Dynamic mode decomposition (DMD) has emerged as a powerful data-driven technique for extracting dominant spatiotemporal patterns from complex signals. However, conventional DMD methods are limited to matrix-based formulations, which might be inefficient or inadequate for modeling inherently multidimensional data including images, videos, and multiway sensor data. In this letter, we propose tensor dynamic mode decomposition (TDMD), a novel extension of DMD to third-order tensors using recently developed transform-based tensor products. By leveraging tensor factorization techniques, TDMD provides a compact and structure-preserving representation of multidimensional signals, enabling more accurate and efficient signal reconstruction and mode extraction compared to conventional DMD approaches that require data flattening. We demonstrate the effectiveness of TDMD using numerical examples.

Accurate quantification of both aleatoric and epistemic uncertainties is essential when deploying Graph Neural Networks (GNNs) in high-stakes applications such as drug discovery and financial fraud detection, where reliable predictions are critical. Although Evidential Deep Learning (EDL) efficiently quantifies uncertainty using a Dirichlet distribution over predictive probabilities, existing EDL-based GNN (EGNN) models require modifications to the network architecture and retraining, failing to take advantage of pre-trained models. We propose a plug-and-play framework for uncertainty quantification in GNNs that works with pre-trained models without the need for retraining. Our Evidential Probing Network (EPN) uses a lightweight Multi-Layer-Perceptron (MLP) head to extract evidence from learned representations, allowing efficient integration with various GNN architectures. We further introduce evidence-based regularization techniques, referred to as EPN-reg, to enhance the estimation of epistemic uncertainty with theoretical justifications. Extensive experiments demonstrate that the proposed EPN-reg achieves state-of-the-art performance in accurate and efficient uncertainty quantification, making it suitable for real-world deployment.

This paper focuses on recovering an underlying matrix from its noisy partial entries, a problem commonly known as matrix completion. We delve into the investigation of a non-convex regularization, referred to as transformed $L_1$ (TL1), which interpolates between the rank and the nuclear norm of matrices through a hyper-parameter $a \in (0, \infty)$. While some literature adopts such regularization for matrix completion, it primarily addresses scenarios with uniformly missing entries and focuses on algorithmic advances. To fill in the gap in the current literature, we provide a comprehensive statistical analysis for the estimator from a TL1-regularized recovery model under general sampling distribution. In particular, we show that when $a$ is sufficiently large, the matrix recovered by the TL1-based model enjoys a convergence rate measured by the Frobenius norm, comparable to that of the model based on the nuclear norm, despite the challenges posed by the non-convexity of the TL1 regularization. When $a$ is small enough, we show that the rank of the estimated matrix remains a constant order when the true matrix is exactly low-rank. A trade-off between controlling the error and the rank is established through different choices of tuning parameters. The appealing practical performance of TL1 regularization is demonstrated through a simulation study that encompasses various sampling mechanisms, as well as two real-world applications. Additionally, the role of the hyper-parameter $a$ on the TL1-based model is explored via experiments to offer guidance in practical scenarios.

Hyperspectral imaging (HSI) technology captures spectral information across a broad wavelength range, providing richer pixel features compared to traditional color images with only three channels. Although pixel classification in HSI has been extensively studied, especially using graph convolution neural networks (GCNs), quantifying epistemic and aleatoric uncertainties associated with the HSI classification (HSIC) results remains an unexplored area. These two uncertainties are effective for out-of-distribution (OOD) and misclassification detection, respectively. In this paper, we adapt two advanced uncertainty quantification models, evidential GCNs (EGCN) and graph posterior networks (GPN), designed for node classifications in graphs, into the realm of HSIC. We first reveal theoretically that a popular uncertainty cross-entropy (UCE) loss function is insufficient to produce good epistemic uncertainty when learning EGCNs. To mitigate the limitations, we propose two regularization terms. One leverages the inherent property of HSI data where each feature vector is a linear combination of the spectra signatures of the confounding materials, while the other is the total variation (TV) regularization to enforce the spatial smoothness of the evidence with edge-preserving. We demonstrate the effectiveness of the proposed regularization terms on both EGCN and GPN on three real-world HSIC datasets for OOD and misclassification detection tasks. The code is available at GitHub.

