Saiprasad Ravishankar

Hui Li, Jinghan Jia, Shijun Liang et al.

Although deep learning (DL) has gained much popularity for accelerated magnetic resonance imaging (MRI), recent studies have shown that DL-based MRI reconstruction models could be oversensitive to tiny input perturbations (that are called 'adversarial perturbations'), which cause unstable, low-quality reconstructed images. This raises the question of how to design robust DL methods for MRI reconstruction. To address this problem, we propose a novel image reconstruction framework, termed SMOOTHED UNROLLING (SMUG), which advances a deep unrolling-based MRI reconstruction model using a randomized smoothing (RS)-based robust learning operation. RS, which improves the tolerance of a model against input noises, has been widely used in the design of adversarial defense for image classification. Yet, we find that the conventional design that applies RS to the entire DL process is ineffective for MRI reconstruction. We show that SMUG addresses the above issue by customizing the RS operation based on the unrolling architecture of the DL-based MRI reconstruction model. Compared to the vanilla RS approach and several variants of SMUG, we show that SMUG improves the robustness of MRI reconstruction with respect to a diverse set of perturbation sources, including perturbations to the input measurements, different measurement sampling rates, and different unrolling steps. Code for SMUG will be available at https://github.com/LGM70/SMUG.

Yixuan Jia, Siyi Chen, Yida Pan et al.

Data assimilation (DA) estimates the state of an evolving dynamical system from noisy, partial observations, and is widely used in scientific simulation as well as weather and climate science. In practice, filtering methods rely on frame-to-frame transition models. However, these models are fragile when observations are non-Markovian (when they form only a partial slice of a higher-dimensional latent state as in real-world weather data): they tend to accumulate errors over long horizons. At the same time, learned DA methods typically commit to a single regime, either filtering (nowcasting, real-time forecasting) or smoothing (retrospective reanalysis), which splits what should be a shared prior across application-specific pipelines. To address both issues, we introduce ForcingDAS, a unified and robust DA framework. Built on Diffusion Forcing with an independent noise level assigned to each frame, ForcingDAS learns a joint-trajectory prior instead of frame-to-frame transitions. This allows it to capture long-horizon temporal dependencies and reduce error accumulation. In addition, the same trained model spans the full filtering to smoothing spectrum at inference time. Specifically, nowcasting, fixed-lag smoothing, and batch reanalysis are selected through the inference schedule alone, without retraining. We evaluate ForcingDAS on 2D Navier-Stokes vorticity, precipitation nowcasting, and global atmospheric state estimation. Across all settings, a single model is competitive with or outperforms both learned and classical baselines that are specialized for individual regimes, with the largest gains observed on real-world weather benchmarks.

Chong Wang, Rongkai Zhang, Saiprasad Ravishankar et al.

Image restoration schemes based on the pre-trained deep models have received great attention due to their unique flexibility for solving various inverse problems. In particular, the Plug-and-Play (PnP) framework is a popular and powerful tool that can integrate an off-the-shelf deep denoiser for different image restoration tasks with known observation models. However, obtaining the observation model that exactly matches the actual one can be challenging in practice. Thus, the PnP schemes with conventional deep denoisers may fail to generate satisfying results in some real-world image restoration tasks. We argue that the robustness of the PnP framework is largely limited by using the off-the-shelf deep denoisers that are trained by deterministic optimization. To this end, we propose a novel deep reinforcement learning (DRL) based PnP framework, dubbed RePNP, by leveraging a light-weight DRL-based denoiser for robust image restoration tasks. Experimental results demonstrate that the proposed RePNP is robust to the observation model used in the PnP scheme deviating from the actual one. Thus, RePNP can generate more reliable restoration results for image deblurring and super resolution tasks. Compared with several state-of-the-art deep image restoration baselines, RePNP achieves better results subjective to model deviation with fewer model parameters.

Avrajit Ghosh, Michael T. McCann, Madeline Mitchell et al.

We present a method for supervised learning of sparsity-promoting regularizers for denoising signals and images. Sparsity-promoting regularization is a key ingredient in solving modern signal reconstruction problems; however, the operators underlying these regularizers are usually either designed by hand or learned from data in an unsupervised way. The recent success of supervised learning (mainly convolutional neural networks) in solving image reconstruction problems suggests that it could be a fruitful approach to designing regularizers. Towards this end, we propose to denoise signals using a variational formulation with a parametric, sparsity-promoting regularizer, where the parameters of the regularizer are learned to minimize the mean squared error of reconstructions on a training set of ground truth image and measurement pairs. Training involves solving a challenging bilievel optimization problem; we derive an expression for the gradient of the training loss using the closed-form solution of the denoising problem and provide an accompanying gradient descent algorithm to minimize it. Our experiments with structured 1D signals and natural images show that the proposed method can learn an operator that outperforms well-known regularizers (total variation, DCT-sparsity, and unsupervised dictionary learning) and collaborative filtering for denoising. While the approach we present is specific to denoising, we believe that it could be adapted to the larger class of inverse problems with linear measurement models, giving it applicability in a wide range of signal reconstruction settings.

