Bregman Plug-and-Play Priors

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.

Learning-based Motion Artifact Removal Networks (LEARN) for Quantitative $R_2^\ast$ Mapping

Purpose: To introduce two novel learning-based motion artifact removal networks (LEARN) for the estimation of quantitative motion- and $B0$-inhomogeneity-corrected $R_2^\ast$ maps from motion-corrupted multi-Gradient-Recalled Echo (mGRE) MRI data. Methods: We train two convolutional neural networks (CNNs) to correct motion artifacts for high-quality estimation of quantitative $B0$-inhomogeneity-corrected $R_2^\ast$ maps from mGRE sequences. The first CNN, LEARN-IMG, performs motion correction on complex mGRE images, to enable the subsequent computation of high-quality motion-free quantitative $R_2^\ast$ (and any other mGRE-enabled) maps using the standard voxel-wise analysis or machine-learning-based analysis. The second CNN, LEARN-BIO, is trained to directly generate motion- and $B0$-inhomogeneity-corrected quantitative $R_2^\ast$ maps from motion-corrupted magnitude-only mGRE images by taking advantage of the biophysical model describing the mGRE signal decay. We show that both CNNs trained on synthetic MR images are capable of suppressing motion artifacts while preserving details in the predicted quantitative $R_2^\ast$ maps. Significant reduction of motion artifacts on experimental in vivo motion-corrupted data has also been achieved by using our trained models. Conclusion: Both LEARN-IMG and LEARN-BIO can enable the computation of high-quality motion- and $B0$-inhomogeneity-corrected $R_2^\ast$ maps. LEARN-IMG performs motion correction on mGRE images and relies on the subsequent analysis for the estimation of $R_2^\ast$ maps, while LEARN-BIO directly performs motion- and $B0$-inhomogeneity-corrected $R_2^\ast$ estimation. Both LEARN-IMG and LEARN-BIO jointly process all the available gradient echoes, which enables them to exploit spatial patterns available in the data. The high computational speed of LEARN-BIO is an advantage that can lead to a broader clinical application.

MoDIR: Motion-Compensated Training for Deep Image Reconstruction without Ground Truth

Deep neural networks for medical image reconstruction are traditionally trained using high-quality ground-truth images as training targets. Recent work onNoise2Noise (N2N) has shown the potential of using multiple noisy measurements of the same object as an alternative to having a ground truth. However, existing N2N-based methods cannot exploit information from various motion states, limiting their ability to learn on moving objects. This paper addresses this issue by proposing a novel motion-compensated deep image reconstruction (MoDIR) method that can use information from several unregistered and noisy measurements for training. MoDIR deals with object motion by including a deep registration module jointly trained with the deep reconstruction network without any ground-truth supervision. We validate MoDIR on both simulated and experimentally collected magnetic resonance imaging (MRI) data and show that it significantly improves imaging quality.

Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition

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.

CoIL: Coordinate-based Internal Learning for Imaging Inverse Problems

We propose Coordinate-based Internal Learning (CoIL) as a new deep-learning (DL) methodology for the continuous representation of measurements. Unlike traditional DL methods that learn a mapping from the measurements to the desired image, CoIL trains a multilayer perceptron (MLP) to encode the complete measurement field by mapping the coordinates of the measurements to their responses. CoIL is a self-supervised method that requires no training examples besides the measurements of the test object itself. Once the MLP is trained, CoIL generates new measurements that can be used within a majority of image reconstruction methods. We validate CoIL on sparse-view computed tomography using several widely-used reconstruction methods, including purely model-based methods and those based on DL. Our results demonstrate the ability of CoIL to consistently improve the performance of all the considered methods by providing high-fidelity measurement fields.

SGD-Net: Efficient Model-Based Deep Learning with Theoretical Guarantees

Deep unfolding networks have recently gained popularity in the context of solving imaging inverse problems. However, the computational and memory complexity of data-consistency layers within traditional deep unfolding networks scales with the number of measurements, limiting their applicability to large-scale imaging inverse problems. We propose SGD-Net as a new methodology for improving the efficiency of deep unfolding through stochastic approximations of the data-consistency layers. Our theoretical analysis shows that SGD-Net can be trained to approximate batch deep unfolding networks to an arbitrary precision. Our numerical results on intensity diffraction tomography and sparse-view computed tomography show that SGD-Net can match the performance of the batch network at a fraction of training and testing complexity.

Joint Reconstruction and Calibration using Regularization by Denoising

Regularization by denoising (RED) is a broadly applicable framework for solving inverse problems by using priors specified as denoisers. While RED has been shown to provide state-of-the-art performance in a number of applications, existing RED algorithms require exact knowledge of the measurement operator characterizing the imaging system, limiting their applicability in problems where the measurement operator has parametric uncertainties. We propose a new method, called Calibrated RED (Cal-RED), that enables joint calibration of the measurement operator along with reconstruction of the unknown image. Cal-RED extends the traditional RED methodology to imaging problems that require the calibration of the measurement operator. We validate Cal-RED on the problem of image reconstruction in computerized tomography (CT) under perturbed projection angles. Our results corroborate the effectiveness of Cal-RED for joint calibration and reconstruction using pre-trained deep denoisers as image priors.

Async-RED: A Provably Convergent Asynchronous Block Parallel Stochastic Method using Deep Denoising Priors

Regularization by denoising (RED) is a recently developed framework for solving inverse problems by integrating advanced denoisers as image priors. Recent work has shown its state-of-the-art performance when combined with pre-trained deep denoisers. However, current RED algorithms are inadequate for parallel processing on multicore systems. We address this issue by proposing a new asynchronous RED (ASYNC-RED) algorithm that enables asynchronous parallel processing of data, making it significantly faster than its serial counterparts for large-scale inverse problems. The computational complexity of ASYNC-RED is further reduced by using a random subset of measurements at every iteration. We present complete theoretical analysis of the algorithm by establishing its convergence under explicit assumptions on the data-fidelity and the denoiser. We validate ASYNC-RED on image recovery using pre-trained deep denoisers as priors.

Deep Image Reconstruction using Unregistered Measurements without Groundtruth

One of the key limitations in conventional deep learning based image reconstruction is the need for registered pairs of training images containing a set of high-quality groundtruth images. This paper addresses this limitation by proposing a novel unsupervised deep registration-augmented reconstruction method (U-Dream) for training deep neural nets to reconstruct high-quality images by directly mapping pairs of unregistered and artifact-corrupted images. The ability of U-Dream to circumvent the need for accurately registered data makes it widely applicable to many biomedical image reconstruction tasks. We validate it in accelerated magnetic resonance imaging (MRI) by training an image reconstruction model directly on pairs of undersampled measurements from images that have undergone nonrigid deformations.

Scalable Plug-and-Play ADMM with Convergence Guarantees

Plug-and-play priors (PnP) is a broadly applicable methodology for solving inverse problems by exploiting statistical priors specified as denoisers. Recent work has reported the state-of-the-art performance of PnP algorithms using pre-trained deep neural nets as denoisers in a number of imaging applications. However, current PnP algorithms are impractical in large-scale settings due to their heavy computational and memory requirements. This work addresses this issue by proposing an incremental variant of the widely used PnP-ADMM algorithm, making it scalable to large-scale datasets. We theoretically analyze the convergence of the algorithm under a set of explicit assumptions, extending recent theoretical results in the area. Additionally, we show the effectiveness of our algorithm with nonsmooth data-fidelity terms and deep neural net priors, its fast convergence compared to existing PnP algorithms, and its scalability in terms of speed and memory.