Dynamic imaging involves the reconstruction of a spatio-temporal object at all times using its undersampled measurements. In particular, in dynamic computed tomography (dCT), only a single projection at one view angle is available at a time, making the inverse problem very challenging. Moreover, ground-truth dynamic data is usually either unavailable or too scarce to be used for supervised learning techniques. To tackle this problem, we propose RSR-NF, which uses a neural field (NF) to represent the dynamic object and, using the Regularization-by-Denoising (RED) framework, incorporates an additional static deep spatial prior into a variational formulation via a learned restoration operator. We use an ADMM-based algorithm with variable splitting to efficiently optimize the variational objective. We compare RSR-NF to three alternatives: NF with only temporal regularization; a recent method combining a partially-separable low-rank representation with RED using a denoiser pretrained on static data; and a deep-image prior-based model. The first comparison demonstrates the reconstruction improvements achieved by combining the NF representation with static restoration priors, whereas the other two demonstrate the improvement over state-of-the art techniques for dCT.

The Bayesian Cram\'er-Rao bound (BCRB) is a crucial tool in signal processing for assessing the fundamental limitations of any estimation problem as well as benchmarking within a Bayesian frameworks. However, the BCRB cannot be computed without full knowledge of the prior and the measurement distributions. In this work, we propose a fully learned Bayesian Cram\'er-Rao bound (LBCRB) that learns both the prior and the measurement distributions. Specifically, we suggest two approaches to obtain the LBCRB: the Posterior Approach and the Measurement-Prior Approach. The Posterior Approach provides a simple method to obtain the LBCRB, whereas the Measurement-Prior Approach enables us to incorporate domain knowledge to improve the sample complexity and {interpretability}. To achieve this, we introduce a Physics-encoded score neural network which enables us to easily incorporate such domain knowledge into a neural network. We {study the learning} errors of the two suggested approaches theoretically, and validate them numerically. We demonstrate the two approaches on several signal processing examples, including a linear measurement problem with unknown mixing and Gaussian noise covariance matrices, frequency estimation, and quantized measurement. In addition, we test our approach on a nonlinear signal processing problem of frequency estimation with real-world underwater ambient noise.

Dynamic imaging addresses the recovery of a time-varying 2D or 3D object at each time instant using its undersampled measurements. In particular, in the case of dynamic tomography, only a single projection at a single view angle may be available at a time, making the problem severely ill-posed. In this work, we propose an approach, RED-PSM, which combines for the first time two powerful techniques to address this challenging imaging problem. The first, are partially separable models, which have been used to efficiently introduce a low-rank prior for the spatio-temporal object. The second is the recent \textit{Regularization by Denoising (RED)}, which provides a flexible framework to exploit the impressive performance of state-of-the-art image denoising algorithms, for various inverse problems. We propose a partially separable objective with RED and a computationally efficient and scalable optimization scheme with variable splitting and ADMM. Theoretical analysis proves the convergence of our objective to a value corresponding to a stationary point satisfying the first-order optimality conditions. Convergence is accelerated by a particular projection-domain-based initialization. We demonstrate the performance and computational improvements of our proposed RED-PSM with a learned image denoiser by comparing it to a recent deep-prior-based method known as TD-DIP. Although the main focus is on dynamic tomography, we also show performance advantages of RED-PSM in a cardiac dynamic MRI setting.

Scatter due to interaction of photons with the imaged object is a fundamental problem in X-ray Computed Tomography (CT). It manifests as various artifacts in the reconstruction, making its abatement or correction critical for image quality. Despite success in specific settings, hardware-based methods require modification in the hardware, or increase in the scan time or dose. This accounts for the great interest in software-based methods, including Monte-Carlo based scatter estimation, analytical-numerical, and kernel-based methods, with data-driven learning-based approaches demonstrated recently. In this work, two novel physics-inspired deep-learning-based methods, PhILSCAT and OV-PhILSCAT, are proposed. The methods estimate and correct for the scatter in the acquired projection measurements. Different from previous works, they incorporate both an initial reconstruction of the object of interest and the scatter-corrupted measurements related to it, and use a deep neural network architecture and cost function, both specifically tailored to the problem. Numerical experiments with data generated by Monte-Carlo simulations of the imaging of phantoms reveal consistent improvement over a recent purely projection-domain deep neural network scatter correction method.

