Deep Network Regularization for Phasebased Magnetic Resonance Electrical Properties Tomography with Stein's Unbiased Risk Estimator

Magnetic resonance imaging (MRI) can extract the tissue conductivity values from in vivo data using the so-called phase-based magnetic resonance electrical properties tomography (MR-EPT). However, this procedure suffers from noise amplification caused by the use of the Laplacian operator. To counter this issue, we propose a novel preprocessing denoiser for magnetic resonance transceive phase images, operating in an unsupervised manner. Inspired by the deep image prior approach, we apply the random initialization of a convolutional neural network, which enforces an implicit regularization. Additionally, we introduce Stein's unbiased risk estimator, which is the unbiased estimator of the mean square error for optimizing the network without the need for label images. This modification not only tackles the overfitting problem inherent in the deep image prior approach but also operates within a purely unsupervised framework. In addition, instead of using phase images, we use real and imaginary images, which aligns with the theoretical model of the risk estimator. Our generative model needs neither the preparation of training datasets nor prior training procedure, and it maintains adaptability across various resolutions and signal-to-noise ratio levels. In testing. our method significantly diminished residual error remaining in phase maps, quantitatively as well as qualitatively, for both phantom and simulated brain data. Furthermore, it outperformed other denoising methods in reducing noise amplification and boundary error. When applied to healthy volunteer and patient data, the proposed method revealed reduced error in the reconstructed conductivity maps, with conductivity values aligning well with established literature values. To the best of our knowledge, this is the first blind approach using a purely unsupervised denoising framework that can implement a 2D phase-based MR-EPT reconstruction algorithm. The source code is available at https://github.com/Yonsei-MILab/Implicit-Regularization-forMREPT-with-SURE.