Wavelet Algorithms for High-Resolution Image Reconstruction
SIAM Journal on Scientific Computing
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Deconvolution: a wavelet frame approach
Numerische Mathematik
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
Exact Regularization of Convex Programs
SIAM Journal on Optimization
Restoration of Chopped and Nodded Images by Framelets
SIAM Journal on Scientific Computing
Inpainting and Zooming Using Sparse Representations
The Computer Journal
Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
SIAM Journal on Optimization
Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing
SIAM Journal on Imaging Sciences
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
SIAM Journal on Imaging Sciences
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
SIAM Journal on Imaging Sciences
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 Singular Value Thresholding Algorithm for Matrix Completion
SIAM Journal on Optimization
Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction
SIAM Journal on Imaging Sciences
Analysis and Generalizations of the Linearized Bregman Method
SIAM Journal on Imaging Sciences
NESTA: A Fast and Accurate First-Order Method for Sparse Recovery
SIAM Journal on Imaging Sciences
An EM algorithm for wavelet-based image restoration
IEEE Transactions on Image Processing
Adaptive Multiresolution Analysis Structures and Shearlet Systems
SIAM Journal on Numerical Analysis
Robust Video Restoration by Joint Sparse and Low Rank Matrix Approximation
SIAM Journal on Imaging Sciences
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Frame-based image restoration by using the balanced approach has been developed over the last decade. Many recently developed algorithms for image restoration can be viewed as an acceleration of the proximal forward-backward splitting algorithm. Accelerated proximal gradient (APG) algorithms studied by Nesterov, Nemirovski, and others have been demonstrated to be efficient in solving various regularized convex optimization problems arising in compressed sensing, machine learning, and control. In this paper, we adapt the APG algorithm to solve the $\ell_1$-regularized linear least squares problem in the balanced approach in frame-based image restoration. This algorithm terminates in $O(1/\sqrt{\epsilon})$ iterations with an $\epsilon$-optimal solution, and we demonstrate that this single algorithmic framework can universally handle several image restoration problems, such as image deblurring, denoising, inpainting, and cartoon-texture decomposition. Our numerical results suggest that the APG algorithms are efficient and robust in solving large-scale image restoration problems. The algorithms we implemented are able to restore $512\times512$ images in various image restoration problems in less than 50 seconds on a modest PC. We also compare the numerical performance of our proposed algorithms applied to image restoration problems by using one frame-based system with that by using cartoon and texture systems for image deblurring, denoising, and inpainting.