ENSURE: ENSEMBLE STEIN'S UNBIASED RISK ESTIMATOR FOR UNSUPERVISED LEARNING

Clean Self-Supervised MRI Reconstruction from Noisy, Sub-Sampled Training Data with Robust SSDU

Most existing methods for magnetic resonance imaging (MRI) reconstruction with deep learning use fully supervised training, which assumes that a fully sampled dataset with a high signal-to-noise ratio (SNR) is available for training. In many circumstances, however, such a dataset is highly impractical or even technically infeasible to acquire. Recently, a number of self-supervised methods for MRI reconstruction have been proposed, which use sub-sampled data only. However, the majority of such methods, such as Self-Supervised Learning via Data Undersampling (SSDU), are susceptible to reconstruction errors arising from noise in the measured data. In response, we propose Robust SSDU, which provably recovers clean images from noisy, sub-sampled training data by simultaneously estimating missing k-space samples and denoising the available samples. Robust SSDU trains the reconstruction network to map from a further noisy and sub-sampled version of the data to the original, singly noisy, and sub-sampled data and applies an additive Noisier2Noise correction term upon inference. We also present a related method, Noiser2Full, that recovers clean images when noisy, fully sampled data are available for training. Both proposed methods are applicable to any network architecture, are straightforward to implement, and have a similar computational cost to standard training. We evaluate our methods on the multi-coil fastMRI brain dataset with novel denoising-specific architecture and find that it performs competitively with a benchmark trained on clean, fully sampled data.

A Theoretical Framework for Self-Supervised MR Image Reconstruction Using Sub-Sampling via Variable Density Noisier2Noise

In recent years, there has been attention on leveraging the statistical modeling capabilities of neural networks for reconstructing sub-sampled Magnetic Resonance Imaging (MRI) data. Most proposed methods assume the existence of a representative fully-sampled dataset and use fully-supervised training. However, for many applications, fully sampled training data is not available, and may be highly impractical to acquire. The development and understanding of self-supervised methods, which use only sub-sampled data for training, are therefore highly desirable. This work extends the Noisier2Noise framework, which was originally constructed for self-supervised denoising tasks, to variable density sub-sampled MRI data. We use the Noisier2Noise framework to analytically explain the performance of Self-Supervised Learning via Data Undersampling (SSDU), a recently proposed method that performs well in practice but until now lacked theoretical justification. Further, we propose two modifications of SSDU that arise as a consequence of the theoretical developments. Firstly, we propose partitioning the sampling set so that the subsets have the same type of distribution as the original sampling mask. Secondly, we propose a loss weighting that compensates for the sampling and partitioning densities. On the fastMRI dataset we show that these changes significantly improve SSDU's image restoration quality and robustness to the partitioning parameters.

Physics-Driven Deep Learning for Computational Magnetic Resonance Imaging: Combining physics and machine learning for improved medical imaging

Physics-driven deep learning methods have emerged as a powerful tool for computational magnetic resonance imaging (MRI) problems, pushing reconstruction performance to new limits. This article provides an overview of the recent developments in incorporating physics information into learning-based MRI reconstruction. We consider inverse problems with both linear and non-linear forward models for computational MRI, and review the classical approaches for solving these. We then focus on physics-driven deep learning approaches, covering physics-driven loss functions, plug-and-play methods, generative models, and unrolled networks. We highlight domain-specific challenges such as real- and complex-valued building blocks of neural networks, and translational applications in MRI with linear and non-linear forward models. Finally, we discuss common issues and open challenges, and draw connections to the importance of physics-driven learning when combined with other downstream tasks in the medical imaging pipeline.

MRI RECOVERY WITH A SELF-CALIBRATED DENOISER

Plug-and-play (PnP) methods that employ application-specific denoisers have been proposed to solve inverse problems, including MRI reconstruction. However, training application-specific denoisers is not feasible for many applications due to the lack of training data. In this work, we propose a PnP-inspired recovery method that does not require data beyond the single, incomplete set of measurements. The proposed self-supervised method, called recovery with a self-calibrated denoiser (ReSiDe), trains the denoiser from the patches of the image being recovered. The denoiser training and a call to the denoising subroutine are performed in each iteration of a PnP algorithm, leading to a progressive refinement of the reconstructed image. For validation, we compare ReSiDe with a compressed sensing-based method and a PnP method with BM3D denoising using single-coil MRI brain data.

ENSURE: ENSEMBLE STEIN'S UNBIASED RISK ESTIMATOR FOR UNSUPERVISED LEARNING

Deep learning algorithms are emerging as powerful alternatives to compressed sensing methods, offering improved image quality and computational efficiency. Unfortunately, fully sampled training images may not be available or are difficult to acquire in several applications, including high-resolution and dynamic imaging. Previous studies in image reconstruction have utilized Stein's Unbiased Risk Estimator (SURE) as a mean square error (MSE) estimate for the image denoising step in an unrolled network. Unfortunately, the end-to-end training of a network using SURE remains challenging since the projected SURE loss is a poor approximation to the MSE, especially in the heavily undersampled setting. We propose an ENsemble SURE (ENSURE) approach to train a deep network only from undersampled measurements. In particular, we show that training a network using an ensemble of images, each acquired with a different sampling pattern, can closely approximate the MSE. Our preliminary experimental results show that the proposed ENSURE approach gives comparable reconstruction quality to supervised learning and a recent unsupervised learning method.