L1-L2 Optimization in Signal and Image Processing

@article{Zibulevsky2010L1L2OI,
  title={L1-L2 Optimization in Signal and Image Processing},
  author={Michael Zibulevsky and Michael Elad},
  journal={IEEE Signal Processing Magazine},
  year={2010},
  volume={27},
  pages={76-88}
}
Sparse, redundant representations offer a powerful emerging model for signals. This model approximates a data source as a linear combination of few atoms from a prespecified and over-complete dictionary. Often such models are fit to data by solving mixed ¿1-¿2 convex optimization problems. Iterative-shrinkage algorithms constitute a new family of highly effective numerical methods for handling these problems, surpassing traditional optimization techniques. In this article, we give a broad view… 
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References

SHOWING 1-10 OF 59 REFERENCES
Sparse Representation-based Image Deconvolution by iterative Thresholding
TLDR
Recent advances in fast solution algorithms of deconvolution problems, i.e. Stagewise Iterative Thresholding, are exploited to solve the optimization problem and provide fast and good image recovery results.
Block coordinate relaxation methods for nonparametric signal denoising with wavelet dictionaries
TLDR
It is shown that the B CR algorithm is globally convergent, and empirically, the BCR algorithm is faster than the IP algorithm for a variety of signal denoising problems.
A bound optimization approach to wavelet-based image deconvolution
TLDR
An algorithm based on the bound optimization approach is proposed; this approach allows deriving EM-type algorithms without using the concept of missing/hidden data and has provable monotonicity both with orthogonal or redundant wavelet transforms.
Signal Recovery by Proximal Forward-Backward Splitting
We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation…
Iterative Thresholding for Sparse Approximations
TLDR
This paper studies two iterative algorithms that are minimising the cost functions of interest and adapts the algorithms and shows on one example that this adaptation can be used to achieve results that lie between those obtained with Matching Pursuit and those found with Orthogonal Matching pursuit, while retaining the computational complexity of the Matching pursuit algorithm.
Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization
  • D. Donoho, Michael Elad
  • Computer Science
    Proceedings of the National Academy of Sciences of the United States of America
  • 2003
TLDR
This article obtains parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems, and sketches three applications: separating linear features from planar ones in 3D data, noncooperative multiuser encoding, and identification of over-complete independent component models.
Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization
Sparse MRI: The application of compressed sensing for rapid MR imaging
TLDR
Practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference and demonstrate improved spatial resolution and accelerated acquisition for multislice fast spin‐echo brain imaging and 3D contrast enhanced angiography.
Majorization–Minimization Algorithms for Wavelet-Based Image Restoration
TLDR
Several new results are proved which strongly support the claim that the SI does not compromise the usefulness of this class of algorithms, and introduce a new algorithm, resulting from using bounds for nonconvex regularizers, which confirms the superior performance of this method, when compared to the one based on quadratic majorization.
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
TLDR
The aim of this paper is to introduce a few key notions and applications connected to sparsity, targeting newcomers interested in either the mathematical aspects of this area or its applications.
...
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