Properties of Higher Order Nonlinear Diffusion Filtering

Didas, Stephan; Weickert, Joachim; Burgeth, Bernhard

doi:10.1007/s10851-009-0166-x

Cited by 84 publications

(81 citation statements)

References 46 publications

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“…Since variational approaches have found their success in a variety of scientific and engineering fields [6, 16, 18–20, 45, 60, 62], a variational derivation of the PDE transform has also been presented [55, 76]. Here we briefly review the variational derivation of the PDE transform.…”

Section: Theory and Algorithmmentioning

confidence: 99%

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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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Section: Theory and Algorithmmentioning

confidence: 99%

“…Some of the most commonly used penalty functions include the Tikhonov form [20, 34]

normalΛ false(x^{2} false) = x^{2},

the mean curvature form

normalΛ false(x^{2} false) = {false(σ^{2} + x^{2} false)}^{\frac{1}{2}},

and the Gaussian form

normalΛ false(x^{2} false) = e^{x^{2} / 2 σ^{2}} .

…”

Section: Theory and Algorithmmentioning

confidence: 99%

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

Chen

Wei

2012

Computational and Mathematical Biophysics

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“…All of these quantities measure a comparison between the denoised image and the original clean image. A stopping time could be determined by optimizing one of these quantities (as is often done, see, e.g., [15,20]). However, it should not be assumed that the clean image is known, and the stopping time Furthermore, these quantities do not necessarily give an accurate measure of the quality of a denoised image, as they may not adequately penalize certain undesirable properties, such as excessive noise, splotchiness, or blurriness.…”

Section: Quantitative Evaluationmentioning

confidence: 99%

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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“…Nevertheless, only in a very limited number of simple cases, EL PDE that corresponds to the target energy functional can be analytically solved [11,12]. Thus, in all related works, the actual minimization is conducted by the transition from an elliptic EL PDE, to a parabolic PDE with the artificial time.…”

Section: Introductionmentioning

confidence: 99%

“…A similar problem appears with the choice of the optimal time step. Actually, a parabolic PDE model is obtained from the particular EL equation in the limiting process (j-j 0 )/λ 0 as λ 0, where j 0 is the noisy image, and λ is a Lagrange multiplier (see [11] or [13]). The role of λ, as the trade of between image smoothness and preservation of image features is lost: it becomes just a time step in the filtering process.…”

Section: Introductionmentioning

confidence: 99%

Image denoising by a direct variational minimization

Janev

Atanacković

Pilipović

et al. 2011

EURASIP J. Adv. Signal Process.

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In this article we introduce a novel method for the image de-noising which combines a mathematically wellposdenes of the variational modeling with the efficiency of a patch-based approach in the field of image processing. It based on a direct minimization of an energy functional containing a minimal surface regularizer that uses fractional gradient. The minimization is obtained on every predefined patch of the image, independently. By doing so, we avoid the use of an artificial time PDE model with its inherent problems of finding optimal stopping time, as well as the optimal time step. Moreover, we control the level of image smoothing on each patch (and thus on the whole image) by adapting the Lagrange multiplier using the information on the level of discontinuities on a particular patch, which we obtain by pre-processing. In order to reduce the average number of vectors in the approximation generator and still to obtain the minimal degradation, we combine a Ritz variational method for the actual minimization on a patch, and a complementary fractional variational principle. Thus, the proposed method becomes computationally feasible and applicable for practical purposes. We confirm our claims with experimental results, by comparing the proposed method with a couple of PDE-based methods, where we get significantly better denoising results specially on the oscillatory regions.

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Properties of Higher Order Nonlinear Diffusion Filtering

Cited by 84 publications

References 46 publications

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

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

Two Enhanced Fourth Order Diffusion Models for Image Denoising

Image denoising by a direct variational minimization

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