Shijun Liang, Anish Lahiri, Saiprasad Ravishankar

Recent medical image reconstruction techniques focus on generating high-quality medical images suitable for clinical use at the lowest possible cost and with the fewest possible adverse effects on patients. Recent works have shown significant promise for reconstructing MR images from sparsely sampled k-space data using deep learning. In this work, we propose a technique that rapidly estimates deep neural networks directly at reconstruction time by fitting them on small adaptively estimated neighborhoods of a training set. In brief, our algorithm alternates between searching for neighbors in a data set that are similar to the test reconstruction, and training a local network on these neighbors followed by updating the test reconstruction. Because our reconstruction model is learned on a dataset that is in some sense similar to the image being reconstructed rather than being fit on a large, diverse training set, it is more adaptive to new scans. It can also handle changes in training sets and flexible scan settings, while being relatively fast. Our approach, dubbed LONDN-MRI, was validated on multiple data sets using deep unrolled reconstruction networks. Reconstructions were performed at four fold and eight fold undersampling of k-space with 1D variable-density random phase-encode undersampling masks. Our results demonstrate that our proposed locally-trained method produces higher-quality reconstructions compared to models trained globally on larger datasets as well as other scan-adaptive methods.

Ling Chen, Zhishen Huang, Yong Long et al.

Traditional model-based image reconstruction (MBIR) methods combine forward and noise models with simple object priors. Recent application of deep learning methods for image reconstruction provides a successful data-driven approach to addressing the challenges when reconstructing images with undersampled measurements or various types of noise. In this work, we propose a hybrid supervised-unsupervised learning framework for X-ray computed tomography (CT) image reconstruction. The proposed learning formulation leverages both sparsity or unsupervised learning-based priors and neural network reconstructors to simulate a fixed-point iteration process. Each proposed trained block consists of a deterministic MBIR solver and a neural network. The information flows in parallel through these two reconstructors and is then optimally combined. Multiple such blocks are cascaded to form a reconstruction pipeline. We demonstrate the efficacy of this learned hybrid model for low-dose CT image reconstruction with limited training data, where we use the NIH AAPM Mayo Clinic Low Dose CT Grand Challenge dataset for training and testing. In our experiments, we study combinations of supervised deep network reconstructors and MBIR solver with learned sparse representation-based priors or analytical priors. Our results demonstrate the promising performance of the proposed framework compared to recent low-dose CT reconstruction methods.

Xikai Yang, Zhishen Huang, Yong Long et al.

The recently proposed sparsifying transform models incur low computational cost and have been applied to medical imaging. Meanwhile, deep models with nested network structure reveal great potential for learning features in different layers. In this study, we propose a network-structured sparsifying transform learning approach for X-ray computed tomography (CT), which we refer to as multi-layer clustering-based residual sparsifying transform (MCST) learning. The proposed MCST scheme learns multiple different unitary transforms in each layer by dividing each layer's input into several classes. We apply the MCST model to low-dose CT (LDCT) reconstruction by deploying the learned MCST model into the regularizer in penalized weighted least squares (PWLS) reconstruction. We conducted LDCT reconstruction experiments on XCAT phantom data and Mayo Clinic data and trained the MCST model with 2 (or 3) layers and with 5 clusters in each layer. The learned transforms in the same layer showed rich features while additional information is extracted from representation residuals. Our simulation results demonstrate that PWLS-MCST achieves better image reconstruction quality than the conventional FBP method and PWLS with edge-preserving (EP) regularizer. It also outperformed recent advanced methods like PWLS with a learned multi-layer residual sparsifying transform prior (MARS) and PWLS with a union of learned transforms (ULTRA), especially for displaying clear edges and preserving subtle details.

Tyler Wheeler, Michelle P. Kuchera, Raghuram Ramanujan et al.

