L0 Gradient-Regularization and Scale Space Representation Model for Cartoon and Texture Decomposition.

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 L0 -based total variation norm for the cartoon component and an L2 norm for the scale space representation of the texture component. We show that the texture component has a small L2 norm in the scale space representation. We apply a quadratic penalty function to handle the non-separable L0 norm minimization problem. Numerical experiments are given to illustrate the efficiency and effectiveness of our approach.

A primal-dual method for total-variation-based wavelet domain inpainting.

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.

Parameter selection for total-variation-based image restoration using discrepancy principle.

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.

Fast splitting algorithm for multiframe total variation blind video deconvolution.

We consider the recovery of degraded videos without complete knowledge about the degradation. A spatially shift-invariant but temporally shift-varying video formation model is used. This leads to a simple multiframe degradation model that relates each original video frame with multiple observed frames and point spread functions (PSFs). We propose a variational method that simultaneously reconstructs each video frame and the associated PSFs from the corresponding observed frames. Total variation (TV) regularization is used on both the video frames and the PSFs to further reduce the ill-posedness and to better preserve edges. In order to make TV minimization practical for video sequences, we propose an efficient splitting method that generalizes some recent fast single-image TV minimization methods to the multiframe case. Both synthetic and real videos are used to show the performance of the proposed method.

A fast optimization transfer algorithm for image inpainting in wavelet domains.

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.

Efficient total variation minimization methods for color image restoration.

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.