Elizabeth Cole^{1}, John Pauly^{1}, Shreyas Vasanawala^{2}, and Joseph Cheng^{2}

To improve MRI reconstruction accuracy, we propose various complex-valued frameworks for reconstructions using convolutional neural networks. By introducing complex-valued convolution and activation functions, we improve reconstruction of our subsampled images and achieve competitive results compared to the real-valued counterpart of our model.

Recent
work suggests that complex-valued CNNs could improve accuracy in comparison to real-valued
CNNs when dealing with complex-valued data. Research on this topic remains
scarce, because traditionally, CNNs have been applied to real-valued data. Initial
work with CNN for complex data consists of feeding the data into CNNs by using
a 2-channel architecture where the channels contain the real and imaginary
components of the data. However, this architecture does not accurately
represent the data because it disregards phase information, which is valuable
in many MRI applications including
blood flow, quantitative susceptibility mapping (QSM), fat-water separation,
disease detection, and brain segmentation. For example, in a two-channel
CNN, the rectifier activation function (ReLU) is applied separately to the real
and imaginary components of the data, which does not preserve the phase
component. Recent work in applying complex-valued CNNs to computer vision tasks
as well as music and speech spectrum demonstrates that complex models are
highly competitive with their real two-channel counterparts.^{9}
Additionally, complex-valued neural networks have been applied to MRI
fingerprinting, the task of identifying tissue parameters, with improvements in
accuracy in comparison to real models.^{10}

^{ }In this work, we apply the concept
of complex-valued CNNs to the problem of subsampled image reconstruction by
modifying components of our current CNN within our deep unrolled architecture,
as described by ^{2}, to be complex-valued. The structure of our
network is displayed in Figure 1. Specifically, we perform complex convolution,
which relies on the distributive property of convolution. This reduces the
number of parameters the network learns. We also explore training the network
with complex-valued convolution using various complex-valued activation
functions which keep the pre-activated phase intact as well as activation
functions which are based on the phase component. These activation functions include
modReLU and zReLU, as described by ^{9}, and as well as the cardioid
activation function, as described by ^{10}. We evaluate the performance
in terms of accuracy of the complex-valued models compared to their real-valued
counterpart.

1. Chen, Feiyu et al. “Variable-Density Single-Shot Fast Spin-Echo MRI with Deep Learning Reconstruction by Using Variational Networks.” Radiology (2018): 180445.

2. Cheng, Joseph. “Highly Scalable Image Reconstruction Using Deep Neural Networks with Bandpass Filtering.” arXiv:1805.03300 [cs.CV], 8 May 2018, arxiv.org/abs/1805.03300.

3. Daval-Frerot, G, et. al., “Exploring Complex-Valued Neural Networks with Trainable Activation Functions for Magnetic Resonance Imaging.” ISMRM. Workshop on Machine Learning, 26 Oct. 2018.

4. Griswold, M, et. al., “Generalized autocalibrating partially parallel acquisitions (GRAPPA),” Magnetic Resonance in Medicine, vol. 47, no. 6, pp. 1202–1210, Jun. 2002. [Online]. Available: http://www.ncbi.nlm.nih.gov/pubmed/12111967

5. Hammernik K, et. al., “Learning a Variational Network for Reconstruction of Accelerated MRI Data,” arXiv:1704.00447 [cs.CV], Apr. 2017. [Online]. Available: http://arxiv.org/abs/1704.00447

6. Lustig, M, et al., “Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging.” Magnetic Resonance in Medicine, vol. 58, no. 6, pp. 1182–1195, Dec. 2007. [Online]. Available: http://www.ncbi.nlm.nih.gov/pubmed/17969013

7. Pruessmann, K P, et al., “SENSE: sensitivity encoding for fast MRI.” Magnetic Resonance in Medicine, vol. 42, no. 5, pp. 952–62, Nov. 1999. [Online]. Available: http://www.ncbi.nlm.nih.gov/pubmed/10542355

8. Scardapane, Simone, et al., “Kafnets: Kernel-Based Non-Parametric Activation Functions for Neural Networks.” Neural Networks (Elsevier), 23 Nov. 2017.

9. Trabelsi, Chiheb, et. al., “Deep Complex Networks.” ICLR 2018, 25 Feb. 2018.

10. Virtue, Patrick, et. al., “Better than Real: Complex-Valued Neural Nets for MRI Fingerprinting.” 2017 IEEE International Conference on Image Processing (ICIP), 1 July 2017.

Unrolled Complex Network Architecture

Validation Loss Comparison Between Complex and Real Convolution

Set of Test Magnitude Images: Input, Complex Convolution, Real Convolution, Truth

Set of Test Phase Images: Input, Complex Convolution, Real Convolution, Truth

Comparison Between
Various Activation Functions