Abstract. In this paper, we consider a low-rank tensor recovery problem. Based on the tensor singular value decomposition (t-SVD), we propose the ratio of the tensor nuclear norm and the tensor Frobenius norm (TNF) as a novel nonconvex surrogate of tensor’s tubal rank. The rationale of the proposed model for enforcing a low-rank structure is analyzed as its theoretical properties. Specifically, we introduce a null space property (NSP) type condition, under which a low-rank tensor is a local minimum for the proposed TNF recovery model. Numerically, we consider a low-rank tensor completion problem as a specific application of tensor recovery and employ the alternating direction method of multipliers (ADMM) to secure a model solution with guaranteed subsequential convergence under mild conditions. Extensive experiments demonstrate the superiority of our proposed model over state-of-the-art methods.

Tensor completion is an important problem in modern data analysis. In this work, we investigate a specific sampling strategy, referred to as tubal sampling. We propose two novel non-convex tensor completion frameworks that are easy to implement, named tensor L1-L2(TL12) and tensor completion via CUR (TCCUR). We test the efficiency of both methods on synthetic data and a color image inpainting problem. Empirical results reveal a trade-off between the accuracy and time efficiency of these two methods in a low sampling ratio. Each of them outperforms some classical completion methods in at least one aspect.

Poisson noise commonly occurs in images captured by photon-limited imaging systems such as in astronomy and medicine. As the distribution of Poisson noise depends on the pixel intensity value, noise levels vary from pixels to pixels. Hence, denoising a Poisson-corrupted image while preserving important details can be challenging. In this paper, we propose a Poisson denoising model by incorporating the weighted anisotropic–isotropic total variation (AITV) as a regularization. We then develop an alternating direction method of multipliers with a combination of a proximal operator for an efficient implementation. Lastly, numerical experiments demonstrate that our algorithm outperforms other Poisson denoising methods in terms of image quality and computational efficiency.

Deep neural networks have achieved significant success in the last decades, but they are not well-calibrated and often produce unreliable predictions. A large number of literature relies on uncertainty quantification to evaluate the reliability of a learning model, which is particularly important for applications of out-of-distribution (OOD) detection and misclassification detection. We are interested in uncertainty quantification for interdependent node-level classification. We start our analysis based on graph posterior networks (GPNs) that optimize the uncertainty cross-entropy (UCE)-based loss function. We describe the theoretical limitations of the widely-used UCE loss. To alleviate the identified drawbacks, we propose a distance-based regularization that encourages clustered OOD nodes to remain clustered in the latent space. We conduct extensive comparison experiments on eight standard datasets and demonstrate that the proposed regularization outperforms the state-of-the-art in both OOD detection and misclassification detection.

Hyperspectral unmixing is an effective tool to ascertain the material composition of each pixel in a hyperspectral image with typically hundreds of spectral channels. In this paper, we propose two graph-based semi-supervised unmixing methods. The first one directly applies graph learning to the unmixing problem, while the second one solves an optimization problem that combines the linear unmixing model and a graph-based regularization term. Following a semi-supervised framework, our methods require a very small number of training pixels that can be selected by a graph-based active learning method. We assume to obtain the ground truth information at these selected pixels, which can be either the exact abundance value or the one-hot pseudo label. In practice, the latter is much easier to obtain, which can be achieved by minimally involving a human in the loop. Compared to other popular blind unmixing methods, our methods significantly improve performance with minimal supervision. Specifically, the experiments demonstrate that the proposed methods improve the state-of-the-art blind unmixing approaches by 50% or more using only 0.4% of training pixels.

We analyze the impact of the Los Angeles Mayor’s Office of Gang Reduction Youth Development (GRYD) prevention programming using quasi-experimental data. We model the evolution of questionnaire scores and apply Dynamic Mode Decomposition (DMD) to describe the asymptotic behavior of the dynamical system. The analysis indicates that risk decreased for youth who enrolled in GRYD prevention services, while it increased or remained the same for those who were in the control group. We augment these observations using a difference-in-differences (DID) model, showing that the decrease in risk can be attributed to enrolment in prevention services. We draw a connection between DMD and DID using both mathematical analysis and empirical evidence from the questionnaire data. Combining DMD and DID with factor analysis, we investigate the effectiveness of prevention services with respect to different attitudinal domains. We conclude that gang prevention is most effective in impacting attitudes towards negative peer obedience and least effective in impacting attitudes towards violence for self defense. Our analytical approach can be extended to other types of repeated questionnaires.