Time Projection Chambers (TPCs) are versatile detectors that reconstruct charged-particle tracks in an ionizing medium, enabling sensitive measurements across a wide range of nuclear physics experiments. We explore sparse convolutional networks for representation learning on TPC data, finding that a sparse ResNet architecture, even with randomly set weights, provides useful structured vector embeddings of events. Pre-training this architecture on a simple physics-motivated binary classification task further improves the embedding quality. Using data from the GAseous Detector with GErmanium Tagging (GADGET) II TPC, a detector optimized for measuring low-energy $β$-delayed particle decays, we represent raw pad-level signals as sparse tensors, train Minkowski Engine ResNet models, and probe the resulting event-level embeddings which reveal rich event structure. As a cross-detector test, we embed data from the Active-Target TPC (AT-TPC) -- a detector designed for nuclear reaction studies in inverse kinematics -- using the same encoder. We find that even an untrained sparse ResNet model provides useful embeddings of AT-TPC data, and we observe improvements when the model is trained on GADGET data. Together, these results highlight the potential of sparse convolutional techniques as a general tool for representation learning in diverse TPC experiments.

Zhishen Huang, Saiprasad Ravishankar

There is much recent interest in techniques to accelerate the data acquisition process in MRI by acquiring limited measurements. Often sophisticated reconstruction algorithms are deployed to maintain high image quality in such settings. In this work, we propose a data-driven sampler using a convolutional neural network, MNet, to provide object-specific sampling patterns adaptive to each scanned object. The network observes very limited low-frequency k-space data for each object and rapidly predicts the desired undersampling pattern in one go that achieves high image reconstruction quality. We propose an accompanying alternating-type training framework with a mask-backward procedure that efficiently generates training labels for the sampler network and jointly trains an image reconstruction network. Experimental results on the fastMRI knee dataset demonstrate the ability of the proposed learned undersampling network to generate object-specific masks at fourfold and eightfold acceleration that achieve superior image reconstruction performance than several existing schemes. The source code for the proposed joint sampling and reconstruction learning framework is available at https://github.com/zhishenhuang/mri.

Lanqing Guo, Zhiyuan Zha, Saiprasad Ravishankar et al.

Constructing effective image priors is critical to solving ill-posed inverse problems in image processing and imaging. Recent works proposed to exploit image non-local similarity for inverse problems by grouping similar patches and demonstrated state-of-the-art results in many applications. However, compared to classic methods based on filtering or sparsity, most of the non-local algorithms are time-consuming, mainly due to the highly inefficient and redundant block matching step, where the distance between each pair of overlapping patches needs to be computed. In this work, we propose a novel Self-Convolution operator to exploit image non-local similarity in a self-supervised way. The proposed Self-Convolution can generalize the commonly-used block matching step and produce equivalent results with much cheaper computation. Furthermore, by applying Self-Convolution, we propose an effective multi-modality image restoration scheme, which is much more efficient than conventional block matching for non-local modeling. Experimental results demonstrate that (1) Self-Convolution can significantly speed up most of the popular non-local image restoration algorithms, with two-fold to nine-fold faster block matching, and (2) the proposed multi-modality image restoration scheme achieves superior denoising results in both efficiency and effectiveness on RGB-NIR images. The code is publicly available at \href{https://github.com/GuoLanqing/Self-Convolution}.

Shijun Liang, Evan Bell, Qing Qu et al.

The ability of deep image prior (DIP) to recover high-quality images from incomplete or corrupted measurements has made it popular in inverse problems in image restoration and medical imaging including magnetic resonance imaging (MRI). However, conventional DIP suffers from severe overfitting and spectral bias effects. In this work, we first provide an analysis of how DIP recovers information from undersampled imaging measurements by analyzing the training dynamics of the underlying networks in the kernel regime for different architectures. This study sheds light on important underlying properties for DIP-based recovery. Current research suggests that incorporating a reference image as network input can enhance DIP's performance in image reconstruction compared to using random inputs. However, obtaining suitable reference images requires supervision, and raises practical difficulties. In an attempt to overcome this obstacle, we further introduce a self-driven reconstruction process that concurrently optimizes both the network weights and the input while eliminating the need for training data. Our method incorporates a novel denoiser regularization term which enables robust and stable joint estimation of both the network input and reconstructed image. We demonstrate that our self-guided method surpasses both the original DIP and modern supervised methods in terms of MR image reconstruction performance and outperforms previous DIP-based schemes for image inpainting.

Huijie Zhang, Yifu Lu, Ismail Alkhouri et al.