Magnetic resonance imaging (MRI) is widely used in clinical practice for visualizing both biological structure and function, but its use has been traditionally limited by its slow data acquisition. Recent advances in compressed sensing (CS) techniques for MRI that exploit sparsity models of images reduce acquisition time while maintaining high image quality. Whereas classical CS assumes the images are sparse in a known analytical dictionary or transform domain, methods that use learned image models for reconstruction have become popular in recent years. The model could be learned from a dataset and used for reconstruction or learned simultaneously with the reconstruction, a technique called blind CS (BCS). While the well-known synthesis dictionary model has been exploited for MRI reconstruction, recent advances in transform learning (TL) provide an efficient alternative framework for sparse modeling in MRI. TL-based methods enjoy numerous advantages including exact sparse coding, transform update, and clustering solutions, cheap computation, and convergence guarantees, and provide high quality results in MRI as well as in other inverse problems compared to popular competing methods. This paper provides a review of key works in MRI reconstruction from limited data, with focus on the recent class of TL-based reconstruction methods. A unified framework for incorporating various TL-based models is presented. We discuss the connections between transform learning and convolutional or filterbank models and corresponding multi-layer extensions, as well as connections to unsupervised and supervised deep learning. Finally, we discuss recent trends in MRI, open problems, and future directions for the field.

A Generative Adversarial Network (GAN) with generator $G$ trained to model the prior of images has been shown to perform better than sparsity-based regularizers in ill-posed inverse problems. In this work, we propose a new method of deploying a GAN-based prior to solve linear inverse problems using projected gradient descent (PGD). Our method learns a network-based projector for use in the PGD algorithm, eliminating the need for expensive computation of the Jacobian of $G$. Experiments show that our approach provides a speed-up of $30\text{-}40\times$ over earlier GAN-based recovery methods for similar accuracy in compressed sensing. Our main theoretical result is that if the measurement matrix is moderately conditioned for range($G$) and the projector is $\delta$-approximate, then the algorithm is guaranteed to reach $O(\delta)$ reconstruction error in $O(log(1/\delta))$ steps in the low noise regime. Additionally, we propose a fast method to design such measurement matrices for a given $G$. Extensive experiments demonstrate the efficacy of this method by requiring $5\text{-}10\times$ fewer measurements than random Gaussian measurement matrices for comparable recovery performance.

Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from convolutional measurements $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from convolutional measurements $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

Data is said to follow the transform (or analysis) sparsity model if it becomes sparse when acted on by a linear operator called a sparsifying transform. Several algorithms have been designed to learn such a transform directly from data, and data-adaptive sparsifying transforms have demonstrated excellent performance in signal restoration tasks. Sparsifying transforms are typically learned using small sub-regions of data called patches, but these algorithms often ignore redundant information shared between neighboring patches. We show that many existing transform and analysis sparse representations can be viewed as filter banks, thus linking the local properties of patch-based model to the global properties of a convolutional model. We propose a new transform learning framework where the sparsifying transform is an undecimated perfect reconstruction filter bank. Unlike previous transform learning algorithms, the filter length can be chosen independently of the number of filter bank channels. Numerical results indicate filter bank sparsifying transforms outperform existing patch-based transform learning for image denoising while benefiting from additional flexibility in the design process.

Many machine learning problems, especially multi-modal learning problems, have two sets of distinct features (e.g., image and text features in news story classification, or neuroimaging data and neurocognitive data in cognitive science research). This paper addresses the joint dimensionality reduction of two feature vectors in supervised learning problems. In particular, we assume a discriminative model where low-dimensional linear embeddings of the two feature vectors are sufficient statistics for predicting a dependent variable. We show that a simple algorithm involving singular value decomposition can accurately estimate the embeddings provided that certain sample complexities are satisfied, without specifying the nonlinear link function (regressor or classifier). The main results establish sample complexities under multiple settings. Sample complexities for different link functions only differ by constant factors.