ENSURE: A General Approach for Unsupervised Training of Deep Image Reconstruction Algorithms.

Image reconstruction using deep learning algorithms offers improved reconstruction quality and lower reconstruction time than classical compressed sensing and model-based algorithms. Unfortunately, clean and fully sampled ground-truth data to train the deep networks is often unavailable in several applications, restricting the applicability of the above methods. We introduce a novel metric termed the ENsemble Stein's Unbiased Risk Estimate (ENSURE) framework, which can be used to train deep image reconstruction algorithms without fully sampled and noise-free images. The proposed framework is the generalization of the classical SURE and GSURE formulation to the setting where the images are sampled by different measurement operators, chosen randomly from a set. We evaluate the expectation of the GSURE loss functions over the sampling patterns to obtain the ENSURE loss function. We show that this loss is an unbiased estimate for the true mean-square error, which offers a better alternative to GSURE, which only offers an unbiased estimate for the projected error. Our experiments show that the networks trained with this loss function can offer reconstructions comparable to the supervised setting. While we demonstrate this framework in the context of MR image recovery, the ENSURE framework is generally applicable to arbitrary inverse problems.

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.

Deep Generalization of Structured Low-Rank Algorithms (Deep-SLR).

Structured low-rank (SLR) algorithms, which exploit annihilation relations between the Fourier samples of a signal resulting from different properties, is a powerful image reconstruction framework in several applications. This scheme relies on low-rank matrix completion to estimate the annihilation relations from the measurements. The main challenge with this strategy is the high computational complexity of matrix completion. We introduce a deep learning (DL) approach to significantly reduce the computational complexity. Specifically, we use a convolutional neural network (CNN)-based filterbank that is trained to estimate the annihilation relations from imperfect (under-sampled and noisy) k-space measurements of Magnetic Resonance Imaging (MRI). The main reason for the computational efficiency is the pre-learning of the parameters of the non-linear CNN from exemplar data, compared to SLR schemes that learn the linear filterbank parameters from the dataset itself. Experimental comparisons show that the proposed scheme can enable calibration-less parallel MRI; it can offer performance similar to SLR schemes while reducing the runtime by around three orders of magnitude. Unlike pre-calibrated and self-calibrated approaches, the proposed uncalibrated approach is insensitive to motion errors and affords higher acceleration. The proposed scheme also incorporates image domain priors that are complementary, thus significantly improving the performance over that of SLR schemes.

J-MoDL: Joint Model-Based Deep Learning for Optimized Sampling and Reconstruction.

Modern MRI schemes, which rely on compressed sensing or deep learning algorithms to recover MRI data from undersampled multichannel Fourier measurements, are widely used to reduce the scan time. The image quality of these approaches is heavily dependent on the sampling pattern. We introduce a continuous strategy to optimize the sampling pattern and the network parameters jointly. We use a multichannel forward model, consisting of a non-uniform Fourier transform with continuously defined sampling locations, to realize the data consistency block within a model-based deep learning image reconstruction scheme. This approach facilitates the joint and continuous optimization of the sampling pattern and the CNN parameters to improve image quality. We observe that the joint optimization of the sampling patterns and the reconstruction module significantly improves the performance of most deep learning reconstruction algorithms. The source code is available at https://github.com/hkaggarwal/J-MoDL.

Improved MUSSELS reconstruction for high-resolution multi-shot diffusion weighted imaging.

PURPOSE: MUSSELS is a one-step iterative reconstruction method for multishot diffusion weighted (msDW) imaging. The current work presents an efficient implementation, termed IRLS MUSSELS, that enables faster reconstruction to enhance its utility for high-resolution diffusion MRI studies. METHODS: The recently proposed MUSSELS reconstruction belongs to a new class of parallel imaging-based methods that recover artifact-free DWIs from msDW data without needing phase compensation. The reconstruction is achieved via structured low-rank matrix completion algorithms, which are computationally demanding due to the large size of the Hankel matrices and their associated computations involving singular value decompositions. Because of this, computational demands of the MUSSELS reconstruction scales as the matrix size and the number of shots increases, which hinders its practical utility for high-resolution applications. In this work, we derive a computationally efficient MUSSELS formulation by modifying the iterative reweighted least squares (IRLS) method that were proposed earlier to solve such problems. Using whole-brain in vivo data, we show the utility of the IRLS MUSSELS for routine high-resolution studies with reduced computational burden. RESULTS: IRLS MUSSELS provides about five times faster reconstruction for matrix sizes 192 × 192 and 256 × 256 compared to the earlier MUSSELS implementation. The widely employed conjugate symmetry priors can also be incorporated into IRLS MUSSELS to reduce blurring of the partial Fourier acquisitions, without incurring much computational burden. CONCLUSIONS: The proposed method is observed to be computationally efficient to enable routine high-resolution studies. The computational complexity matches the traditional msDWI reconstruction methods and provides improved reconstruction results with the additional constraints.

MODEL BASED IMAGE RECONSTRUCTION USING DEEP LEARNED PRIORS (MODL).

We introduce a model-based image reconstruction framework, where we use a deep convolution neural network (CNN) based regularization prior. We rely on a recursive algorithm, which alternates between a CNN based denoising step and enforcement of data consistency. Unrolling the recursive algorithm yields a deep network that is trained using backpropagation. The unique aspect of this method is the use of the same CNN weights at each iteration, which makes the resulting structure consistent with the model-based formulation. Also, this approach reduces the number of trainable parameters, which hence lower the amount of training data needed. The use of a forward model also reduces the size of the network and enables the exploitation additional prior information available from calibration data. The use of the framework for multichannel MRI reconstruction provides improved reconstructions, compared to other state-of-the-art methods.