Diffusion models, emerging as powerful deep generative tools, excel in various applications. They operate through a two-steps process: introducing noise into training samples and then employing a model to convert random noise into new samples (e.g., images). However, their remarkable generative performance is hindered by slow training and sampling. This is due to the necessity of tracking extensive forward and reverse diffusion trajectories, and employing a large model with numerous parameters across multiple timesteps (i.e., noise levels). To tackle these challenges, we present a multi-stage framework inspired by our empirical findings. These observations indicate the advantages of employing distinct parameters tailored to each timestep while retaining universal parameters shared across all time steps. Our approach involves segmenting the time interval into multiple stages where we employ custom multi-decoder U-net architecture that blends time-dependent models with a universally shared encoder. Our framework enables the efficient distribution of computational resources and mitigates inter-stage interference, which substantially improves training efficiency. Extensive numerical experiments affirm the effectiveness of our framework, showcasing significant training and sampling efficiency enhancements on three state-of-the-art diffusion models, including large-scale latent diffusion models. Furthermore, our ablation studies illustrate the impact of two important components in our framework: (i) a novel timestep clustering algorithm for stage division, and (ii) an innovative multi-decoder U-net architecture, seamlessly integrating universal and customized hyperparameters.

Shijun Liang, Van Hoang Minh Nguyen, Jinghan Jia et al.

As the popularity of deep learning (DL) in the field of magnetic resonance imaging (MRI) continues to rise, recent research has indicated that DL-based MRI reconstruction models might be excessively sensitive to minor input disturbances, including worst-case additive perturbations. This sensitivity often leads to unstable, aliased images. This raises the question of how to devise DL techniques for MRI reconstruction that can be robust to train-test variations. To address this problem, we propose a novel image reconstruction framework, termed Smoothed Unrolling (SMUG), which advances a deep unrolling-based MRI reconstruction model using a randomized smoothing (RS)-based robust learning approach. RS, which improves the tolerance of a model against input noises, has been widely used in the design of adversarial defense approaches for image classification tasks. Yet, we find that the conventional design that applies RS to the entire DL-based MRI model is ineffective. In this paper, we show that SMUG and its variants address the above issue by customizing the RS process based on the unrolling architecture of a DL-based MRI reconstruction model. Compared to the vanilla RS approach, we show that SMUG improves the robustness of MRI reconstruction with respect to a diverse set of instability sources, including worst-case and random noise perturbations to input measurements, varying measurement sampling rates, and different numbers of unrolling steps. Furthermore, we theoretically analyze the robustness of our method in the presence of perturbations.

Avrajit Ghosh, Bai Cong, Rio Yokota et al.

Variational Learning (VL) has recently gained popularity for training deep neural networks. Part of its empirical success can be explained by theories such as PAC-Bayes bounds, minimum description length and marginal likelihood, but little has been done to unravel the implicit regularization in play. Here, we analyze the implicit regularization of VL through the Edge of Stability (EoS) framework. EoS has previously been used to show that gradient descent can find flat solutions and we extend this result to show that VL can find even flatter solutions. This result is obtained by controlling the shape of the variational posterior as well as the number of posterior samples used during training. The derivation follows in a similar fashion as in the standard EoS literature for deep learning, by first deriving a result for a quadratic problem and then extending it to deep neural networks. We empirically validate these findings on a wide variety of large networks, such as ResNet and ViT, to find that the theoretical results closely match the empirical ones. Ours is the first work to analyze the EoS dynamics of VL.

Shijun Liang, Evan Bell, Avrajit Ghosh et al.

Deep learning methods are highly effective for many image reconstruction tasks. However, the performance of supervised learned models can degrade when applied to distinct experimental settings at test time or in the presence of distribution shifts. In this study, we demonstrate that pruning deep image reconstruction networks at training time can improve their robustness to distribution shifts. In particular, we consider unrolled reconstruction architectures for accelerated magnetic resonance imaging and introduce a method for pruning unrolled networks (PUN) at initialization. Our experiments demonstrate that when compared to traditional dense networks, PUN offers improved generalization across a variety of experimental settings and even slight performance gains on in-distribution data.

Anish Lahiri, Marc Klasky, Jeffrey A. Fessler et al.

