Computing Sparse Representation in a Highly Coherent Dictionary Based on Difference of $$L_1$$ L 1 and $$L_2$$ L 2

Lou, Yifei; Yin, Pengbin; He, Qi; Xin, Jack

doi:10.1007/s10915-014-9930-1

Cited by 152 publications

(104 citation statements)

References 21 publications

Supporting

Mentioning

100

Contrasting

Unclassified

Order By: Relevance

“…The former is recently proposed in [22,40] as an alternative to L 1 for CS, and the latter is often used in statistics and machine learning [33,41]. Numerical simulations show that L 1 minimization often fails when MSF < 1, in which case we demonstrate that both L 1−2 and CL 1 outperform the classical L 1 method.…”

Section: Our Contributionsmentioning

confidence: 73%

See 1 more Smart Citation

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

2016

Self Cite

View full text Add to dashboard Cite

We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in (1, 2) of the Rayleigh length (physical resolution limit), L 1 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the L 1 certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two L 1 based nonconvex penalties, the difference of L 1 and L 2 norms (L 1−2 ) and capped L 1 (CL 1 ), subject to the measurement constraints. In one and two dimensional numerical SR examples, the local optimal solutions from difference of convex function algorithms outperform the global L 1 solutions near or below Rayleigh length scales either in the accuracy of ground truth recovery or in finding a sparse solution satisfying the constraints more accurately.

show abstract

Section: Our Contributionsmentioning

confidence: 73%

“…This method gives better results than the classical L 1 approaches in the RIP regime and/or incoherent scenario, but it does not work so well for highly coherent CS, as observed in [21,22,40].…”

Section: P Via Irlsmentioning

confidence: 94%

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

2016

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, since the ℓ 0 -regularized problem is computational NP-hard, its ℓ 1 -norm relaxed version is usually considered in practice. Recently ℓ 1−2 regularization has been proposed (Esser et al, 2013; Lou et al, 2014; Yin et al, 2014), and has been shown to provide a sparser result than the widely used ℓ 1 -norm regularization.…”

Section: Methodsmentioning

confidence: 99%

“…which has shown potential in image processing and compressive sensing reconstruction (Lou et al, 2014; Yin et al, 2015) in terms of sparsity and fast convergence. It promotes sparsity of an image, and achieves the smallest value when only one voxel in the image is non-zero.…”

Section: Methodsmentioning

confidence: 99%

“…vTGV enhances the smoothness of the brain image and reconstructs the spatial distribution of the current density more precisely. Meanwhile, motivated by the performance of the ℓ 1−2 regularization in compressive sensing reconstruction and other image processing problems (Esser et al, 2013; Lou et al, 2014; Yin et al, 2015), we incorporate the ℓ 1−2 regularization into the objective function. Numerical experiments show that ℓ 1−2 regularization provides faster convergence and yields sparser source image than the ℓ 1 -norm regularization.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

s-SMOOTH: Sparsity and Smoothness Enhanced EEG Brain Tomography

et al. 2016

View full text Add to dashboard Cite

EEG source imaging enables us to reconstruct current density in the brain from the electrical measurements with excellent temporal resolution (~ ms). The corresponding EEG inverse problem is an ill-posed one that has infinitely many solutions. This is due to the fact that the number of EEG sensors is usually much smaller than that of the potential dipole locations, as well as noise contamination in the recorded signals. To obtain a unique solution, regularizations can be incorporated to impose additional constraints on the solution. An appropriate choice of regularization is critically important for the reconstruction accuracy of a brain image. In this paper, we propose a novel Sparsity and SMOOthness enhanced brain TomograpHy (s-SMOOTH) method to improve the reconstruction accuracy by integrating two recently proposed regularization techniques: Total Generalized Variation (TGV) regularization and ℓ1−2 regularization. TGV is able to preserve the source edge and recover the spatial distribution of the source intensity with high accuracy. Compared to the relevant total variation (TV) regularization, TGV enhances the smoothness of the image and reduces staircasing artifacts. The traditional TGV defined on a 2D image has been widely used in the image processing field. In order to handle 3D EEG source images, we propose a voxel-based Total Generalized Variation (vTGV) regularization that extends the definition of second-order TGV from 2D planar images to 3D irregular surfaces such as cortex surface. In addition, the ℓ1−2 regularization is utilized to promote sparsity on the current density itself. We demonstrate that ℓ1−2 regularization is able to enhance sparsity and accelerate computations than ℓ1 regularization. The proposed model is solved by an efficient and robust algorithm based on the difference of convex functions algorithm (DCA) and the alternating direction method of multipliers (ADMM). Numerical experiments using synthetic data demonstrate the advantages of the proposed method over other state-of-the-art methods in terms of total reconstruction accuracy, localization accuracy and focalization degree. The application to the source localization of event-related potential data further demonstrates the performance of the proposed method in real-world scenarios.

show abstract

Sparse Unmixing for Hyperspectral Image with Nonlocal Low-Rank Prior

Zheng

Sun

2019

Intelligence Science and Big Data Engineering. Visual Data Engineering

View full text Add to dashboard Cite

Hyperspectral unmixing is a key preprocessing technique for hyperspectral image analysis. To further improve the unmixing performance, in this paper, a nonlocal low-rank prior associated with spatial smoothness and spectral collaborative sparsity are integrated together for unmixing the hyperspectral data. The proposed method is based on a fact that hyperspectral images have self-similarity in nonlocal sense and smoothness in local sense. To explore the spatial self-similarity, nonlocal cubic patches are grouped together to compose a low-rank matrix. Then, based on the linear mixed model framework, the nuclear norm is constrained to the abundance matrix of these similar patches to enforce low-rank property. In addition, the local spatial information and spectral characteristic are also taken into account by introducing TV regularization and collaborative sparse terms, respectively. Finally, the results of the experiments on two simulated data sets and two real data sets show that the proposed algorithm produces better performance than other state-of-the-art algorithms.

show abstract

Computing Sparse Representation in a Highly Coherent Dictionary Based on Difference of $$L_1$$ L 1 and $$L_2$$ L 2

Cited by 152 publications

References 21 publications

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

s-SMOOTH: Sparsity and Smoothness Enhanced EEG Brain Tomography

Sparse Unmixing for Hyperspectral Image with Nonlocal Low-Rank Prior

Contact Info

Product

Resources

About