Two Enhanced Fourth Order Diffusion Models for Image Denoising

Guidotti, Patrick; Longo, Kate

doi:10.1007/s10851-010-0256-9

Cited by 42 publications

(45 citation statements)

References 32 publications

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“…The authors propose in [21] two modifications of 5 in which the edge detector is replaced with the fractional gradient used in 3 and the fractional Laplacian operator used in 4, respectively. These two new equations read…”

Section: P Guidotti and K Longo Nodeamentioning

confidence: 99%

“…Denoising experiments performed with some of the above equations (see, in particular, [15,[21][22][23]28,35]) show that fourth order models can avoid the staircasing and cartoonish effects of second order models while still removing noise and preserving edges. However, little analysis has been conducted on these fourth order equations.…”

Section: Well-posednessmentioning

confidence: 99%

“…This paper addresses the well-posedness of two new denoising models proposed in [21]. While the analysis is focused on equations 10 and 11, it can be easily modified to apply to every other equation listed in the previous section, with the exception of 5.…”

Section: Well-posednessmentioning

confidence: 99%

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

Guidotti

Longo

2011

Nonlinear Differ. Equ. Appl.

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Abstract.A number of image denoising models based on higher order parabolic partial differential equations (PDEs) have been proposed in an effort to overcome some of the problems attendant to second order methods such as the famous Perona-Malik model. However, there is little analysis of these equations to be found in the literature. In this paper, methods of maximal regularity are used to prove the existence of unique local solutions to a class of fourth order PDEs for noise removal. The proof is laid out explicitly for two newly proposed fourth order models, and an outline is given for how to apply the techniques to other proposed models. Mathematics Subject Classification (2000). 35K35, 35K59, 35K65, 35Q68.

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Section: P Guidotti and K Longo Nodeamentioning

confidence: 99%

Section: Well-posednessmentioning

confidence: 99%

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

Guidotti

Longo

2011

Nonlinear Differ. Equ. Appl.

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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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“…As a result, several approaches have been proposed for regularization of the method in an attempt to overcome the illposed nature of the problem and attain desirable results. [3][4][5] In a medical setting, it is often the case that the acquired images suffer from low contrast, which may ultimately affect the accuracy of analysis and diagnosis. Contrast enhancement of an image is therefore of critical interest.…”

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confidence: 99%