Reconstruction of CT images from a limited set of projections through an object is important in several applications ranging from medical imaging to industrial settings. As the number of available projections decreases, traditional reconstruction techniques such as the FDK algorithm and model-based iterative reconstruction methods perform poorly. Recently, data-driven methods such as deep learning-based reconstruction have garnered a lot of attention in applications because they yield better performance when enough training data is available. However, even these methods have their limitations when there is a scarcity of available training data. This work focuses on image reconstruction in such settings, i.e., when both the number of available CT projections and the training data is extremely limited. We adopt a sequential reconstruction approach over several stages using an adversarially trained shallow network for 'destreaking' followed by a data-consistency update in each stage. To deal with the challenge of limited data, we use image subvolumes to train our method, and patch aggregation during testing. To deal with the computational challenge of learning on 3D datasets for 3D reconstruction, we use a hybrid 3D-to-2D mapping network for the 'destreaking' part. Comparisons to other methods over several test examples indicate that the proposed method has much potential, when both the number of projections and available training data are highly limited.

Avrajit Ghosh, Michael T. Mccann, Saiprasad Ravishankar

We present a method for supervised learning of sparsity-promoting regularizers, a key ingredient in many modern signal reconstruction problems. The parameters of the regularizer are learned to minimize the mean squared error of reconstruction on a training set of ground truth signal and measurement pairs. Training involves solving a challenging bilevel optimization problem with a nonsmooth lower-level objective. We derive an expression for the gradient of the training loss using the implicit closed-form solution of the lower-level variational problem given by its dual problem, and provide an accompanying gradient descent algorithm (dubbed BLORC) to minimize the loss. Our experiments on simple natural images and for denoising 1D signals show that the proposed method can learn meaningful operators and the analytical gradients calculated are faster than standard automatic differentiation methods. While the approach we present is applied to denoising, we believe that it can be adapted to a wide-variety of inverse problems with linear measurement models, thus giving it applicability in a wide range of scenarios.

Zhishen Huang, Marc Klasky, Trevor Wilcox et al.

Object density reconstruction from projections containing scattered radiation and noise is of critical importance in many applications. Existing scatter correction and density reconstruction methods may not provide the high accuracy needed in many applications and can break down in the presence of unmodeled or anomalous scatter and other experimental artifacts. Incorporating machine-learned models could prove beneficial for accurate density reconstruction particularly in dynamic imaging, where the time-evolution of the density fields could be captured by partial differential equations or by learning from hydrodynamics simulations. In this work, we demonstrate the ability of learned deep neural networks to perform artifact removal in noisy density reconstructions, where the noise is imperfectly characterized. We use a Wasserstein generative adversarial network (WGAN), where the generator serves as a denoiser that removes artifacts in densities obtained from traditional reconstruction algorithms. We train the networks from large density time-series datasets, with noise simulated according to parametric random distributions that may mimic noise in experiments. The WGAN is trained with noisy density frames as generator inputs, to match the generator outputs to the distribution of clean densities (time-series) from simulations. A supervised loss is also included in the training, which leads to improved density restoration performance. In addition, we employ physics-based constraints such as mass conservation during network training and application to further enable highly accurate density reconstructions. Our preliminary numerical results show that the models trained in our frameworks can remove significant portions of unknown noise in density time-series data.

Zhishen Huang, Siqi Ye, Michael T. McCann et al.

Many techniques have been proposed for image reconstruction in medical imaging that aim to recover high-quality images especially from limited or corrupted measurements. Model-based reconstruction methods have been particularly popular (e.g., in magnetic resonance imaging and tomographic modalities) and exploit models of the imaging system's physics together with statistical models of measurements, noise and often relatively simple object priors or regularizers. For example, sparsity or low-rankness based regularizers have been widely used for image reconstruction from limited data such as in compressed sensing. Learning-based approaches for image reconstruction have garnered much attention in recent years and have shown promise across biomedical imaging applications. These methods include synthesis dictionary learning, sparsifying transform learning, and different forms of deep learning involving complex neural networks. We briefly discuss classical model-based reconstruction methods and then review reconstruction methods at the intersection of model-based and learning-based paradigms in detail. This review includes many recent methods based on unsupervised learning, and supervised learning, as well as a framework to combine multiple types of learned models together.

Xikai Yang, Yong Long, Saiprasad Ravishankar

Achieving high-quality reconstructions from low-dose computed tomography (LDCT) measurements is of much importance in clinical settings. Model-based image reconstruction methods have been proven to be effective in removing artifacts in LDCT. In this work, we propose an approach to learn a rich two-layer clustering-based sparsifying transform model (MCST2), where image patches and their subsequent feature maps (filter residuals) are clustered into groups with different learned sparsifying filters per group. We investigate a penalized weighted least squares (PWLS) approach for LDCT reconstruction incorporating learned MCST2 priors. Experimental results show the superior performance of the proposed PWLS-MCST2 approach compared to other related recent schemes.

