Analysis of deep complex‐valued convolutional neural networks for MRI reconstruction and phase‐focused applications

Abstract

Deep learning has had success with MRI reconstruction, but previously published works use real-valued networks. The few works which have tried complexvalued networks have not fully assessed their impact on phase. Therefore, the purpose of this work is to fully investigate end-to-end complex-valued convolutional neural networks (CNNs) for accelerated MRI reconstruction and in several phasebased applications in comparison to 2-channel real-valued networks. Methods: Several complex-valued activation functions for MRI reconstruction were implemented, and their performance was compared. Complex-valued convolution was implemented and tested on an unrolled network architecture and a U-Net-based architecture over a wide range of network widths and depths with knee, body, and phase-contrast datasets. Results: Quantitative and qualitative results demonstrated that complex-valued CNNs with complex-valued convolutions provided superior reconstructions compared to real-valued convolutions with the same number of trainable parameters for both an unrolled network architecture and a U-Net-based architecture, and for 3 different datasets. Complex-valued CNNs consistently had superior normalized RMS error, structural similarity index, and peak SNR compared to real-valued CNNs. Conclusion: Complex-valued CNNs can enable superior accelerated MRI reconstruction and phase-based applications such as fat-water separation, and flow quantification compared to real-valued convolutional neural networks.

Prospective Deployment of Deep Learning in MRI: A Framework for Important Considerations, Challenges, and Recommendations for Best Practices

“…70 A particularly exciting opportunity that remains for improving model-based DL robustness is the incorporation of other intrinsic signal structures such as complex-valued convolutional layers. 71,72 Most DL frameworks do not support complex-valued DL and networks generally decompose the real and imaginary components into two separate real-valued channels as inputs into reconstruction models. MRI is inherently complex-valued and conserving phase information is important during reconstruction, as well for downstream fat/water separation, phase-contrast imaging, displacement encoding, and quantitative susceptibility mapping applications.…”

“…In addition, complex-valued convolutional operations have greater expressivity, possibly reducing the number of model parameters and the likelihood of overfitting on training data. 71,72 Novel activation functions such as the zReLU, cReLU, modReLU, and cardioid have been developed for maintaining phase data along with magnitude activations during gradient descent of the DL networks. 72,73 These activations ensure that not only is the network input complexvalued, but also that the internal representation of the network is complex-valued and consistent with the acquired data (Fig.…”

Real‐time deep artifact suppression using recurrent U‐Nets for low‐latency cardiac MRI

“…This is in keeping with previous studies that show that tiny golden angle radial sampling enables accurate reconstruction, even as a postprocessing step. 14 Nevertheless, further optimizations (use of data consistency terms, unrolled optimizations, 11 , 12 and complex valued networks 32 , 33 ) might increase reconstruction accuracy and further improve image quality. 17 , 32 However, these will be heavily constrained by reconstruction times and the lack of large amounts of raw k‐space data for training.…”

Machine learning in Magnetic Resonance Imaging: Image reconstruction

“…These approaches do not necessarily maintain the phase information of the data. Development of complex-valued networks remains an area of active research [89][90][91]. However, PyTorch has recently (year: 2020) introduced full complex value support, which means that more studies may use complex-valued data in the future.…”