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The purpose of the current study was to introduce a Deep learning‐based Accelerated and Noise‐Suppressed Estimation (DANSE) method for reconstructing quantitative maps of biological tissue cellular‐specific,R2t*, and hemodynamic‐specific,R2’, metrics of quantitative gradient‐recalled echo (qGRE) MRI. The DANSE method adapts a supervised learning paradigm to train a convolutional neural network for robust estimation ofR2t*andR2’maps with significantly reduced sensitivity to noise and the adverse effects of macroscopic (B₀) magnetic field inhomogeneities directly from the gradient‐recalled echo (GRE) magnitude images. TheR2t*andR2’maps for training were generated by means of a voxel‐by‐voxel fitting of a previously developed biophysical quantitative qGRE model accounting for tissue, hemodynamic, and B₀‐inhomogeneities contributions to multigradient‐echo GRE signal using a nonlinear least squares (NLLS) algorithm. We show that the DANSE model efficiently estimates the aforementioned qGRE maps and preserves all the features of the NLLS approach with significant improvements including noise suppression and computation speed (from many hours to seconds). The noise‐suppression feature of DANSE is especially prominent for data with low signal‐to‐noise ratio (SNR ~ 50–100), where DANSE‐generatedR2t*andR2’maps had up to three times smaller errors than that of the NLLS method. The DANSE method enables fast reconstruction of qGRE maps with significantly reduced sensitivity to noise and magnetic field inhomogeneities. The DANSE method does not require any information about field inhomogeneities during application. It exploits spatial and gradient echo time‐dependent patterns in the GRE data and previously gained knowledge from the biophysical model, thus producing high quality qGRE maps, even in environments with high noise levels. These features along with fast computational speed can lead to broad qGRE clinical and research applications.

 

The past few years have seen a surge of activity around integration of deep learning networks and optimization algorithms for solving inverse problems. Recent work on plug-and-play priors (PnP), regularization by denoising (RED), and deep unfolding has shown the state-of-the-art performance of such integration in a variety of applications. However, the current paradigm for designing such algorithms is inherently Euclidean, due to the usage of the quadratic norm within the projection and proximal operators. We propose to broaden this perspective by considering a non-Euclidean setting based on the more general Bregman distance. Our new Bregman Proximal Gradient Method variant of PnP (PnP-BPGM) and Bregman Steepest Descent variant of RED (RED-BSD) replace the traditional updates in PnP and RED from the quadratic norms to more general Bregman distance. We present a theoretical convergence result for PnP-BPGM and demonstrate the effectiveness of our algorithms on Poisson linear inverse problems. 

Regularization by denoising (RED) is a widely-used framework for solving inverse problems by leveraging image de-noisers as image priors. Recent work has reported the state-of-the-art performance of RED in a number of imaging applications using pre-trained deep neural nets as denoisers. Despite the recent progress, the stable convergence of RED algorithms remains an open problem. The existing RED theory only guarantees stability for convex data-fidelity terms and nonexpansive denoisers. This work addresses this issue by developing a new monotone RED (MRED) algorithm, whose convergence does not require nonexpansiveness of the deep denoising prior. Simulations on image deblurring and compressive sensing recovery from random matrices show the stability of MRED even when the traditional RED diverges. 

Plug-and-Play Priors (PnP) and Regularization by Denoising (RED) are widely- used frameworks for solving imaging inverse problems by computing fixed-points of operators combining physical measurement models and learned image priors. While traditional PnP/RED formulations have focused on priors specified using image denoisers, there is a growing interest in learning PnP/RED priors that are end-to-end optimal. The recent Deep Equilibrium Models (DEQ) framework has enabled memory-efficient end-to-end learning of PnP/RED priors by implicitly differentiating through the fixed-point equations without storing intermediate activation values. However, the dependence of the computational/memory complexity of the measurement models in PnP/RED on the total number of measurements leaves DEQ impractical for many imaging applications. We propose ODER as a new strategy for improving the efficiency of DEQ through stochastic approximations of the measurement models. We theoretically analyze ODER giving insights into its ability to approximate the traditional DEQ approach for solving inverse problems. Our numerical results suggest the potential improvements in training/testing complexity due to ODER on three distinct imaging applications. 

The plug-and-play priors (PnP) and regularization by denoising (RED) methods have become widely used for solving inverse problems by leveraging pre-trained deep denoisers as image priors. While the empirical imaging performance and the theoretical convergence properties of these algorithms have been widely investigated, their recovery properties have not previously been theoretically analyzed. We address this gap by showing how to establish theoretical recovery guarantees for PnP/RED by assuming that the solution of these methods lies near the fixed-points of a deep neural network. We also present numerical results comparing the recovery performance of PnP/RED in compressive sensing against that of recent compressive sensing algorithms based on generative models. Our numerical results suggest that PnP with a pre-trained artifact removal network provides significantly better results compared to the existing state-of-the-art methods.