Kanghyun Ryu^{1}, Sung-Min Gho^{2}, Yoonho Nam^{3}, Kevin Koch^{4}, and Dong-Hyun Kim^{1}

MRI phase images are increasingly used for susceptibility mapping and distortion correction in function and diffusion MRI. However, acquired values of phase maps are wrapped between [-π π ] and require an additional phase unwrapping process. Here we developed a novel deep learning method that can learn the transformation between the wrapped phase images and the corresponding unwrapped phase images. The method was tested for numerical simulations and on actual MR images.

**Introduction**

Phase images in MRI are used in various applications^{1-4}. Typical phase images require dedicated phase unwrapping algorithms which have trade-off between high computation times and accuracy^{5}. Thus, deep learning (DL) methods which are both time efficient and accurate may have advantages. Conventionally used deep learning architectures^{6} used in medical imaging are difficult to apply directly. Architectures such as UNET^{7 }are based on convolutional layers and have difficulty in learning the global features of the image due to receptive field constraints. In addition, loss functions such as mean squared error (MSE) is insufficient for the problem as the solution is not unique, i.e., if Y is one solution (unwrapped) for arbitrary integer n, Y+2nπ may also be solutions in phase unwrapping. We developed a DL method that can resolve the aforementioned problems. Specifically, we utilized recurrent modules that can efficiently learn global features and an appropriate loss function based on the contrast of the error. Numerical simulations and in vivo studies with comparison to 3D PRELUDE^{8,9}, which is often regarded as the gold standard, is performed.

**Methods**

**[Network architecture]**

Figure 1 (a) shows the proposed deep neural network design. The key feature of the architecture is the bidirectional RNN (recurrent NN) module^{10}. This module reads through each pixel following the four paths (Left to right, right to left, down to up, up to down), enabling global spatial features to be learned. Conventional UNET was also trained and compared.

**[Loss function]**

Loss function was composed of two loss functions, namely total variation loss and variance of error loss. Total variation loss was $$$L_{TV} (\widehat{Y},Y)= E[|∇_x (\widehat{Y}-Y)|+|∇_y (\widehat{Y}-Y)|]$$$ and the variance of error loss was $$$L_V (\widehat{Y},Y)= E[(\widehat{Y}-Y)^2 ]-(E[\widehat{Y}-Y])^2,$$$ where E[X] denotes the mean value of the image X. The final loss function was weighted by $$L(\widehat{Y},Y)=0.1*L_{TV} (\widehat{Y},Y)+ L_V (\widehat{Y},Y).$$ For in-vivo dataset, to reflect the fact that the phase SNR is inversely proportional to the magnitude intensity, the losses were further modified respectively as: $$ L_{TV} (\widehat{Y},Y)= (|∇_x M*(\widehat{Y}-Y)|_F+|∇_y M*(\widehat{Y}-Y)|_F)/∑M $$,

$$ L_{V} (\widehat{Y},Y)= E[M*(\widehat{Y}-Y)^2]/E[M]-(E[M*(\widehat{Y}-Y)]/E[M])^2,$$

where M is the magnitude image and * the element-wise multiplication.

**[Dataset]**

**1. Simulation: **

For numerical simulation, images w generated based on Eq.1 with randomly generated variables σ_{a}, σ_{b}, σ_{c}, M, x_{c}, y_{c}.

$$I(x,y)= σ_a x+σ_b x+σ_c+\sum_{n=1}^Me^{-{\frac{(x-x_c)^2}{2σ_x}+\frac{(y-y_c)^2}{2σ_y}}} [Eq.1]$$

The wrapped phase was synthetically generated by $$$\phi(x,y)= \angle{e^{jI(x,y)}}.$$$ An exemple dataset pair is shown in Fig.1 (b).

**2. In-vivo**

The paired dataset was generated by processing the PRELUDE on actual multi-echo GRE images. Dataset from ten subjects were used for training the network. An example dataset is shown in Fig.1 (c). mGRE images (8-echoes) from one subjects not included in the training were used for the testing.

**Results**

**Discussion and Conclusion**

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and future Planning (NRF-2016R1A2B3016273).

The research was supported by Graduate Student Scholarship Program funded by Hyundai Motor Chung Mong-Koo Foundation.

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3. Kelin, Tilmann A., Markus Ullsperger, and Gerhard Jocham. “Learning relative values in the striatum induces violations of normative decision making.” Nature Communications 8 (2017).

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5. Robinson, Simon Daniel, et al. “An illustrated comparison of processing methods for MR phase imaging and QSM: combining array coil signals and phase unwrapping.” NMR in Biomedicine 30.4 (2017)

6. Dongwook, Lee., et al. "Deep residual learning for compressed sensing MRI.", 2017 IEEE 14th International Symposium on Biomedical Imaging (ISBI 2017)

7. Ronneberger, Olaf et al. "U-Net: Convolutional Networks for Biomedical Image Segmentation.", Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015

8. Jenkinson, Mark “Fast, automated, N-dimensional phase-unwrapping algorithm.” Magnetic Resonance in medicine 49.1 (2003): 193-197

9. FSL Prelude: http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FUGUE

10. Visin, Francesco et al., “ReNet: A Recurrent Neural Network Based Alternative to Convolutional Networks”, arXiv:1505.00393

Figure
1. (a) Network architecture (b) Training Datasets for numerical simulation (c)
Datasets for in-vivo application.

Figure 2. Comparison of results from proposed network and conventional U-Net. (a) for local phase wraps (b) for global phase wraps

Figure 3. Comparison of results from proposed loss function and conventional loss function. (a) Loss evolutions for MSE and proposed. (b) Two different unwrapping results

Figure 4. In-vivo application of a subject. (a) result of an axial image. (b) result of sagittal images