Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology, and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired, and thus, how to recover the full signal becomes an active research topic, but existing approaches cannot efficiently recover $N$ -dimensional exponential signals with $N\geq 3$ . In this paper, we study the problem of recovering $N$ -dimensional (particularly $N\geq 3$ ) exponential signals from partial observations, and formulate this problem as a low-rank tensor completion problem with exponential factor vectors. The full signal is reconstructed by simultaneously exploiting the CANDECOMP/PARAFAC tensor structure and the exponential structure of the associated factor vectors. The latter is promoted by minimizing an objective function involving the nuclear norm of Hankel matrices. Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank.

Many signals are modeled as a superposition of exponential functions in spectroscopy of chemistry, biology, and medical imaging. This paper studies the problem of recovering exponential signals from a random subset of samples. We exploit the Vandermonde structure of the Hankel matrix formed by the exponential signal and formulate signal recovery as Hankel matrix completion with Vandermonde factorization (HVaF). A numerical algorithm is developed to solve the proposed model and its sequence convergence is analyzed theoretically. Experiments on synthetic data demonstrate that HVaF succeeds over a wider regime than the state-of-the-art nuclear-norm-minimization-based Hankel matrix completion method, while it has a less restriction on frequency separation than the state-of-the-art atomic norm minimization and fast iterative hard thresholding methods. The effectiveness of HVaF is further validated on biological magnetic resonance spectroscopy data.

In magnetic resonance imaging (MRI), the super-resolution technology has played a great role in improving image quality. The aim of this paper is to improve edges of brain MRI by incorporating the gradient information of another contrast high-resolution image. Multi-contrast images are assumed to possess the same gradient direction in a local pattern. We proposed to establish a relation model of gradient value between different contrast images to restore a high-resolution image from its input low-resolution version. The similarity of image patches is employed to estimate intensity parameters, leading a more accurate reconstructed image. Then, an iterative back-projection filter is applied to the reconstructed image to further increase the image quality. The new approach is verified on synthetic and real brain MRI images and achieves higher visual quality and higher objective quality criteria than the compared state-of-the-art super-resolution approaches. The gradient information of the multi-contrast MRI images is very useful. With a proper relation model, the proposed method enhances image edges in MRI image super-resolution. Improving the MRI image resolution from very low-resolution observations is challenging. We tackle this problem by first modeling the relation of gradient value in multi-contrast MRI and then performing fast supper-resolution methods. This relation model may be helpful for other MRI reconstruction problems.

G oal: The two dimensional magnetic resonance spectroscopy (MRS) possesses many important applications in bioengineering but suffers from long acquisition duration. Non-uniform sampling has been applied to the spatiotemporally encoded ultrafast MRS, but results in missing data in the hybrid time and frequency plane. An approach is proposed to recover this missing signal, of which enables high quality spectrum reconstruction. M ethods: The natural exponential characteristic of MRS is exploited to recover the hybrid time and frequency signal. The reconstruction issue is formulated as a low rank enhanced Hankel matrix completion problem and is solved by a fast numerical algorithm. R esults: Experiments on synthetic and real MRS data show that the proposed method provides faithful spectrum reconstruction, and outperforms the state-of-the-art compressed sensing approach on recovering low-intensity spectral peaks and robustness to different sampling patterns. C onclusion: The exponential signal property serves as an useful tool to model the time-domain MRS signals and even allows missing data recovery. The proposed method has been shown to reconstruct high quality MRS spectra from non-uniformly sampled data in the hybrid time and frequency plane. S ignificance: Low-intensity signal reconstruction is generally challenging in biological MRS and we provide a solution to this problem. The proposed method may be extended to recover signals that generally can be modeled as a sum of exponential functions in biomedical engineering applications, e.g., signal enhancement, feature extraction, and fast sampling.

Inferring the covariance matrix of multivariate data is of great interest in statistics, finance, and data science. It is often carried out via the maximum likelihood estimation (MLE) principle, which seeks a covariance matrix estimator maximizing the observed data likelihood. However, such estimator is usually poor when number of samples is not sufficiently larger than the number of variables. With the assumption that a covariance matrix can be decomposed into a low-rank matrix and a diagonal matrix, factor analysis (FA) model is a popular dimensionality reduction technique in improving the estimation performance. Recently, more and more evidence shows that the covariance matrix of real-valued data may admit the nonnegative correlation structure, which has attracted a lot of interest in some areas like finance and psychometrics. There does not exist any work estimating the covariance matrix simultaneously satisfying both structures. In this paper, we propose an MLE problem formulation for covariance matrix considering jointly the low-rank FA model and nonnegative correlation structures. Since the proposed problem formulation is an intractable non-convex problem, a block coordinate descent algorithm is further proposed to solve a relaxed version of our proposed formulation. The superior performance of our proposed formulation and the algorithm are verified through numerical simulations on both synthetic data and real market data.

This paper introduces Polynomial Graphical Lasso (PGL), a new approach to learning graph structures from nodal signals. Our key contribution lies in modeling the signals as Gaussian and stationary on the graph, enabling the development of a graph learning formulation that combines the strengths of graphical lasso with a more encompassing model. Specifically, we assume that the precision matrix can take any polynomial form of the sought graph, allowing for increased flexibility in modeling nodal relationships. Given the inherent complexity and nonconvexity of the optimization problem, we (i) propose a low-complexity algorithm that alternates between estimating the graph and precision matrices, and (ii) characterize its convergence. We evaluate the performance of PGL through comprehensive numerical simulations using both synthetic and real data, demonstrating its superiority over several alternatives. Overall, this approach presents a significant advancement in graph learning and holds promise for various applications in graph-aware signal analysis and beyond.