Well-posedness for a class of fourth order diffusions for image processing

Guidotti, Patrick; Longo, Kate

doi:10.1007/s00030-011-0101-x

Cited by 14 publications

(16 citation statements)

References 28 publications

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“…For small ε, (4.1) offers at best a modest improvement to the performance of (2.7), but unlike (2.7), it can be shown to possess a unique short time solution [19]. While the Laplacian | u| in (2.7) is effective at detecting edges, the gradient |∇u| is generally a better edge detector, especially on images with very sharp edges.…”

Section: Proposed Modelsmentioning

confidence: 99%

“…This allows solutions to transiently relax to smoother states where sharp but smooth gradients are preserved as opposed to being further sharpened into jumps, thus avoiding the blocky effects associated with (1.1). The fourth order (2.7) exhibits similar analytical behavior to (1.1), and is likely also ill-posed [19]. One could consider instead the equation…”

Section: Introductionmentioning

confidence: 99%

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Two Enhanced Fourth Order Diffusion Models for Image Denoising

Guidotti

Longo

2011

J Math Imaging Vis

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This paper presents two new higher order diffusion models for removing noise from images. The models employ fractional derivatives and are modifications of an existing fourth order partial differential equation (PDE) model which was developed by You and Kaveh as a generalization of the well-known second order Perona-Malik equation. The modifications serve to cure the ill-posedness of the You-Kaveh model without sacrificing performance. Also proposed in this paper is a simple smoothing technique which can be used in numerical experiments to improve denoising and reduce processing time. Numerical experiments are shown for comparison.

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Section: Proposed Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Two Enhanced Fourth Order Diffusion Models for Image Denoising

Guidotti

Longo

2011

J Math Imaging Vis

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“…were also investigated by Guidotti and Longo [8] recently. Using the theory of maximal regularity, they proved the local existence of solutions u ∈ W 1,p (0, T ; L p (Ω)) to the above two equations with periodic boundary conditions.…”

Section: Introductionmentioning

confidence: 97%

Well-posedness for a fourth order nonlinear equation related to image processing

Min

Yang

2014

Nonlinear Analysis: Real World Applications

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“…These authors proved the existence of the strong solution of Wei’s fourth order equation. Most recently, Guidotti and Longo have shown the existence of the solution to a class of fourth order diffusion operators [31] and proposed two enhanced fourth order diffusion models for image denoising [30]. Due to the stiffness of high-order nonlinear PDEs, computational techniques for solving higher order geometric PDEs are important issues.…”

Section: Introductionmentioning

confidence: 99%

High-order fractional partial differential equation transform for molecular surface construction

Chen

Wei

2012

Computational and Mathematical Biophysics

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Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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Well-posedness for a class of fourth order diffusions for image processing

Cited by 14 publications

References 28 publications

Two Enhanced Fourth Order Diffusion Models for Image Denoising

Two Enhanced Fourth Order Diffusion Models for Image Denoising

Well-posedness for a fourth order nonlinear equation related to image processing

High-order fractional partial differential equation transform for molecular surface construction

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