises 4 layers. The first 3 layers contain 1280 neurons each, followed by Rectified Linear Unit (ReLU). The final layer contains 512 neurons, followed by linear activation to predict the desired output.

1. Pruessmann, Klaas P., et al. "SENSE: sensitivity encoding for fast MRI." Magnetic resonance in medicine 42.5 (1999): 952-962.

2. Griswold, Mark A., et al. "Generalized autocalibrating partially parallel acquisitions (GRAPPA)." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 47.6 (2002): 1202-1210.

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.

5. Epperson K, Sawyer AM, Lustig M, Alley M, Uecker M. Creation Of Fully Sampled MR Data Repository For Compressed Sensing Of The Knee. In: Proceedings of the 22nd Annual Meeting for Section for Magnetic Resonance Technologists, Salt Lake City, Utah, USA, 2013.

Figure.1. (a) Line-by-line under-sampled MR image reconstruction
framework on image domain. (b) The proposed line-by-line under-sampled MR image
reconstruction on the k-x domain, i.e.,1D inverse Fourier transformed k-space. Nch denotes the number of channels.

Figure.2. (a) Under-sampled MR knee image. (b) Reconstructed MR knee
image using the framework in Figure.1-(a). (c) Reconstructed MR knee image
using the framework in Figure.1-(b). (d) Ground Truth label image. The white
arrows in (a)~(d) denote the aliasing artifacts that remain on image domain
learning but removed on the proposed method. (e)~(h) Magnified images of the
orange boxes in (a)~(d) showing the performance of reconstruction in recovering
the details of the MR images.

Figure.3. (a) Under-sampled MR knee image. (b) Reconstructed MR knee
image using the framework in Figure.1-(a). (c) Reconstructed MR knee image
using the framework in Figure.1-(b). (d) Ground Truth label image. (e)~(h)
Magnified images of the orange boxes in (a)~(d) highlighting the horizontal
artifacts occurred on image domain learning. (f) and (g) clearly shows the
difference. (i)~(k) The difference images between (h) and (e)~(g).