Mode Decomposition Evolution Equations

Yang, Wang; Wei, Guo-Wei; Yang, Siyang

doi:10.1007/s10915-011-9509-z

Cited by 23 publications

(32 citation statements)

References 80 publications

(140 reference statements)

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“…The PDE transform of arbitrarily high integer order PDEs has been introduced in our earlier work [55, 58]. 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].…”

Section: Theory and Algorithmmentioning

confidence: 99%

“…The residue of the trend is an edge function, i.e., a high frequency mode, including possible noisy. By systematically repeating the low-pass PDE transform (37), one can extract all the desirable higher order mode functions step by step [58]. …”

Section: Theory and Algorithmmentioning

confidence: 99%

“…In our previous work, high-order nonlinear evolution PDEs have made vital contributions to mode decomposition [58], image analysis [55], and molecular surface generation [76]. However, solving high-order PDE in Eq.…”

Section: Theory and Algorithmmentioning

confidence: 99%

“…A major extension of our earlier nonlinear PDE based high-pass filters is the introduction of the PDE transform [55, 56, 58], which is a systematic method for decomposing images, signals, and data into various functional modes. Functional modes are mode components which share the same band of frequency as well as same category, i.e., trend, edge, texture, noise etc.…”

Section: Introductionmentioning

confidence: 99%

“…Once the intrinsic functional modes are obtained by the PDE transform, subsequent processing or secondary processing of the individual functional modes enables us to achieve our goals of signal, image, surface and data analysis. Typical tasks include edge detection, feature extraction, trend estimation, enhancement, denoising, texture analysis, segmentation, pattern recognition, etc [55, 56, 58]. In addition to these tasks, the PDE transform has also been applied to the surface construction of proteins and viruses [76].…”

Section: Introductionmentioning

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%

Section: Theory and Algorithmmentioning

confidence: 99%

Section: Theory and Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

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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Partial differential equation transform—Variational formulation and Fourier analysis

Yang

Guo

Yang

2011

Numer Methods Biomed Eng

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Nonlinear partial differential equation (PDE) models are established approaches for image/signal processing, data analysis and surface construction. Most previous geometric PDEs are utilized as low-pass filters which give rise to image trend information. In an earlier work, we introduced mode decomposition evolution equations (MoDEEs), which behave like high-pass filters and are able to systematically provide intrinsic mode functions (IMFs) of signals and images. Due to their tunable time-frequency localization and perfect reconstruction, the operation of MoDEEs is called a PDE transform. By appropriate selection of PDE transform parameters, we can tune IMFs into trends, edges, textures, noise etc., which can be further utilized in the secondary processing for various purposes. This work introduces the variational formulation, performs the Fourier analysis, and conducts biomedical and biological applications of the proposed PDE transform. The variational formulation offers an algorithm to incorporate two image functions and two sets of low-pass PDE operators in the total energy functional. Two low-pass PDE operators have different signs, leading to energy disparity, while a coupling term, acting as a relative fidelity of two image functions, is introduced to reduce the disparity of two energy components. We construct variational PDE transforms by using Euler-Lagrange equation and artificial time propagation. Fourier analysis of a simplified PDE transform is presented to shed light on the filter properties of high order PDE transforms. Such an analysis also offers insight on the parameter selection of the PDE transform. The proposed PDE transform algorithm is validated by numerous benchmark tests. In one selected challenging example, we illustrate the ability of PDE transform to separate two adjacent frequencies of sin(x) and sin(1.1x). Such an ability is due to PDE transform’s controllable frequency localization obtained by adjusting the order of PDEs. The frequency selection is achieved either by diffusion coefficients or by propagation time. Finally, we explore a large number of practical applications to further demonstrate the utility of proposed PDE transform.

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Biomolecular surface construction by PDE transform

Zheng

Yang

Guo

2011

Numer Methods Biomed Eng

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This work proposes a new framework for the surface generation based on the partial differential equation (PDE) transform. The PDE transform has recently been introduced as a general approach for the mode decomposition of images, signals, and data. It relies on the use of arbitrarily high order PDEs to achieve the time-frequency localization, control the spectral distribution, and regulate the spatial resolution. The present work provides a new variational derivation of high order PDE transforms. The fast Fourier transform is utilized to accomplish the PDE transform so as to avoid stringent stability constraints in solving high order PDEs. As a consequence, the time integration of high order PDEs can be done efficiently with the fast Fourier transform. The present approach is validated with a variety of test examples in two and three-dimensional settings. We explore the impact of the PDE transform parameters, such as the PDE order and propagation time, on the quality of resulting surfaces. Additionally, we utilize a set of 10 proteins to compare the computational efficiency of the present surface generation method and the MSMS approach in Cartesian meshes. Moreover, we analyze the present method by examining some benchmark indicators of biomolecular surface, i.e., surface area, surface enclosed volume, solvation free energy and surface electrostatic potential. A test set of 13 protein molecules is used in the present investigation. The electrostatic analysis is carried out via the Poisson-Boltzmann equation model. To further demonstrate the utility of the present PDE transform based surface method, we solve the Poisson-Nernst-Planck (PNP) equations with a PDE transform surface of a protein. Second order convergence is observed for the electrostatic potential and concentrations. Finally, to test the capability and efficiency of the present PDE transform based surface generation method, we apply it to the construction of an excessively large biomolecule, a virus surface capsid. Virus surface morphologies of different resolutions are attained by adjusting the propagation time. Therefore, the present PDE transform provides a multiresolution analysis in the surface visualization. Extensive numerical experiment and comparison with an established surface model indicate that the present PDE transform is a robust, stable and efficient approach for biomolecular surface generation in Cartesian meshes.

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Mode Decomposition Evolution Equations

Cited by 23 publications

References 80 publications

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

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

Partial differential equation transform—Variational formulation and Fourier analysis

Biomolecular surface construction by PDE transform

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