Parallel and distributed computation: numerical methods
Parallel and distributed computation: numerical methods
Ten lectures on wavelets
Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Mathematical Programming: Series A and B
Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming
Mathematics of Operations Research
Geometric partial differential equations and image analysis
Geometric partial differential equations and image analysis
Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures
Wavelet Algorithms for High-Resolution Image Reconstruction
SIAM Journal on Scientific Computing
Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (Applied Mathematical Sciences)
Inpainting and Zooming Using Sparse Representations
The Computer Journal
Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Linearized Bregman Iterations for Frame-Based Image Deblurring
SIAM Journal on Imaging Sciences
The Split Bregman Method for L1-Regularized Problems
SIAM Journal on Imaging Sciences
A New Alternating Minimization Algorithm for Total Variation Image Reconstruction
SIAM Journal on Imaging Sciences
Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data
ISBI'09 Proceedings of the Sixth IEEE international conference on Symposium on Biomedical Imaging: From Nano to Macro
Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction
SIAM Journal on Imaging Sciences
Sparse Signal Reconstruction via Iterative Support Detection
SIAM Journal on Imaging Sciences
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
De-noising by soft-thresholding
IEEE Transactions on Information Theory
An EM algorithm for wavelet-based image restoration
IEEE Transactions on Image Processing
Image decomposition via the combination of sparse representations and a variational approach
IEEE Transactions on Image Processing
Fast Algorithms for Image Reconstruction with Application to Partially Parallel MR Imaging
SIAM Journal on Imaging Sciences
Hybrid regularization image deblurring in the presence of impulsive noise
Journal of Visual Communication and Image Representation
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Wavelet frame based models for image restoration have been extensively studied for the past decade (Chan et al. in SIAM J. Sci. Comput. 24(4):1408---1432, 2003; Cai et al. in Multiscale Model. Simul. 8(2):337---369, 2009; Elad et al. in Appl. Comput. Harmon. Anal. 19(3):340---358, 2005; Starck et al. in IEEE Trans. Image Process. 14(10):1570---1582, 2005; Shen in Proceedings of the international congress of mathematicians, vol. 4, pp. 2834---2863, 2010; Dong and Shen in IAS lecture notes series, Summer program on "The mathematics of image processing", Park City Mathematics Institute, 2010). The success of wavelet frames in image restoration is mainly due to their capability of sparsely approximating piecewise smooth functions like images. Most of the wavelet frame based models designed in the past are based on the penalization of the 驴 1 norm of wavelet frame coefficients, which, under certain conditions, is the right choice, as supported by theories of compressed sensing (Candes et al. in Appl. Comput. Harmon. Anal., 2010; Candes et al. in IEEE Trans. Inf. Theory 52(2):489---509, 2006; Donoho in IEEE Trans. Inf. Theory 52:1289---1306, 2006). However, the assumptions of compressed sensing may not be satisfied in practice (e.g. for image deblurring and CT image reconstruction). Recently in Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), the authors propose to penalize the l 0 "norm" of the wavelet frame coefficients instead, and they have demonstrated significant improvements of their method over some commonly used l 1 minimization models in terms of quality of the recovered images. In this paper, we propose a new algorithm, called the mean doubly augmented Lagrangian (MDAL) method, for l 0 minimizations based on the classical doubly augmented Lagrangian (DAL) method (Rockafellar in Math. Oper. Res. 97---116, 1976). Our numerical experiments show that the proposed MDAL method is not only more efficient than the method proposed by Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), but can also generate recovered images with even higher quality. This study reassures the feasibility of using the l 0 "norm" for image restoration problems.