Robust Principal Component Analysis (RPCA) is a powerful framework for separating low-rank and sparse structures in data, with widespread applications in image processing, video surveillance, and anomaly detection. Classical RPCA models typically employ ℓ 1 -norm regularization to promote sparsity; however, such convex relaxations often introduce estimation bias and perform poorly in the presence of strong noise or large outliers. While non-convex penalties can alleviate these issues, they are often computationally expensive and sensitive to initialization. In this paper, we propose a novel RPCA framework that incorporates Adaptive Weighted Least Squares (AWLS) and Low-Rank Matrix Factorization (LRMF) to address these limitations. A key innovation of our method lies in its dynamic weight updating strategy, which enables the weight matrix to automatically adapt during each iteration, thereby suppressing outliers more effectively. The sparse component is regularized using a weighted Frobenius norm, which reduces bias and simplifies the optimization process through closed-form updates. An alternating minimization algorithm is adopted to solve the model efficiently. Extensive numerical experiments demonstrate that the proposed method achieves superior robustness and accuracy compared to state-of-the-art non-convex RPCA methods, while maintaining low computational complexity.

Abstract. The Mumford–Shah model is a classical segmentation model, but its objective function is nonconvex. The smoothing and thresholding (SaT) approach is a convex variant of the Mumford–Shah model, which seeks a smoothed approximation solution to the Mumford–Shah model. The SaT approach separates the segmentation into two stages: first, a convex energy function is minimized to obtain a smoothed image; then, a thresholding technique is applied to segment the smoothed image. The energy function consists of three weighted terms and the weights are called the regularization parameters. Selecting appropriate regularization parameters is crucial to achieving effective segmentation results. Traditionally, the regularization parameters are chosen by trial-and-error, which is a very time-consuming procedure and is not practical in real applications. In this paper, we apply a Bayesian inference approach to infer the regularization parameters and estimate the smoothed image. We analyze the convex variant Mumford–Shah variational model from a statistical perspective and then construct a hierarchical Bayesian model. A mean field variational family is used to approximate the posterior distribution. The variational density of the smoothed image is assumed to have a Gaussian density, and the hyperparameters are assumed to have Gamma variational densities. All the parameters in the Gaussian density and Gamma densities are iteratively updated. Experimental results show that the proposed approach is capable of generating high-quality segmentation results. Although the proposed approach contains an inference step to estimate the regularization parameters, it requires less CPU running time to obtain the smoothed image than previous methods.

In this paper, we consider decomposing an image into its cartoon and texture components. Traditional methods, which mainly rely on the gradient amplitude of images to distinguish between these components, often show limitations in decomposing small-scale, high-contrast texture patterns and large-scale, low-contrast structural components. Specifically, these methods tend to decompose the former to the cartoon image and the latter to the texture image, neglecting the scale features inherent in both components. To overcome these challenges, we introduce a new variational model which incorporates an$L_{0}$-based total variation norm for the cartoon component and an$L_{2}$norm for the scale space representation of the texture component. We show that the texture component has a small$L_{2}$norm in the scale space representation. We apply a quadratic penalty function to handle the non-separable$L_{0}$norm minimization problem. Numerical experiments are given to illustrate the efficiency and effectiveness of our approach.

In this paper, we consider the problem of image restoration with box-constraints. Image restoration problem is ill-conditioned and the regularization approach has widely been used to stabilize the solution. The restored image highly depends on the choice of the regularization parameter. The regularization parameter is generally determined by trial-and-error method when no true original image is available. Obviously, it is time consuming. The main aim in this paper is to develop an algorithm to choose the regularization parameter automatically when the box-constraints are imposed. In the proposed algorithm, the regularization parameter is adaptively determined by the previous iterative solution. Numerical simulations are used to demonstrate the performance of the proposed method.

There are two key issues in successfully solving the image restoration problem: 1) estimation of the regularization parameter that balances data fidelity with the regularity of the solution and 2) development of efficient numerical techniques for computing the solution. In this paper, we derive a fast algorithm that simultaneously estimates the regularization parameter and restores the image. The new approach is based on the total-variation (TV) regularized strategy and Morozov's discrepancy principle. The TV norm is represented by the dual formulation that changes the minimization problem into a minimax problem. A proximal point method is developed to compute the saddle point of the minimax problem. By adjusting the regularization parameter adaptively in each iteration, the solution is guaranteed to satisfy the discrepancy principle. We will give the convergence proof of our algorithm and numerically show that it is better than some state-of-the-art methods in terms of both speed and accuracy.

Loss of information in a wavelet domain can occur during storage or transmission when the images are formatted and stored in terms of wavelet coefficients. This calls for image inpainting in wavelet domains. In this paper, a variational approach is used to formulate the reconstruction problem. We propose a simple but very efficient iterative scheme to calculate an optimal solution and prove its convergence. Numerical results are presented to show the performance of the proposed algorithm.

Multiplicative noise removal problems have attracted much attention in recent years. Unlike additive noise removal problems, the noise is multiplied to the orginal image, so almost all information of the original image may disappear in the observed image. The main aim of this paper is to propose and study a strictly convex objective function for multiplicative noise removal problems. We also incorporate the modified total variation regularization in the objective function to recover image edges. We develop an alternating minimization algorithm to find the minimizer of such an objective function efficiently and also show the convergence of the minimizing method. Our experimental results show that the quality of images denoised by the proposed method is quite good.

In this paper, we study the restoration of blurred images corrupted by impulse noise or mixed impulse plus Gaussian noises. In the proposed method, we use the modified total variation minimization scheme to regularize the deblurred image and fill in suitable values for noisy image pixels where these are detected by median-type filters. An alternating minimization algorithm is employed to solve the proposed total variation minimization problem. Our experimental results show the proposed algorithm is very efficient and the quality of restored images by the proposed method is competitive with those restored by the existing variational image restoration methods.

A wavelet inpainting problem refers to the problem of filling in missing wavelet coefficients in an image. A variational approach was used by Chan et al. The resulting functional was minimized by the gradient descent method. In this paper, we use an optimization transfer technique which involves replacing their univariate functional by a bivariate functional by adding an auxiliary variable. Our bivariate functional can be minimized easily by alternating minimization: for the auxiliary variable, the minimum has a closed form solution, and for the original variable, the minimization problem can be formulated as a classical total variation (TV) denoising problem and, hence, can be solved efficiently using a dual formulation. We show that our bivariate functional is equivalent to the original univariate functional. We also show that our alternating minimization is convergent. Numerical results show that the proposed algorithm is very efficient and outperforms that of Chan et al.

In this paper, we consider and study a total variation minimization model for color image restoration. In the proposed model, we use the color total variation minimization scheme to denoise the deblurred color image. An alternating minimization algorithm is employed to solve the proposed total variation minimization problem. We show the convergence of the alternating minimization algorithm and demonstrate that the algorithm is very efficient. Our experimental results show that the quality of restored color images by the proposed method are competitive with the other tested methods.