In a class of piecewise-constant image segmentation models, we propose to incorporate a weighted difference of anisotropic and isotropic total variation (AITV) to regularize the partition boundaries in an image. In particular, we replace the total variation regularization in the Chan--Vese segmentation model and a fuzzy region competition model by the proposed AITV. To deal with the nonconvex nature of AITV, we apply the difference-of-convex algorithm (DCA), in which the subproblems can be minimized by the primal-dual hybrid gradient method with linesearch. The convergence of the DCA scheme is analyzed. In addition, a generalization to color image segmentation is discussed. In the numerical experiments, we compare the proposed models with the classic convex approaches and the two-stage segmentation methods (smoothing and then thresholding) on various images, showing that our models are effective in image segmentation and robust with respect to impulsive noises.

In this paper, we consider minimizing the $L_1/L_2$ term on the gradient for a limited-angle scanning problem in computed tomography (CT) reconstruction. We design a specific splitting framework for an unconstrained optimization model so that the alternating direction method of multipliers (ADMM) has guaranteed convergence under certain conditions. In addition, we incorporate a box constraint that is reasonable for imaging applications, and the convergence for the additional box constraint can also be established. Numerical results on both synthetic and experimental datasets demonstrate the effectiveness and efficiency of our proposed approach, showing significant improvements over the state-of-the-art methods in the limited-angle CT reconstruction.

Remote sensing data from hyperspectral cameras suffer from limited spatial resolution, in which a single pixel of a hyperspectral image may contain information from several materials in the field of view. Blind hyperspectral image unmixing is the process of identifying the pure spectra of individual materials (i.e., endmembers) and their proportions (i.e., abundances) at each pixel. In this article, we propose a novel blind hyperspectral unmixing model based on the graph total variation (gTV) regularization, which can be solved efficiently by the alternating direction method of multipliers (ADMM). To further alleviate the computational cost, we apply the Nyström method to approximate a fully connected graph by a small subset of sampled points. Furthermore, we adopt the Merriman-Bence-Osher (MBO) scheme to solve the gTV-involved subproblem in ADMM by decomposing a gray-scale image into a bitwise form. A variety of numerical experiments on synthetic and real hyperspectral images are conducted, showcasing the potential of the proposed method in terms of identification accuracy and computational efficiency.

Brain source imaging is an important method for noninvasively characterizing brain activity using Electroencephalogram (EEG) or Magnetoencephalography (MEG) recordings. Traditional EEG/MEG Source Imaging (ESI) methods usually assume the source activities at different time points are unrelated, and do not utilize the temporal structure in the source activation, making the ESI analysis sensitive to noise. Some methods may encourage very similar activation patterns across the entire time course and may be incapable of accounting the variation along the time course. To effectively deal with noise while maintaining flexibility and continuity among brain activation patterns, we propose a novel probabilistic ESI model based on a hierarchical graph prior. Under our method, a spanning tree constraint ensures that activity patterns have spatiotemporal continuity. An efficient algorithm based on an alternating convex search is presented to solve the resulting problem of the proposed model with guaranteed convergence. Comprehensive numerical studies using synthetic data on a realistic brain model are conducted under different levels of signal-to-noise ratio (SNR) from both sensor and source spaces. We also examine the EEG/MEG datasets in two real applications, in which our ESI reconstructions are neurologically plausible. All the results demonstrate significant improvements of the proposed method over benchmark methods in terms of source localization performance, especially at high noise levels.

Phase retrieval plays an important role in vast industrial and scientific applications. We consider a noisy phase retrieval problem in which the magnitudes of the Fourier transform (or a general linear transform) of an underling object are corrupted by Poisson noise, since any optical sensors detect photons, and the number of detected photons follows the Poisson distribution. We propose a variational model for phase retrieval based on a total variation regularization as an image prior and maximum a posteriori estimation of a Poisson noise model, which is referred to as “TV-PoiPR”. We also propose an efficient numerical algorithm based on an alternating direction method of multipliers and establish its convergence. Extensive experiments for coded diffraction, holographic, and ptychographic patterns are conducted using both real- and complex-valued images to demonstrate the effectiveness of our proposed methods.

