Oscillating Patterns in Image Processing and Nonlinear Evolution Equations

Meyer, Yves

doi:10.1090/ulect/022

Cited by 867 publications

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“…As Meyer [19] analyzed, the   -norm applied to the residual    in the TV model (4) On the other hand, Osher et al [23] recently suggested an iterative refinement procedure:…”

Section: 2mentioning

confidence: 99%

“…Thus image restoration has occupied a peculiar position in IP, computer graphics, and their applications [2,13,19,20,21,24,27] .…”

Section: ⅰ Introductionmentioning

confidence: 99%

“…Thus the reduction of nonphysical dissipation becomes an interesting problem in image restoration, requiring challenges and innovative ideas. Although advanced models have been recently suggested for the preservation of fine structures [19,23] , more effective strategies have yet to be developed. In this article, we will first analyze sources of nonphysical dissipation for popular PDE-based restoration models.…”

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confidence: 99%

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Equalized Net Diffusion (END) for the Preservation of Fine Structures in PDE-based Image Restoration

Cha

Kim

2013

The Journal of Korean Institute of Communications and Informati

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The article is concerned with a mathematical modeling which can improve performances of PDE-based restoration models. Most PDE-based restoration models tend to lose fine structures due to certain degrees of nonphysical dissipation. Sources of such an undesirable dissipation are analyzed for total variation-based restoration models. Based on the analysis, the so-called equalized net diffusion (END) modeling is suggested in order for PDE-based restoration models to significantly reduce nonphysical dissipation. It has been numerically verified that the END-incorporated models can preserve and recover fine structures satisfactorily, outperforming the basic models for both quality and efficiency. Various numerical examples are shown to demonstrate effectiveness of the END modeling.Key Words : Fine structures, nonphysical dissipation, total variation (TV) model, non-convex (NC) model, equalized net diffusion (END). [2,13,19,20,21,24,27] .There have been various partial differential equation (PDE)-based restoration models such as the Perona-Malik model [25] , the total variation (TV) model [18,26] , and color restoration models [3,10,15,17,28] . These PDE-based models have been extensively studied to answer fundamental questions in image restoration and have allowed researchers and practitioners not only to introduce new mathematical models but also to improve traditional algorithms [1,4,9,22,30] .However, most PDE-based restoration models and their numerical realizations show a common drawback: loss of fine structures. Such an undesirable loss is due to an excessive numerical dissipation introduced particularly on regions where the image content changes rapidly such as on edges and textures. Thus the reduction of nonphysical dissipation becomes an interesting problem in image restoration, requiring challenges and innovative ideas. Although advanced models have been recently suggested for the preservation of fine structures [19,23] , more effective strategies have yet to be developed. In this article, we will first analyze sources of nonphysical dissipation for popular PDE-based restoration models. Based on the analysis, we will study the so-called equalized net diffusion (END) modeling for PDE-based restoration models, first introduced in [14] . Here we will study mathematical 논문 / Equalized Net Diffusion (END) for the Preservation of Fine Structures in PDE-based Image Restoration 999properties of the END in detail and suggest a heuristic method for parameter choices. It has been numerically verified that with the new parameters, the END modeling can reduce nonphysical dissipation significantly and more effectively for most of natural images we have tested.The article is organized as follows. In the next section, we review TV-based models along with recent studies for the reduction of nonphysical dissipation. Section III analyzes sources of nonphysical dissipation for PDE-based models: nonphysical dissipation occurs more excessively at pixels where the diffusion term evaluates larger in modulus. In Section IV...

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Section: 2mentioning

confidence: 99%

“…Thus image restoration has occupied a peculiar position in IP, computer graphics, and their applications [2,13,19,20,21,24,27] .…”

Section: ⅰ Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Equalized Net Diffusion (END) for the Preservation of Fine Structures in PDE-based Image Restoration

Cha

Kim

2013

The Journal of Korean Institute of Communications and Informati

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“…In 2001, Y. Meyer in [1] has proposed an extension of the famous image restoration model proposed by Rudin, Osher and Fatemi [2], based on the BV 3 space:…”

Section: Decomposition Modelsmentioning

confidence: 99%

“…The first goal was image restoration and denoising; but following the ideas of Yes Meyer [1], in the Total Variation minimization framework of L. Rudin, S. Osher and E. Fatemi [2], image decomposition into geometrical and oscillatory components grants access to the textured part of an image, rejected by the previous model. We only cite, among many others, most recent works which we appear like most relevant and useful papers.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Texture Extraction and Video Denoising

Lugiez

Ménard

El-Hamidi

2009

Advanced Concepts for Intelligent Vision Systems

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Abstract. According to recent works, introduced by Y.Meyer [1] the decomposition models based on Total Variation (TV) appear as a very good way to extract texture from image sequences. Indeed, videos show up characteristic variations along the temporal dimension which can be catched in the decomposition framework. However, there are very few works in literature which deal with spatio-temporal decompositions. Thus, we devote this paper to spatio-temporal extension of the spatial color decomposition model. We provide a relevant method to accurately catch Dynamic Textures (DT) present in videos. Moreover, we obtain the spatio-temporal regularized part (the geometrical component), and we distinctly separate the highly oscillatory variations, (the noise). Furthermore, we present some elements of comparison between several models in denoising purpose.

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Sobolev gradient preconditioning for image‐processing PDEs

Richardson

2006

Commun. Numer. Meth. Engng.

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SUMMARYThe article explores the relationship between Sobolev gradients and H −1 mixed methods for a variety of partial differential equations (PDEs) from image processing. A first-order system least-squares problem is used to introduce the method and compare the Euclidean with the Sobolev gradient. The standard two-term decomposition of an image as f = u + v with u ∈ H 1 and v ∈ L 2 = H 0 yields a second-order linear PDE, while minimizing other L p norms give nonlinear PDEs. Finally, a three-term decomposition f = u + v + w with u ∈ H 1 , v ∈ H −1 , w ∈ H 0 requires the solution of a fourth-order system with the biharmonic operator.

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Oscillating Patterns in Image Processing and Nonlinear Evolution Equations

Cited by 867 publications

References 0 publications

Equalized Net Diffusion (END) for the Preservation of Fine Structures in PDE-based Image Restoration

Equalized Net Diffusion (END) for the Preservation of Fine Structures in PDE-based Image Restoration

Dynamic Texture Extraction and Video Denoising

Sobolev gradient preconditioning for image‐processing PDEs

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