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.

The recovery of magnetic resonance (MR) images from undersampled measurements is a key problem that has been the subject of extensive research in recent years. Unrolled approaches, which rely on end-to-end training of convolutional neural network (CNN) blocks within iterative reconstruction algorithms, offer state-of-the-art performance. These algorithms require a large amount of memory during training, making them difficult to employ in high-dimensional applications. Deep equilibrium (DEQ) models and the recent monotone operator learning (MOL) approach were introduced to eliminate the need for unrolling, thus reducing the memory demand during training. Both approaches require a Lipschitz constraint on the network to ensure that the forward and backpropagation iterations converge. Unfortunately, the constraint often results in reduced performance compared to the unrolled methods. The main focus of this work is to relax the constraint on the CNN block in two different ways. Inspired by convex-non-convex regularization strategies, we now impose the monotone constraint on the sum of the gradient of the data term and the CNN block, rather than constrain the CNN itself to be a monotone operator. This approach enables the CNN to learn possibly non-monotone score functions, which can translate to improved performance. In addition, we only restrict the operator to be monotone in a local neighborhood around the image manifold. Our theoretical results show that the proposed algorithm is guaranteed to converge to the fixed point and that the solution is robust to input perturbations, provided that it is initialized close to the true solution. Our empirical results show that the relaxed constraints translate to improved performance and that the approach enjoys robustness to input perturbations similar to MOL.

Purpose: To develop a deep neural network-based, AIF-free, perfusion estimation method (QTMnet) for improved performance on glioma classification. Methods: A globally defined arterial input function (AIF) is needed to recover perfusion parameters in the two-compartment exchange model (2CXM). We have developed Quantitative Transport Mapping (QTM) to create an AIF-independent estimation method. QTM estimation can be formulated using deep neural networks trained on synthetic DCE-MRI data (QTMnet). Here, we provide a fluid mechanics-based DCE-MRI simulation with exchange between the capillaries and extravascular extracellular space. We implemented tumor ROI generation to morphologically characterize tissue perfusion. We compared our QTMnet implementation with 2CXM on 30 glioma human subjects, 15 of which had low-grade gliomas, and 15 with high-grade glioblastomas. Results: QTMnet outperforms (best AUC: 0.973) traditional 2CXM (best AUC: 0.911) in a glioma grading task. Conclusion: The AIF-independent QTMnet estimation provides a quantitative delineation between low-grade and high-grade gliomas.