Correspondence between frame shrinkage and high-order nonlinear diffusion

Jiang, Qingtang

doi:10.1016/j.apnum.2011.10.002

Cited by 6 publications

(4 citation statements)

References 40 publications

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“…for the image deblurring and deconvolution requires a delicate analysis of the boundary conditions [46] and is treated elsewhere. Other avenues of exploration are treating higher order models [54,28], multi grid [47] and FEM [29] based schemes and their convergence analysis.…”

Section: Comparison Resultsmentioning

confidence: 99%

On convergent finite difference schemes for variational–PDE-based image processing

Prasath

Moreno

2017

Comp. Appl. Math.

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We study an adaptive anisotropic Huber functional based image restoration scheme. By using a combination of L2-L1 regularization functions, an adaptive Huber functional based energy minimization model provides denoising with edge preservation in noisy digital images. We study a convergent finite difference scheme based on continuous piecewise linear functions and use a variable splitting scheme, namely the Split Bregman [25], to obtain the discrete minimizer. Experimental results are given in image denoising and comparison with additive operator splitting, dual fixed point, and projected gradient schemes illustrate that the best convergence rates are obtained for our algorithm.

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Section: Comparison Resultsmentioning

confidence: 99%

On convergent finite difference schemes for variational–PDE-based image processing

Prasath

Moreno

2017

Comp. Appl. Math.

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“…Note that the first approximation of (2.37) was observed earlier in [8], while both approximations of (2.37) in the one-dimensional setting were used in [46]. Using the observation (2.37), we can discretize (2.36) as…”

Section: Wavelet Frame Filter Banksmentioning

confidence: 99%

“…Motivated by [8] and [46], this paper establishes a fundamental connection between wavelet frame shrinkage and nonlinear evolution PDEs, provides interpretations and analytical studies of this connection, and proposes new algorithms for image restoration based on the new understandings. This connection automatically gives the wavelet frame based approach a geometric explanation through nonlinear PDEs, and, at the same time, it equips the PDE based approach with a time frequency analysis by the nature of the two approaches.…”

mentioning

confidence: 99%

“…Some earlier results in [64,65,66] showed the correspondence between Haar wavelet shrinkage and second-order nonlinear diffusions. This work was recently generalized to wavelet frame shrinkage and higher order nonlinear diffusions for the one-dimensional case in [46]. However, it is not clear how two-dimensional (2D) wavelet frame shrinkage can be related to nonlinear PDEs.…”

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confidence: 99%

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Image Restoration: Wavelet Frame Shrinkage, Nonlinear Evolution PDEs, and Beyond

Dong¹,

Jiang²,

Shen³

2017

Multiscale Model. Simul.

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In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems; the partial differential equation (PDE) based approach (e.g., the total variation model [L. ], etc.) and wavelet frame based approach are some successful examples. These approaches were developed through different paths and generally provided understanding from different angles of the same problem. As shown in numerical simulations, implementations of the wavelet frame based approach and the PDE based approach quite often end up solving a similar numerical problem with similar numerical behaviors, even though different approaches have advantages in different applications. Since wavelet frame based and PDE based approaches have all been modeling the same types of problems with success, it is natural to ask whether the wavelet frame based approach is fundamentally connected with the PDE based approach when we trace them all the way back to their roots. A fundamental connection of a wavelet frame based approach with a total variation model and its generalizations was established in [J. Cai, B. Dong, S. Osher, and Z. Shen, J. Amer. Math. Soc., 25 (2012), pp. 1033-1089]. This connection gives the wavelet frame based approach a geometric explanation and, at the same time, it equips a PDE based approach with a time frequency analysis. Cai et al. showed that a special type of wavelet frame model using generic wavelet frame systems can be regarded as an approximation of a generic variational model (with the total variation model as a special case) in the discrete setting. A systematic convergence analysis, as the resolution of the image goes to infinity, which is the key step in linking the two approaches, is also given in Cai et al. Motivated by Cai et al. and [Q. Jiang, Appl. Numer. Math., 62 (2012), pp. 51-66], this paper establishes a fundamental connection between the wavelet frame based approach and nonlinear evolution PDEs, provides interpretations and analytical studies of such connections, and proposes new algorithms for image restoration based on the new understandings. Together with the results in [J. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089], we now have a better picture of how the wavelet frame based approach can be used to interpret the general PDE based approach (e.g., the variational models or nonlinear evolution PDEs) and can be used as a new and useful tool in numerical analysis to discretize and solve various variational and PDE models. To be more precise, we shall establish the following: (1) The connections between wavelet frame shrinkage and nonlinear evolution PDEs provide new and inspiring interpretations of both approaches that enable us to derive new PDE models and (better) wavelet frame shrinkage algorithms for image restoration.(2) A generic nonlinear evolution PDE (of parabolic or hyperbolic type) can be approximated by wavelet frame shrinkage with properly chosen wavelet frame systems and carefully designed shrinkage functions. (3) The main idea of this work i...

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Correspondence Between Multiscale Frame Shrinkage and High-Order Nonlinear Diffusion

Wang

Huang

Meng

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Correspondence between frame shrinkage and high-order nonlinear diffusion

Cited by 6 publications

References 40 publications

On convergent finite difference schemes for variational–PDE-based image processing

On convergent finite difference schemes for variational–PDE-based image processing

Image Restoration: Wavelet Frame Shrinkage, Nonlinear Evolution PDEs, and Beyond

Correspondence Between Multiscale Frame Shrinkage and High-Order Nonlinear Diffusion

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