Asad Aali^{1}, Marius Arvinte^{1,2}, Sidharth Kumar^{1}, Yamin Ishraq Arefeen^{1}, and Jonathan I. Tamir^{1}

^{1}Chandra Family Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX, United States, ^{2}Intel Corporation, Hillsboro, OR, United States

**Keywords:** AI/ML Image Reconstruction, Image Reconstruction, Deep Generative Models, Inverse Problems, Unsupervised Learning, Denoising

Motivation: Publicly available k-space data used for training are inherently noisy with no available ground truth.

Goal(s): To denoise k-space data in an unsupervised manner for downstream applications.

Approach: We use Generalized Stein’s Unbiased Risk Estimate (GSURE) applied to multi-coil MRI to denoise images without access to ground truth. Subsequently, we train a generative model to show improved accelerated MRI reconstruction.

Results: We demonstrate: (1) GSURE can successfully remove noise from k-space; (2) generative priors learned on GSURE-denoised samples produce realistic synthetic samples; and (3) reconstruction performance on subsampled MRI improves using priors trained on denoised images in comparison to training on noisy samples.

Impact: This abstract shows that we can denoise multi-coil data without ground truth and train deep generative models directly on noisy k-space in an unsupervised manner, for improved accelerated reconstruction.

We arrive at $$$y^* = FSx^*+w$$$, where $$$w\sim N(0,I)$$$:

We first apply noise pre-whitening to the fully sampled multi-coil k-space using BART$$$^{10,11}$$$. We (b) apply the GSURE loss function $$$^7$$$ to train a denoiser without ground-truth, that can be used to estimate a coil-combined denoised image. The GSURE denoiser takes as input the adjoint of the measurements scaled by the estimated noise variance. The GSURE loss matches the supervised denoising loss function in expectation, as proved by$$$^{7}$$$. $$$g_{phi}$$$ is the learned denoising model, and we approximate the divergence term in the loss function using monte-carlo SURE$$$^{12}$$$. Using the denoised images, (c) we train a score-based generative model using denoising score-matching as outlined in $$$^{13}$$$. We then reconstruct subsampled MRI data through posterior sampling via annealed Langevin dynamic.

To illustrate the effect, we use T2-weighted brain (average SNR 32 dB) and fat-suppressed proton-density knee (average SNR 24 dB) FastMRI data$$$^{1}$$$, which is inherently noisy. We apply our approach directly to this data, and after adding additional noise to show the impact at lower inherent SNR – 22 dB and 14 dB for the brain and knee, respectively. Using the brain MRI, we train “naive” score-based generative models on the inherently noisy data, and “GSURE-Score” models on the GSURE-denoised data. We retrospectively subsample 57 test samples with an acceleration factor of 5 and compare reconstruction quality using the various deep generative priors. We use normalized root mean squared error (NRMSE) for quantitative evaluation.

Fig 4 shows the prior samples generated from score-based generative models naively trained on noisy data versus GSURE-denoised data. At high-training SNR, the quality of prior samples remains consistent. At lower training SNR, the naive model generates unrealistic images.

Fig 5 shows the result of reconstruction from R=5 retrospectively subsampled data. The reconstruction using the score-based generative model trained on GSURE-denoised data shows quantitative and qualitative improvement over the naively trained models, where the effect is most pronounced when the training set has a lower SNR. This holds true over the 57 test samples.

J. Zbontar et al., “fastMRI: An open dataset and benchmarks for accelerated MRI,” 2018, arXiv:1811.08839. [Online]. Available: http://arxiv.org/abs/1811.08839

Song, Y., Shen, L., Xing, L., & Ermon, S. (2021). Solving inverse problems in medical imaging with score-based generative models. ICLR 2022.

H. Chung, E. S. Lee, and J. C. Ye, “MR image denoising and superresolution using regularized reverse diffusion,” IEEE Transactions on Medical Imaging,vol. 42, no. 4, pp. 922–934, Apr. 2023.

Jalal, A., Arvinte, M., Daras, G., Price, E., Dimakis, A. G., & Tamir, J. (2021). Robust compressed sensing mri with deep generative priors. Advances in Neural Information Processing Systems, 34, 14938-14954.