Xikai Yang, Yong Long, Saiprasad Ravishankar

Signal models based on sparse representations have received considerable attention in recent years. On the other hand, deep models consisting of a cascade of functional layers, commonly known as deep neural networks, have been highly successful for the task of object classification and have been recently introduced to image reconstruction. In this work, we develop a new image reconstruction approach based on a novel multi-layer model learned in an unsupervised manner by combining both sparse representations and deep models. The proposed framework extends the classical sparsifying transform model for images to a Multi-lAyer Residual Sparsifying transform (MARS) model, wherein the transform domain data are jointly sparsified over layers. We investigate the application of MARS models learned from limited regular-dose images for low-dose CT reconstruction using Penalized Weighted Least Squares (PWLS) optimization. We propose new formulations for multi-layer transform learning and image reconstruction. We derive an efficient block coordinate descent algorithm to learn the transforms across layers, in an unsupervised manner from limited regular-dose images. The learned model is then incorporated into the low-dose image reconstruction phase. Low-dose CT experimental results with both the XCAT phantom and Mayo Clinic data show that the MARS model outperforms conventional methods such as FBP and PWLS methods based on the edge-preserving (EP) regularizer in terms of two numerical metrics (RMSE and SSIM) and noise suppression. Compared with the single-layer learned transform (ST) model, the MARS model performs better in maintaining some subtle details.

Michael T. McCann, Saiprasad Ravishankar

We present a method for supervised learning of sparsity-promoting regularizers for image denoising. Sparsity-promoting regularization is a key ingredient in solving modern image reconstruction problems; however, the operators underlying these regularizers are usually either designed by hand or learned from data in an unsupervised way. The recent success of supervised learning (mainly convolutional neural networks) in solving image reconstruction problems suggests that it could be a fruitful approach to designing regularizers. As a first experiment in this direction, we propose to denoise images using a variational formulation with a parametric, sparsity-promoting regularizer, where the parameters of the regularizer are learned to minimize the mean squared error of reconstructions on a training set of (ground truth image, measurement) pairs. Training involves solving a challenging bilievel optimization problem; we derive an expression for the gradient of the training loss using Karush-Kuhn-Tucker conditions and provide an accompanying gradient descent algorithm to minimize it. Our experiments on a simple synthetic, denoising problem show that the proposed method can learn an operator that outperforms well-known regularizers (total variation, DCT-sparsity, and unsupervised dictionary learning) and collaborative filtering. While the approach we present is specific to denoising, we believe that it can be adapted to the whole class of inverse problems with linear measurement models, giving it applicability to a wide range of image reconstruction problems.

Xikai Yang, Xuehang Zheng, Yong Long et al.

Signal models based on sparse representation have received considerable attention in recent years. Compared to synthesis dictionary learning, sparsifying transform learning involves highly efficient sparse coding and operator update steps. In this work, we propose a Multi-layer Residual Sparsifying Transform (MRST) learning model wherein the transform domain residuals are jointly sparsified over layers. In particular, the transforms for the deeper layers exploit the more intricate properties of the residual maps. We investigate the application of the learned MRST model for low-dose CT reconstruction using Penalized Weighted Least Squares (PWLS) optimization. Experimental results on Mayo Clinic data show that the MRST model outperforms conventional methods such as FBP and PWLS methods based on edge-preserving (EP) regularizer and single-layer transform (ST) model, especially for maintaining some subtle details.

Zhipeng Li, Siqi Ye, Yong Long et al.

Recent years have witnessed growing interest in machine learning-based models and techniques for low-dose X-ray CT (LDCT) imaging tasks. The methods can typically be categorized into supervised learning methods and unsupervised or model-based learning methods. Supervised learning methods have recently shown success in image restoration tasks. However, they often rely on large training sets. Model-based learning methods such as dictionary or transform learning do not require large or paired training sets and often have good generalization properties, since they learn general properties of CT image sets. Recent works have shown the promising reconstruction performance of methods such as PWLS-ULTRA that rely on clustering the underlying (reconstructed) image patches into a learned union of transforms. In this paper, we propose a new Supervised-UnsuPERvised (SUPER) reconstruction framework for LDCT image reconstruction that combines the benefits of supervised learning methods and (unsupervised) transform learning-based methods such as PWLS-ULTRA that involve highly image-adaptive clustering. The SUPER model consists of several layers, each of which includes a deep network learned in a supervised manner and an unsupervised iterative method that involves image-adaptive components. The SUPER reconstruction algorithms are learned in a greedy manner from training data. The proposed SUPER learning methods dramatically outperform both the constituent supervised learning-based networks and iterative algorithms for LDCT, and use much fewer iterations in the iterative reconstruction modules.

