Partial differential equation transform—Variational formulation and Fourier analysis

Yang, Wang; Guo, Wei; Yang, Siyang

doi:10.1002/cnm.1452

Cited by 19 publications

(48 citation statements)

References 67 publications

(126 reference statements)

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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%

“…Equation (36) is essentially equivalent to our earlier variational derivation of the PDE transform [55], however, in our Ref. [55], there is a typo, specifically, in page 2003, Λ uj (·) and Λ vj (·) should be defined as Λ uj (·) = (−1) j +1 ∂Λ u /∂| D j u | 2 and Λ vj (·) = (−1) j +1 ∂Λ v /∂| D j v | 2 respectively.…”

Section: Theory and Algorithmmentioning

confidence: 99%

“…[55], there is a typo, specifically, in page 2003, Λ uj (·) and Λ vj (·) should be defined as Λ uj (·) = (−1) j +1 ∂Λ u /∂| D j u | 2 and Λ vj (·) = (−1) j +1 ∂Λ v /∂| D j v | 2 respectively.…”

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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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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Geometric modeling of subcellular structures, organelles, and multiprotein complexes

Feng

Xia

Tong

et al. 2012

Numer Methods Biomed Eng

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SUMMARY Recently, the structure, function, stability, and dynamics of subcellular structures, organelles, and multi-protein complexes have emerged as a leading interest in structural biology. Geometric modeling not only provides visualizations of shapes for large biomolecular complexes but also fills the gap between structural information and theoretical modeling, and enables the understanding of function, stability, and dynamics. This paper introduces a suite of computational tools for volumetric data processing, information extraction, surface mesh rendering, geometric measurement, and curvature estimation of biomolecular complexes. Particular emphasis is given to the modeling of cryo-electron microscopy data. Lagrangian-triangle meshes are employed for the surface presentation. On the basis of this representation, algorithms are developed for surface area and surface-enclosed volume calculation, and curvature estimation. Methods for volumetric meshing have also been presented. Because the technological development in computer science and mathematics has led to multiple choices at each stage of the geometric modeling, we discuss the rationales in the design and selection of various algorithms. Analytical models are designed to test the computational accuracy and convergence of proposed algorithms. Finally, we select a set of six cryo-electron microscopy data representing typical subcellular complexes to demonstrate the efficacy of the proposed algorithms in handling biomolecular surfaces and explore their capability of geometric characterization of binding targets. This paper offers a comprehensive protocol for the geometric modeling of subcellular structures, organelles, and multiprotein complexes.

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Partial differential equation transform—Variational formulation and Fourier analysis

Cited by 19 publications

References 67 publications

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

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

Biomolecular surface construction by PDE transform

Geometric modeling of subcellular structures, organelles, and multiprotein complexes

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