We reformulate the original phase retrieval problem into two variational models (with and without regularization), both containing a globally Lipschitz differentiable term. These two models can be efficiently solved via the proposed Partially Preconditioned Proximal Alternating Linearized Minimization (P${}^3$ALM) for masked Fourier measurements. Thanks to the Lipschitz differentiable term, we prove the global convergence of P${}^3$ALM for solving the nonconvex phase retrieval problems. Extensive experiments are conducted to show the effectiveness of the proposed methods.

Multi-energy computed tomography (CT) is an emerging medical image modality with a number of potential applications in diagnosis and therapy. However, high system cost and technical barriers obstruct its step into routine clinical practice. In this study, we propose a framework to realize multi-energy cone beam CT (ME-CBCT) on the CBCT system that is widely available and has been routinely used for radiotherapy image guidance. In our method, a kVp switching technique is realized, which acquires x-ray projections with kVp levels cycling through a number of values. For this kVp-switching based ME-CBCT acquisition, x-ray projections of each energy channel are only a subset of all the acquired projections. This leads to an undersampling issue, posing challenges to the reconstruction problem. We propose a spatial spectral non-local means (ssNLM) method to reconstruct ME-CBCT, which employs image correlations along both spatial and spectral directions to suppress noisy and streak artifacts. To address the intensity scale difference at different energy channels, a histogram matching method is incorporated. Our method is different from conventionally used NLM methods in that spectral dimension is included, which helps to effectively remove streak artifacts appearing at different directions in images with different energy channels. Convergence analysis of our algorithm is provided. A comprehensive set of simulation and real experimental studies demonstrate feasibility of our ME-CBCT scheme and the capability of achieving superior image quality compared to conventional filtered backprojection-type (FBP) and NLM reconstruction methods.

We propose a novel image inpainting model that can effectively estimate missing pixels in an observed image. The latent image is characterized by a group-based low-rank prior, which assumes that a group of vectorized similar image patches can be well approximated by a low-rank matrix. We enforce the low-rankness of each group by penalizing a truncated difference of the l1and the l2norms of its singular values, which achieves a close approximation to the matrix rank. We apply a difference of convex algorithm (DCA) to solve the proposed model efficiently. Our method is validated on filling missing blocks and randomly missing pixels, with superior performance over the state-of-the-art.

We study a truncated difference of $l_1$ and $l_2$ norms as a nonconvex metric for recovering sparse vectors and low-rank matrices from linear measurements. By discarding large magnitude entries/singular values in penalization, the proposed metric, denoted as truncated $l_{1-2}$, achieves a nearly unbiased approximation of the vector sparsity/matrix rank. We establish exact and stable recovery conditions of truncated $l_{1-2}$ minimization under the restricted isometry property (RIP) framework. Computationally, we apply the difference of convex functions algorithm (DCA) to efficiently solve truncated $l_{1-2}$ minimization with guaranteed convergence. Our method is validated on sparse vector recovery, matrix completion, and magnetic resonance imaging (MRI) reconstruction, with performance comparable to the state of the art. Particularly for MRI reconstruction, it succeeds in reconstructing the $256\times 256$ Shepp--Logan phantom image from merely 7 radial lines.

EEG source imaging integrates temporal and spatial components of EEG to localize the generating source of electrical potentials based on recorded EEG data on the scalp. As EEG sensors can't directly measure activated brain sources, many approaches were proposed to estimate brain source activation pattern given EEG data. However, since most part of the brain activity is composed of the spontaneous non-task related activations, true task caused activation sources will be corrupted in strong background signal. For decades, the EEG inverse problem was solved in an unsupervised way without any utilization of the label information that represents different brain states. We propose that by leveraging label information, the task related discriminative sources can be much better retrieved among strong spontaneous background signals. A novel model for solving EEG inverse problem called Laplacian Graph Regularized Discriminative Source Reconstruction which aims to explicitly extract the discriminative sources by implicitly coding the label information into the graph regularization term. The proposed model can be generally extended with different assumptions. The extension of our framework is applied to VB-SCCD model which aim to estimate extended brain sources by including a spatial total variation regularization term. Simulated results show the effectiveness of the proposed framework.