Asad Aali, Marius Arvinte, Sidharth Kumar, and Jonathan I Tamir. Solving inverse problems with score-based generative priors learned from noisy data. in Proceedings of Asilomar Conference on Signals, Systems, and Computers, 2023.

C. M. Stein, “Estimation of the mean of a multivariate normal distribution,” The annals of Statistics, pp. 1135–1151, 1981.

Yonina C Eldar. Generalized SURE for exponential families: Applications to regularization. IEEE Transactions on Signal Processing, 57(2):471–481, 2008.

Uecker Martin, Lai Peng, Murphy Mark J., Virtue Patrick, Elad Michael, Pauly John M., Vasanawala Shreyas S., Lustig Michael. ESPIRiT-an eigenvalue approach to autocalibrating parallel MRI: Where SENSE meets GRAPPA. Magnetic Resonance in Medicine. 2014;71 (3): 990–1001. http://dx.doi.org/10.1002/mrm.24751.

Marques JP, Simonis FFJ, Webb AG. Low-field MRI: An MR physics perspective. J Magn Reson Imaging. 2019 Jun;49(6):1528-1542. doi: 10.1002/jmri.26637. Epub 2019 Jan 13. PMID: 30637943; PMCID: PMC6590434.

Martin Uecker, Frank Ong, Jonathan I Tamir, Dara Bahri, Patrick Virtue, Joseph Y Cheng, Tao Zhang, and Michael Lustig. Berkeley advanced reconstruction toolbox. In Proc. Intl. Soc. Mag. Reson. Med, volume 23, 2015.

Kellman P, McVeigh ER. Image reconstruction in SNR units: a general method for SNR measurement. Magn Reson Med. 2005 Dec;54(6):1439-47. doi: 10.1002/mrm.20713. Erratum in: Magn Reson Med. 2007 Jul;58(1):211-2. PMID: 16261576; PMCID: PMC2570032.

S. Ramani, T. Blu, and M. Unser, “Monte-carlo sure: A black-box optimization of regularization parameters for general denoising algorithms,” IEEE Transactions on image processing, vol. 17, no. 9, pp. 1540–1554, 2008.

Song, Y., & Ermon, S. (2020). Improved techniques for training score-based generative models. Advances in neural information processing systems, 33, 12438-12448.

Aggarwal, H. K., Pramanik, A., John, M., & Jacob, M. (2022). ENSURE: A general approach for unsupervised training of deep image reconstruction algorithms. IEEE Transactions on Medical Imaging, 42(4), 1133-1144.

Kawar, B., Elata, N., Michaeli, T., & Elad, M. (2023). GSURE-Based Diffusion Model Training with Corrupted Data. arXiv preprint arXiv:2305.13128.

Fig 1 (a) Illustration of FastMRI noise estimation and pre-whitening from fully sampled multi-coil k-space samples. (b) After pre-whitening, a denoising network is trained in an unsupervised manner directly from the noisy data using GSURE, which is equivalent in expectation to supervised learning. (c) Using the denoised data, a score-based generative model is trained using denoising score matching.

Fig. 2: Brain FastMRI denoising results comparing supervised and GSURE-based training at two SNR values. The first column for each SNR value depicts the adjoint image of pre-whitened k-space data normalized by the 99th percentile of the RSS image. The second column shows the denoised image obtained through a supervised denoiser network as shown in Fig.1. Third column shows the GSURE denoiser network trained in a self-supervised manner without access to the ground truth image. Panel (b) further adds noise to investigate the low-SNR setting.

Fig. 3: Knee FastMRI denoising results comparing supervised and GSURE-based training at two SNR values. The first column for each SNR value depicts the adjoint image of pre-whitened k-space data normalized by the 99th percentile of the RSS image. The second column shows the denoised image obtained through a supervised denoiser network as shown in Fig.1. Third column shows the GSURE denoiser network trained in a self-supervised manner without access to the ground truth image. Panel (b) further adds noise to investigate the low-SNR setting.

Fig. 4 Sample MR images generated from the deep generative priors trained at different SNR levels and with and without GSURE-denoising as a preprocessing step.

Fig. 5 Retrospective reconstruction results at R=5 using score-based generative models show improved reconstruction performance with GSURE-denoising, even when the data are at high SNR. NRMSE values over 57 validation samples with naive and GSURE-Score for posterior sampling.