Xuehang Zheng, Saiprasad Ravishankar, Yong Long et al.

Signal models based on sparsity, low-rank and other properties have been exploited for image reconstruction from limited and corrupted data in medical imaging and other computational imaging applications. In particular, sparsifying transform models have shown promise in various applications, and offer numerous advantages such as efficiencies in sparse coding and learning. This work investigates pre-learning a two-layer extension of the transform model for image reconstruction, wherein the transform domain or filtering residuals of the image are further sparsified in the second layer. The proposed block coordinate descent optimization algorithms involve highly efficient updates. Preliminary numerical experiments demonstrate the usefulness of a two-layer model over the previous related schemes for CT image reconstruction from low-dose measurements.

Zhipeng Li, Saiprasad Ravishankar, Yong Long et al.

Dual energy computed tomography (DECT) imaging plays an important role in advanced imaging applications due to its material decomposition capability. Image-domain decomposition operates directly on CT images using linear matrix inversion, but the decomposed material images can be severely degraded by noise and artifacts. This paper proposes a new method dubbed DECT-MULTRA for image-domain DECT material decomposition that combines conventional penalized weighted-least squares (PWLS) estimation with regularization based on a mixed union of learned transforms (MULTRA) model. Our proposed approach pre-learns a union of common-material sparsifying transforms from patches extracted from all the basis materials, and a union of cross-material sparsifying transforms from multi-material patches. The common-material transforms capture the common properties among different material images, while the cross-material transforms capture the cross-dependencies. The proposed PWLS formulation is optimized efficiently by alternating between an image update step and a sparse coding and clustering step, with both of these steps having closed-form solutions. The effectiveness of our method is validated with both XCAT phantom and clinical head data. The results demonstrate that our proposed method provides superior material image quality and decomposition accuracy compared to other competing methods.

Saiprasad Ravishankar, Brendt Wohlberg

Learned data models based on sparsity are widely used in signal processing and imaging applications. A variety of methods for learning synthesis dictionaries, sparsifying transforms, etc., have been proposed in recent years, often imposing useful structures or properties on the models. In this work, we focus on sparsifying transform learning, which enjoys a number of advantages. We consider multi-layer or nested extensions of the transform model, and propose efficient learning algorithms. Numerical experiments with image data illustrate the behavior of the multi-layer transform learning algorithm and its usefulness for image denoising. Multi-layer models provide better denoising quality than single layer schemes.

Brian E. Moore, Saiprasad Ravishankar, Raj Rao Nadakuditi et al.

Sparsity and low-rank models have been popular for reconstructing images and videos from limited or corrupted measurements. Dictionary or transform learning methods are useful in applications such as denoising, inpainting, and medical image reconstruction. This paper proposes a framework for online (or time-sequential) adaptive reconstruction of dynamic image sequences from linear (typically undersampled) measurements. We model the spatiotemporal patches of the underlying dynamic image sequence as sparse in a dictionary, and we simultaneously estimate the dictionary and the images sequentially from streaming measurements. Multiple constraints on the adapted dictionary are also considered such as a unitary matrix, or low-rank dictionary atoms that provide additional efficiency or robustness. The proposed online algorithms are memory efficient and involve simple updates of the dictionary atoms, sparse coefficients, and images. Numerical experiments demonstrate the usefulness of the proposed methods in inverse problems such as video reconstruction or inpainting from noisy, subsampled pixels, and dynamic magnetic resonance image reconstruction from very limited measurements.

Saiprasad Ravishankar, Anna Ma, Deanna Needell

Sparsity-based models and techniques have been exploited in many signal processing and imaging applications. Data-driven methods based on dictionary and sparsifying transform learning enable learning rich image features from data, and can outperform analytical models. In particular, alternating optimization algorithms have been popular for learning such models. In this work, we focus on alternating minimization for a specific structured unitary sparsifying operator learning problem, and provide a convergence analysis. While the algorithm converges to the critical points of the problem generally, our analysis establishes under mild assumptions, the local linear convergence of the algorithm to the underlying sparsifying model of the data. Analysis and numerical simulations show that our assumptions hold for standard probabilistic data models. In practice, the algorithm is robust to initialization.

