scholarly journals An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

2004 ◽  
Vol 57 (11) ◽  
pp. 1413-1457 ◽  
Author(s):  
I. Daubechies ◽  
M. Defrise ◽  
C. De Mol
Keyword(s):  
Inverse Problems ◽  
Linear Inverse Problems ◽  
Iterative Thresholding ◽  
Iterative Thresholding Algorithm ◽  
Sparsity Constraint ◽  
Thresholding Algorithm

Convergence of ℓ2/3 Regularization for Sparse Signal Recovery

2015 ◽  
Vol 32 (04) ◽  
pp. 1550023 ◽  
Author(s):  
Lu Liu ◽  
Di-Rong Chen
Keyword(s):  
Fixed Point ◽  
Large Scale ◽  
Sparse Signal ◽  
Sparse Signal Recovery ◽  
Signal Recovery ◽  
Lipschitz Optimization ◽  
Iterative Thresholding ◽  
Point Representation ◽  
Iterative Thresholding Algorithm ◽  
Thresholding Algorithm

In this paper, we consider the problem of finding the sparsest solution to underdetermined linear systems. Unlike the literatures which use the ℓ1 regularization to approximate the original problem, we consider the ℓ2/3 regularization which leads to a better approximation but a nonconvex, nonsmooth, and non-Lipschitz optimization problem. Through developing a fixed point representation theory associated with the two thirds thresholding operator for ℓ2/3 regularization solutions, we propose a fixed point iterative thresholding algorithm based on two thirds norm for solving the k-sparsity problems. Relying on the restricted isometry property, we provide subsequentional convergence guarantee for this fixed point iterative thresholding algorithm on recovering a sparse signal. By discussing the preferred regularization parameters and studying the phase diagram, we get an adequate and efficient algorithm for the high-dimensional sparse signal recovery. Finally, comparing with the existing algorithms, such as the standard ℓ1 minimization, the iterative reweighted ℓ2 minimization, the iterative reweighted ℓ1 minimization, and iterative Half thresholding algorithm, we display the results of the experiment which indicate that the two thirds norm fixed point iterative thresholding algorithm applied to sparse signal recovery and large scale imageries from noisy measurements can be accepted as an effective solver for ℓ2/3 regularization.

IMAGE INPAINTING FROM PARTIAL NOISY DATA BY DIRECTIONAL COMPLEX TIGHT FRAMELETS

2017 ◽  
Vol 58 (3-4) ◽  
pp. 247-255 ◽  
Author(s):  
YI SHEN ◽  
BIN HAN ◽  
ELENA BRAVERMAN
Keyword(s):  
Image Inpainting ◽  
Tight Frame ◽  
The Other ◽  
Tight Frames ◽  
Discrete Cosine Transforms ◽  
Iterative Thresholding ◽  
Iterative Thresholding Algorithm ◽  
Good For ◽  
Thresholding Algorithm ◽  
Frame Systems

Image inpainting methods recover true images from partial noisy observations. Natural images usually have two layers consisting of cartoons and textures. Methods using simultaneous cartoon and texture inpainting are popular in the literature by using two combined tight frames: one (often built from wavelets, curvelets or shearlets) provides sparse representations for cartoons and the other (often built from discrete cosine transforms) offers sparse approximation for textures. Inspired by the recent development on directional tensor product complex tight framelets ($\text{TP}\text{-}\mathbb{C}\text{TF}$s) and their impressive performance for the image denoising problem, we propose an iterative thresholding algorithm using tight frames derived from $\text{TP}\text{-}\mathbb{C}\text{TF}$s for the image inpainting problem. The tight frame $\text{TP}\text{-}\mathbb{C}\text{TF}_{6}$ contains two classes of framelets; one is good for cartoons and the other is good for textures. Therefore, it can handle both the cartoons and the textures well. For the image inpainting problem with additive zero-mean independent and identically distributed Gaussian noise, our proposed algorithm does not require us to tune parameters manually for reasonably good performance. Experimental results show that our proposed algorithm performs comparatively better than several well-known frame systems for the image inpainting problem.

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