Fast estimation of GRAPPA kernel using Meta-learning
Dongwook Lee1 and Jong Chul Ye1

1Bio and Brain Eng., KAIST, Daejeon, Korea, Republic of


This paper proposes an accelerated MR reconstruction method for parallel imaging from uniformly undersampled k-space data by learning scan-specific GRAPPA kernel using the long short-term memory network (LSTM). In particular, the meta-leaner LSTM is redesigned to quickly estimate the GRAPPA kernel for each k-space from its auto-calibration signals (ACS). The proposed method shows improved reconstruction performance with minimum error.


Recently, inspired by the success of deep learning from computer vision research,1-5 deep learning approaches have been extensively studied for accelerated MR imaging.6-9 Most of the existing deep learning approach for MR reconstruction have focused on improving the average reconstruction performance for all training dataset.6-9 Generalized auto-calibrating partially parallel acquisitions (GRAPPA)10 and Robust artificial-neural-networks for k-space interpolation (RAKI)11 tried to find the scan-specific kernel from its auto-calibration signals (ACS). However, they need separate step to find the scan-specific kernels such as matrix inverse or neural network training process for each scan, which is time-consuming. Here, we propose a meta-learning based algorithm to estimate the scan-specific GRAPPA kernel for parallel imaging using the learner extracted by a Long Short-Term Memory network12(LSTM)-based meta-learner optimizer.


Suppose we train the parameters of a leaner CNN, $$$\theta$$$, using the gradient descent algorithm. The $$$t$$$-th update $$$\theta_t$$$ can be represented by


where $$$\alpha$$$ and $$$\nabla_{\theta_{t-1}}\mathcal{L}$$$ are the learning rate and the gradients of the loss with respect to $$$\theta_{t-1}$$$, respectively. Ravi, et al.13 observed that the LSTM could be used as an optimization tool because of the similarity between the equations for gradient update and cell states update as following:

$$C_{t}=f_t\odot C_{t-1}+i_t\odot\tilde{C}$$

if $$$C_{t-1}=\theta_{t-1}$$$, $$$f_t = 1$$$, $$$i_t = \alpha$$$ and $$$\tilde{C} = -\nabla_{\theta_{t-1}} \mathcal{L}$$$. The meta-learner LSTM was successfully applied to the few-shot learning task as a powerful acceleration technique for optimization. Moreover, it is also as a provider for good initialization, resulting in great performance on the few-shot classification task.13 The proposed algorithm consists of two neural networks, a learner CNN and a meta-learner LSTM as shown in Fig.1. The learner CNN reconstructs the missing k-space using the sampled k-space as an input. The meta-learner LSTM has modified LSTM structure as shown in Fig.1 (b). The proposed meta-learner LSTM receives the whole weights of the learner CNN, $$$\theta$$$, as a vectorized cell state, $$$C_{t}$$$, and update it as follows:

$$C_{t}=f_t\odot C_{t-1}-i_t\odot(\nabla_{\theta_{t-1}}\mathcal{L}+H_{t-1})$$

where $$$H_{t-1}$$$ is the hidden state of the LSTM. Here, we redesign the output gate as momentum gate to transfer the previous gradients by hidden state. The hidden state, $$$H_t$$$, is updated by the following form:

$$ H_t=o_t\odot(\nabla_{\theta_{t-1}}\mathcal{L}+H_{t-1})$$

where the forget, input and momentum gates are as followings,




The MR dataset was acquired in Cartesian coordinate with 7T MR scanner (Philips, Achieva). The following parameters were used for multi-slice FFE scan: TR 831ms, TE 5ms, slice thickness 0.75mm, 288$$$\times$$$288 matrix, 32 coils, FOV 240$$$\times$$$240mm, and FA 15 degrees. Total 567 number of the axial brain images were scanned from nine subjects. The scans are divided by 7/1/1 subjects for $$$\mathcal{D}_{meta-train}$$$/ $$$\mathcal{D}_{meta-validation}$$$/ $$$\mathcal{D}_{meta-test}$$$, respectively. Each $$$\mathcal{D}_{meta-set}$$$ consists of $$$D_{train}$$$ and $$$D_{test}$$$. We designed two input/target pairs from each k-space. Specifically, $$$(X_{ACS}$$$, $$$Y_{ACS})\in D_{train}$$$ refer to the sampled/missing k-space from ACS, while $$$(X_{full}$$$, $$$Y_{full})\in D_{test}$$$ denotes the sampled/missing k-space from whole k-space. The learner CNN initially reconstructs the k-space from ACS ($$$D_{train}$$$) and the loss function is calculated as following:


The loss and the gradients of the loss are fed to the meta-learner LSTM, $$$M$$$, and $$$\theta_{t+1}$$$ are estimated:


After the $$$T$$$ number of LSTM updates, the final estimated parameters, $$$\theta_{T+1}$$$, are applied to reconstruct the full k-space ($$$D_{test}$$$) to minimize the following cost:


To train the meta-learner LSTM and to find the best initial state, $$$\theta_0$$$, we minimize the summation of $$$\mathcal{L}_{train}$$$ and $$$\mathcal{L}_{test}$$$. The complex values are handled by concatenating real and imaginary values along the channel direction. The preprocessing method14 for gradients and loss is applied for better performance.

Results and Discussion

The proposed method shows minimum normalized root mean squared error (NRMSE) compared to RAKI and GRAPPA. The artifacts are still visible in the result of RAKI, when the hyper-parameters are not optimally selected by trial and error. The GRAPPA shows great reconstruction results but the noise from the high-frequency components are amplified. However, the hyper-parameter tuning is not necessary in the proposed method, because it utilizes the LSTM gates as the dynamic hyperparameters. The gates are dynamically tuned at each update for proper and fast optimization based on the current loss, the current gradients, and the previous values of each gate. Accordingly, the proposed method produced the best reconstruction result without high-frequency noise amplification.


The proposed method took only 540ms for the reconstruction from 32 coils (Figure 2), which is faster than GRAPPA and RAKI. We presented a fast and scan-specific reconstruction method for the highly under-sampled k-space using LSTM-based meta learner. To our best knowledge, this is the first meta-learning algorithm for accelerated MR imaging.


This work was supported by Institute for Information & Communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) [2016-0-00562(R0124-16-0002), Emotional Intelligence Technology to Infer Human Emotion and Carry on Dialogue Accordingly]


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Figure 1. Architectures of (a) the learner CNN, G, and (b) the LSTM meta-learner, M. The learner consists of three convolution layers and two leaky-ReLU layers. (c) The forward pass of the proposed algorithm consists of T number of cell state updates for the LSTM. After the cell states are optimized by T number of updates using Dtrain(the k-space from ACS), the updated learner CNN is applied to Dtest.

Figure 2. Comparison of the reconstruction performances. The downsampling rate is four and the number of ACS lines is 32 (11% of the total PE). The reconstruction time for a scan and the average of normalized root mean squared errors for test set are displayed for each reconstruction and error image, respectively.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)