We propose to incorporate a weighted difference of anisotropic and isotropic total variation (TV) norms into a relaxed formulation of the two phase Mumford-Shah (MS) model for image segmentation. We show results exceeding those obtained by the MS model when using the standard TV norm to regularize partition boundaries. In particular, examples illustrating the qualitative differences between the proposed model and the standard MS one are shown. A fast numerical method is introduced to minimize the proposed model utilizing the difference-of-convex algorithm (DCA) and the primal dual hybrid gradient (PDHG) method.

We propose a weighted difference of anisotropic and isotropic total variation (TV) as a regularization for image processing tasks, based on the well-known TV model and natural image statistics. Due to the form of our model, it is natural to compute via a difference of convex algorithm (DCA). We draw its connection to the Bregman iteration for convex problems and prove that the iteration generated from our algorithm converges to a stationary point with the objective function values decreasing monotonically. A stopping strategy based on the stable oscillatory pattern of the iteration error from the ground truth is introduced. In numerical experiments on image denoising, image deblurring, and magnetic resonance imaging (MRI) reconstruction, our method improves on the classical TV model consistently and is on par with representative state-of-the-art methods.

This paper addresses the nonstationary out-of-focus (OOF) blur removal in the application of barcode reconstruction. We propose a partially blind deblurring method when partial knowledge of the clean barcode is available. In particular, we consider an image formation model based on geometrical optics, which involves the point-spread function (PSF) for the OOF blur. With the known information, we can estimate a low-dimensional representation of the PSF using the Levenberg--Marquardt algorithm. Once the PSF is obtained, the deblurred image is computed by solving a quadratic program. We find that imposing a [0,1] box constraint is often good enough to enforce binary signal. Experiments on real data demonstrate that the forward model is physically realistic and our partially blind deblurring method can yield good reconstructions.

Unmixing problems in many areas such as hyperspectral imaging and differential optical absorption spectroscopy (DOAS) often require finding sparse nonnegative linear combinations of dictionary elements that match observed data. We show how aspects of these problems, such as misalignment of DOAS references and uncertainty in hyperspectral endmembers, can be modeled by expanding the dictionary with grouped elements and imposing a structured sparsity assumption that the combinations within each group should be sparse or even 1-sparse. If the dictionary is highly coherent, it is difficult to obtain good solutions using convex or greedy methods, such as nonnegative least squares (NNLS) or orthogonal matching pursuit. We use penalties related to the Hoyer measure, which is the ratio of the $l_1$ and $l_2$ norms, as sparsity penalties to be added to the objective in NNLS-type models. For solving the resulting nonconvex models, we propose a scaled gradient projection algorithm that requires solving a sequence of strongly convex quadratic programs. We discuss its close connections to convex splitting methods and difference of convex programming. We also present promising numerical results for DOAS analysis and hyperspectral unmixing problems.

Under-sampling and insufficient data result in a big challenge in the reconstruction of x-ray computed tomographic (CT) images. In addition, patient's respiratory motion also deteriorates this reconstruction process as it normally leads to blurred outputs. In this work, we propose an iterative method with a combination of total variation (TV) regularization and principle component analysis (PCA) regularization. Partial prior knowledge of the CT images, obtained through PCA analysis of training images is incorporated in the reconstruction process. Numerical experiments are performed in the context of a fan-beam CT reconstruction, which shows advantages of our method over the ones with just TV regularization or just PCA regularization.

Most algorithms for reconstructing shape from defocus assume that the images are obtained with a camera that has been previously calibrated so that the aperture, focal plane, and focal length are known. In this manuscript we characterize the set of scenes that can be reconstructed from defocused images regardless of calibration parameters. In lack of knowledge about the camera or about the scene, reconstruction is possible only up to an equivalence class that is described analytically. When weak knowledge about the scene is available, however, we show how it can be exploited in order to auto-calibrate the imaging device. This includes imaging a slanted plane or generic assumptions on the restoration of the deblurred images.