Jian-Feng Cai

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This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task of recovering a large matrix from a small subset of its entries(More)
Split Bregman methods introduced in [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2(2009), pp. 323–343] have been demonstrated to be efficient tools for solving total variation norm minimization problems, which arise from partial differential equation based image restoration such as image denoising and magnetic resonance imaging reconstruction from(More)
In this paper, we propose a two-phase approach to restore images corrupted by blur and impulse noise. In the first phase, we identify the outlier candidates—the pixels that are likely to be corrupted by impulse noise. We consider that the remaining data pixels are essentially free of outliers. Then in the second phase, the image is deblurred and denoised(More)
The variational techniques (e.g., the total variation based method [1]) are well established and effective for image restoration, as well as many other applications, while wavelet frame based approach is relatively new and came from a different school. This paper is designed to establish a connection between these two major approaches for image restoration.(More)
Restoring a clear image from a single motion-blurred image due to camera shake has long been a challenging problem in digital imaging. Existing blind deblurring techniques either only remove simple motion blurring, or need user interactions to work on more complex cases. In this paper, we present an approach to remove motion blurring from a single image by(More)
How to recover a clear image from a single motion-blurred image has long been a challenging open problem in digital imaging. In this paper, we focus on how to recover a motion-blurred image due to camera shake. A regularization-based approach is proposed to remove motion blurring from the image by regularizing the sparsity of both the original image and the(More)
The restoration of blurred images corrupted with impulse noise is a difficult problem which has been considered in a series of recent papers. These papers tackle the problem by using variational methods involving an L1shaped data-fidelity term. Because of this term, the relevant methods exhibit systematic errors at the corrupted pixel locations and require(More)
One of the key steps in compressed sensing is to solve the basis pursuit problem minu∈Rn{‖u‖1 : Au = f}. Bregman iteration was very successfully used to solve this problem in [40]. Also, a simple and fast iterative algorithm based on linearized Bregman iteration was proposed in [40], which is described in detail with numerical simulations in [35]. A(More)
Sparsity based regularization methods for image restoration assume that the underlying image has a good sparse approximation under a certain system. Such a system can be a basis, a frame, or a general over-complete dictionary. One widely used class of such systems in image restoration are wavelet tight frames. There have been enduring efforts on seeking(More)