$\ell _0$ Minimization for wavelet frame based image restoration

Zhang, Yong; Dong, Bin; Lu, Zhaosong

doi:10.1090/s0025-5718-2012-02631-7

Cited by 93 publications

(104 citation statements)

References 52 publications

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“…Recently, l p quasi-norm (0 ≤ p ≤ 1) regularization was further investigated to recover the image with more sharped edges. The authors in [50] proposed to use the l 0 "norm" instead of the l 1 norm in the analysis model:…”

Section: The Single L 1 and L 0 Minimization Modelmentioning

confidence: 99%

“…An algorithm called PD method was proposed to solve the above l 0 minimization problem in [50]. Recently, a more efficient algorithm, called MDAL method is developed for solving the same problem in literature [17].…”

Section: The Single L 1 and L 0 Minimization Modelmentioning

confidence: 99%

“…Therefore, the l 1 norm based models often achieves suboptimal performance. Recently, Zhang et.al [50] proposed to penalized the l 0 "norm" of wavelet frame coefficients instead, and they developed an algorithm called penalty decomposition (PD) to solve the analysis approach with l 0 minimization. However, due to the non-convexity of l 0 "norm", the computational cost of PD method is a little bit high.…”

Section: Introductionmentioning

confidence: 99%

“…Most wavelet frame based models exploit the l 1 norm of frame coefficients for a sparsity constraint in the past. The authors in [50,17] proposed an l 0 minimization model, where the l 0 norm of wavelet frame coefficients is penalized instead, and have demonstrated that significant improvements can be achieved compared to the commonly used l 1 minimization model. Very recently, the authors in [11] proposed l 0 -l 2 minimization model, where the nonlocal prior of frame coefficients is incorporated.…”

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confidence: 99%

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Wavelet frame-based image restoration using sparsity, nonlocal, and support prior of frame coefficients

Wang

Xiang

2017

Vis Comput

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The wavelet frame systems have been widely investigated and applied for image restoration and many other image processing problems over the past decades, attributing to their good capability of sparsely approximating piece-wise smooth functions such as images. Most wavelet frame based models exploit the l 1 norm of frame coefficients for a sparsity constraint in the past. The authors in [50,17] proposed an l 0 minimization model, where the l 0 norm of wavelet frame coefficients is penalized instead, and have demonstrated that significant improvements can be achieved compared to the commonly used l 1 minimization model. Very recently, the authors in [11] proposed l 0 -l 2 minimization model, where the nonlocal prior of frame coefficients is incorporated. This model proved to outperform the single l 0 minimization based model in terms of better recovered image quality. In this paper, we propose a truncated l 0 -l 2 minimization model which combines sparsity, nonlocal and support prior of the frame coefficients. The extensive experiments have shown that the recovery results from the proposed regularization method performs better than existing state-of-the-art wavelet frame based methods, in terms of edge enhancement and texture preserving performance.

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Section: The Single L 1 and L 0 Minimization Modelmentioning

confidence: 99%

Section: The Single L 1 and L 0 Minimization Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Wavelet frame-based image restoration using sparsity, nonlocal, and support prior of frame coefficients

Wang

Xiang

2017

Vis Comput

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“…For the resulting 0 minimization problems, typically iterative thresholding methods are applied; see [5,6] for related problems without constraints as well as [26,1] for related minimization problems with constraints. Another approach to 0 minimization problems are the penalty decomposition methods of [45,46,75]. They deal with more general data terms and constraints by a two-stage iterative method.…”

mentioning

confidence: 99%

Fast Partitioning of Vector-Valued Images

Storath¹,

Weinmann²

2014

SIAM J. Imaging Sci.

138

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Abstract. We propose a fast splitting approach to the classical variational formulation of the image partitioning problem, which is frequently referred to as the Potts or piecewise constant Mumford-Shah model. For vector-valued images, our approach is significantly faster than the methods based on graph cuts and convex relaxations of the Potts model which are presently the state-of-the-art. The computational costs of our algorithm only grow linearly with the dimension of the data space which contrasts the exponential growth of the state-of-the-art methods. This allows us to process images with high-dimensional codomains such as multispectral images. Our approach produces results of a quality comparable with that of graph cuts and the convex relaxation strategies, and we do not need an a priori discretization of the label space. Furthermore, the number of partitions has almost no influence on the computational costs, which makes our algorithm also suitable for the reconstruction of piecewise constant (color or vectorial) images.

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Tight Graph Framelets for Sparse Diffusion MRI q-Space Representation

Yap¹,

Dong

Zhang

et al. 2016

Lecture Notes in Computer Science

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In diffusion MRI, the outcome of estimation problems can often be improved by taking into account the correlation of diffusion-weighted images scanned with neighboring wavevectors in q-space. For this purpose, we propose in this paper to employ tight wavelet frames constructed on non-flat domains for multi-scale sparse representation of diffusion signals. This representation is well suited for signals sampled regularly or irregularly, such as on a grid or on multiple shells, in q-space. Using spectral graph theory, the frames are constructed based on quasi-affine systems (i.e., generalized dilations and shifts of a finite collection of wavelet functions) defined on graphs, which can be seen as a discrete representation of manifolds. The associated wavelet analysis and synthesis transforms can be computed efficiently and accurately without the need for explicit eigen-decomposition of the graph Laplacian, allowing scalability to very large problems. We demonstrate the effectiveness of this representation, generated using what we call tight graph framelets, in two specific applications: denoising and super-resolution in q-space using ℓ0 regularization. The associated optimization problem involves only thresholding and solving a trivial inverse problem in an iterative manner. The effectiveness of graph framelets is confirmed via evaluation using synthetic data with noncentral chi noise and real data with repeated scans.

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$\ell _0$ Minimization for wavelet frame based image restoration

Cited by 93 publications

References 52 publications

Wavelet frame-based image restoration using sparsity, nonlocal, and support prior of frame coefficients

Wavelet frame-based image restoration using sparsity, nonlocal, and support prior of frame coefficients

Fast Partitioning of Vector-Valued Images

Tight Graph Framelets for Sparse Diffusion MRI q-Space Representation

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