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.

The Approximate Message Passing (AMP) algorithm efficiently reconstructs signals which have been sampled with large i.i.d. sub-Gaussian sensing matrices. Central to AMP is its "state evolution", which guarantees that the difference between the current estimate and ground truth (the "aliasing") at every iteration obeys a Gaussian distribution that can be fully characterized by a scalar. However, when Fourier coefficients of a signal with non-uniform spectral density are sampled, such as in Magnetic Resonance Imaging (MRI), the aliasing is intrinsically colored, AMP's scalar state evolution is no longer accurate and the algorithm encounters convergence problems. In response, we propose the Variable Density Approximate Message Passing (VDAMP) algorithm, which uses the wavelet domain to model the colored aliasing. We present empirical evidence that VDAMP obeys a "colored state evolution", where the aliasing obeys a Gaussian distribution that can be fully characterized with one scalar per wavelet subband. A benefit of state evolution is that Stein's Unbiased Risk Estimate (SURE) can be effectively implemented, yielding an algorithm with subband-dependent thresholding that has no free parameters. We empirically evaluate the effectiveness of VDAMP on three variations of Fast Iterative Shrinkage-Thresholding (FISTA) and find that it converges in around 10 times fewer iterations on average than the next-fastest method, and to a comparable mean-squared-error.

The hairless guinea pig (HGP) is used by our laboratory to model the human cutaneous response to sulfur mustard (HD), bis(2-chloroethylsulfide), exposure. We determined the HD content in the skin of HGP after a 7-min exposure to vapors saturated with a mixture of HD and ^l4C-HD. Concentration/time (C_T) values in the range of 2 μg/cm²/min were determined by counting skin ¹⁴C disintegrations per min (dpm) in animals euthanized immediately after exposure. These values are similar to human penetration rates obtained by other investigators. A rate curve monitoring the reduction in skin ¹⁴C dpm was developed for animals euthanized between 0 and 24 hr post- exposure. This curve showed the greatest change after 1 hr. The epidermal (62%) to dermal (38%) ratio of ¹⁴C at 24 hr was measured for two animals. We saw no site preference for HD penetration among the 8 sites used. The ¹⁴C content of template adhesive tape was determined to follow HD distribution. These results contribute to a better understanding of the cutaneous response to HD in the HGP model.

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.