Time-Varying Graph Signal Recovery Using High-Order Smoothness and Adaptive Low-rankness

Time-varying graph signal recovery has been widely used in many applications, including climate change, environmental hazard monitoring, and epidemic studies. It is crucial to choose appropriate regularizations to describe the characteristics of the underlying signals, such as the smoothness of the signal over the graph domain and the low-rank structure of the spatial-temporal signal modeled in a matrix form. As one of the most popular options, the graph Laplacian is commonly adopted in designing graph regularizations for reconstructing signals defined on a graph from partially observed data. In this work, we propose a time-varying graph signal recovery method based on the high-order Sobolev smoothness and an error-function weighted nuclear norm regularization to enforce the low-rankness. Two efficient algorithms based on the alternating direction method of multipliers and iterative reweighting are proposed, and convergence of one algorithm is shown in detail. We conduct various numerical experiments on synthetic and real-world data sets to demonstrate the proposed method's effectiveness compared to the state-of-the-art in graph signal recovery.

Improvements on Uncertainty Quantification for Node Classification via Distance-Based Regularization

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.

Weighted Anisotropic-Isotropic Total Variation for Poisson Denoising

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.

Non-convex approaches for low-rank tensor completion under tubal sampling

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 $L_1$-$L_2$ (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.

Efficient Image Segmentation Framework with Difference of Anisotropic and Isotropic Total Variation for Blur and Poisson Noise Removal

In this paper, we aim to segment an image degraded by blur and Poisson noise. We adopt a smoothing-and-thresholding (SaT) segmentation framework that finds a piecewise-smooth solution, followed by $k$-means clustering to segment the image. Specifically for the image smoothing step, we replace the least-squares fidelity for Gaussian noise in the Mumford-Shah model with a maximum posterior (MAP) term to deal with Poisson noise and we incorporate the weighted difference of anisotropic and isotropic total variation (AITV) as a regularization to promote the sparsity of image gradients. For such a nonconvex model, we develop a specific splitting scheme and utilize a proximal operator to apply the alternating direction method of multipliers (ADMM). Convergence analysis is provided to validate the efficacy of the ADMM scheme. Numerical experiments on various segmentation scenarios (grayscale/color and multiphase) showcase that our proposed method outperforms a number of segmentation methods, including the original SaT.

A Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation

In this paper, we propose a multi-stage image segmentation framework that incorporates a weighted difference of anisotropic and isotropic total variation (AITV). The segmentation framework generally consists of two stages: smoothing and thresholding, thus referred to as SaT. In the first stage, a smoothed image is obtained by an AITV-regularized Mumford-Shah (MS) model, which can be solved efficiently by the alternating direction method of multipliers (ADMM) with a closed-form solution of a proximal operator of the $\ell_1 -\alpha \ell_2$ regularizer. Convergence of the ADMM algorithm is analyzed. In the second stage, we threshold the smoothed image by $k$-means clustering to obtain the final segmentation result. Numerical experiments demonstrate that the proposed segmentation framework is versatile for both grayscale and color images, efficient in producing high-quality segmentation results within a few seconds, and robust to input images that are corrupted with noise, blur, or both. We compare the AITV method with its original convex and nonconvex TV$^p (0<p<1)$ counterparts, showcasing the qualitative and quantitative advantages of our proposed method.

A Lifted $\ell_1 $ Framework for Sparse Recovery

Motivated by re-weighted $\ell_1$ approaches for sparse recovery, we propose a lifted $\ell_1$ (LL1) regularization which is a generalized form of several popular regularizations in the literature. By exploring such connections, we discover there are two types of lifting functions which can guarantee that the proposed approach is equivalent to the $\ell_0$ minimization. Computationally, we design an efficient algorithm via the alternating direction method of multiplier (ADMM) and establish the convergence for an unconstrained formulation. Experimental results are presented to demonstrate how this generalization improves sparse recovery over the state-of-the-art.

Minimizing L1 over L2 norms on the gradient

In this paper, we study the L1/L2 minimization on the gradient for imaging applications. Several recent works have demonstrated that L1/L2 is better than the L1 norm when approximating the L0 norm to promote sparsity. Consequently, we postulate that applying L1/L2 on the gradient is better than the classic total variation (the L1 norm on the gradient) to enforce the sparsity of the image gradient. To verify our hypothesis, we consider a constrained formulation to reveal empirical evidence on the superiority of L1/L2 over L1 when recovering piecewise constant signals from low-frequency measurements. Numerically, we design a specific splitting scheme, under which we can prove the subsequential convergence for the alternating direction method of multipliers (ADMM). Experimentally, we demonstrate visible improvements of L1/L2 over L1 and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of MRI and CT reconstruction. All the numerical results show the efficiency of our proposed approach.

Limited-angle CT reconstruction via the L1/L2 minimization

In this paper, we consider minimizing the L1/L2 term on the gradient for a limit-angle scanning problem in computed tomography (CT) reconstruction. We design a splitting framework for both constrained and unconstrained optimization models. In addition, we can incorporate a box constraint that is reasonable for imaging applications. Numerical schemes are based on the alternating direction method of multipliers (ADMM), and we provide the convergence analysis of all the proposed algorithms (constrained/unconstrained and with/without the box constraint). Experimental results demonstrate the efficiency of our proposed approaches, showing significant improvements over the state-of-the-art methods in the limit-angle CT reconstruction. Specifically worth noticing is an exact recovery of the Shepp-Logan phantom from noiseless projection data with 30 scanning angle.

A Weighted Difference of Anisotropic and Isotropic Total Variation for Relaxed Mumford-Shah Color and Multiphase Image Segmentation

In a class of piecewise-constant image segmentation models, we incorporate a weighted difference of anisotropic and isotropic total variation (TV) to regularize the partition boundaries in an image. To deal with the weighted anisotropic-isotropic TV, we apply the difference-of-convex algorithm (DCA), where the subproblems can be minimized by the primal-dual hybrid gradient method (PDHG). As a result, we are able to design an alternating minimization algorithm to solve the proposed image segmentation models. The models and algorithms are further extended to segment color images and to perform multiphase segmentation. In the numerical experiments, we compare our proposed models with the Chan-Vese models that use either anisotropic or isotropic TV and the two-stage segmentation methods (denoising and then thresholding) on various images. The results demonstrate the effectiveness and robustness of incorporating weighted anisotropic-isotropic TV in image segmentation.