Bihan Wen, Saiprasad Ravishankar, Yoram Bresler

Techniques exploiting the sparsity of images in a transform domain have been effective for various applications in image and video processing. Transform learning methods involve cheap computations and have been demonstrated to perform well in applications such as image denoising and medical image reconstruction. Recently, we proposed methods for online learning of sparsifying transforms from streaming signals, which enjoy good convergence guarantees, and involve lower computational costs than online synthesis dictionary learning. In this work, we apply online transform learning to video denoising. We present a novel framework for online video denoising based on high-dimensional sparsifying transform learning for spatio-temporal patches. The patches are constructed either from corresponding 2D patches in successive frames or using an online block matching technique. The proposed online video denoising requires little memory, and offers efficient processing. Numerical experiments compare the performance to the proposed video denoising scheme but fixing the transform to be 3D DCT, as well as prior schemes such as dictionary learning-based schemes, and the state-of-the-art VBM3D and VBM4D on several video data sets, demonstrating the promising performance of the proposed methods.

Saiprasad Ravishankar, Brian E. Moore, Raj Rao Nadakuditi et al.

Sparsity-based approaches have been popular in many applications in image processing and imaging. Compressed sensing exploits the sparsity of images in a transform domain or dictionary to improve image recovery from undersampled measurements. In the context of inverse problems in dynamic imaging, recent research has demonstrated the promise of sparsity and low-rank techniques. For example, the patches of the underlying data are modeled as sparse in an adaptive dictionary domain, and the resulting image and dictionary estimation from undersampled measurements is called dictionary-blind compressed sensing, or the dynamic image sequence is modeled as a sum of low-rank and sparse (in some transform domain) components (L+S model) that are estimated from limited measurements. In this work, we investigate a data-adaptive extension of the L+S model, dubbed LASSI, where the temporal image sequence is decomposed into a low-rank component and a component whose spatiotemporal (3D) patches are sparse in some adaptive dictionary domain. We investigate various formulations and efficient methods for jointly estimating the underlying dynamic signal components and the spatiotemporal dictionary from limited measurements. We also obtain efficient sparsity penalized dictionary-blind compressed sensing methods as special cases of our LASSI approaches. Our numerical experiments demonstrate the promising performance of LASSI schemes for dynamic magnetic resonance image reconstruction from limited k-t space data compared to recent methods such as k-t SLR and L+S, and compared to the proposed dictionary-blind compressed sensing method.

Saiprasad Ravishankar, Raj Rao Nadakuditi, Jeffrey A. Fessler

The sparsity of natural signals and images in a transform domain or dictionary has been extensively exploited in several applications such as compression, denoising and inverse problems. More recently, data-driven adaptation of synthesis dictionaries has shown promise in many applications compared to fixed or analytical dictionary models. However, dictionary learning problems are typically non-convex and NP-hard, and the usual alternating minimization approaches for these problems are often computationally expensive, with the computations dominated by the NP-hard synthesis sparse coding step. In this work, we investigate an efficient method for $\ell_{0}$ "norm"-based dictionary learning by first approximating the training data set with a sum of sparse rank-one matrices and then using a block coordinate descent approach to estimate the unknowns. The proposed block coordinate descent algorithm involves efficient closed-form solutions. In particular, the sparse coding step involves a simple form of thresholding. We provide a convergence analysis for the proposed block coordinate descent approach. Our numerical experiments show the promising performance and significant speed-ups provided by our method over the classical K-SVD scheme in sparse signal representation and image denoising.

Saiprasad Ravishankar, Raj Rao Nadakuditi, Jeffrey A. Fessler

The sparsity of signals in a transform domain or dictionary has been exploited in applications such as compression, denoising and inverse problems. More recently, data-driven adaptation of synthesis dictionaries has shown promise compared to analytical dictionary models. However, dictionary learning problems are typically non-convex and NP-hard, and the usual alternating minimization approaches for these problems are often computationally expensive, with the computations dominated by the NP-hard synthesis sparse coding step. This paper exploits the ideas that drive algorithms such as K-SVD, and investigates in detail efficient methods for aggregate sparsity penalized dictionary learning by first approximating the data with a sum of sparse rank-one matrices (outer products) and then using a block coordinate descent approach to estimate the unknowns. The resulting block coordinate descent algorithms involve efficient closed-form solutions. Furthermore, we consider the problem of dictionary-blind image reconstruction, and propose novel and efficient algorithms for adaptive image reconstruction using block coordinate descent and sum of outer products methodologies. We provide a convergence study of the algorithms for dictionary learning and dictionary-blind image reconstruction. Our numerical experiments show the promising performance and speed-ups provided by the proposed methods over previous schemes in sparse data representation and compressed sensing-based image reconstruction.