You-Wei Wen

A New Total Variation Method for Multiplicative Noise Removal

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.

A Fast Total Variation Minimization Method for Image Restoration

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

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 Morozovs 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.

Iterative Algorithms Based on Decoupling of Deblurring and Denoising for Image Restoration

In this paper, we propose iterative algorithms for solving image restoration problems. The iterative algorithms are based on decoupling of deblurring and denoising steps in the restoration process. In the deblurring step, an efficient deblurring method using fast transforms can be employed. In the denoising step, effective methods such as the wavelet shrinkage denoising method or the total variation denoising method can be used. The main advantage of this proposal is that the resulting algorithms can be very efficient and can produce better restored images in visual quality and signal-to-noise ratio than those by the restoration methods using the combination of a data-fitting term and a regularization term. The convergence of the proposed algorithms is shown in the paper. Numerical examples are also given to demonstrate the effectiveness of these algorithms.

Fast Image Restoration Methods for Impulse and Gaussian Noises Removal

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.

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.

Exact Histogram Specification for Digital Images Using a Variational Approach

We consider the problem of exact histogram specification for digital (quantized) images. The goal is to transform the input digital image into an output (also digital) image that follows a prescribed histogram. Classical histogram modification methods are designed for real-valued images where all pixels have different values, so exact histogram specification is straightforward. Digital images typically have numerous pixels which share the same value. If one imposes the prescribed histogram to a digital image, usually there are numerous ways of assigning the prescribed values to the quantized values of the image. Therefore, exact histogram specification for digital images is an ill-posed problem. In order to guarantee that any prescribed histogram will be satisfied exactly, all pixels of the input digital image must be rearranged in a strictly ordered way. Further, the obtained strict ordering must faithfully account for the specific features of the input digital image. Such a task can be realized if we are able to extract additional representative information (called auxiliary attributes) from the input digital image. This is a real challenge in exact histogram specification for digital images. We propose a new method that efficiently provides a strict and faithful ordering for all pixel values. It is based on a well designed variational approach. Noticing that the input digital image contains quantization noise, we minimize a specialized objective function whose solution is a real-valued image with slightly reduced quantization noise, which remains very close to the input digital image. We show that all the pixels of this real-valued image can be ordered in a strict way with a very high probability. Then transforming the latter image into another digital image satisfying a specified histogram is an easy task. Numerical results show that our method outperforms by far the existing competing methods.

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.

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.

On inversion of Toeplitz matrices

In 1992, Labahn and Shalom showed that the inverse of a Toeplitz matrix can be reconstructed by two of its columns and by some entries of the original Toeplitz matrix. We here present a modification of this result and another (shorter) proof.