Iterative Filtering Decomposition Based on Local Spectral Evolution Kernel

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

doi:10.1007/s10915-011-9496-0

Cited by 34 publications

(31 citation statements)

References 76 publications

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“…Mode decomposition is an elementary operation in image and signal processing, and enables essentially all the other processing tasks such as noise removal, image edge detection, distortion restoration, feature extraction, enhancement, segmentation, and pattern recognition. Although there are many mode decomposition techniques such as empirical mode decomposition (EMD), 32 iterative filtering decomposition 42, 70, 71 and wavelets, partial differential equation (PDE) approaches have not been discovered. A major obstacle is due to the limited understanding of high order PDEs.…”

Section: Discussionmentioning

confidence: 99%

“…Similar to the IFD algorithm, 42, 70 we extract the highest frequency component first, then residual is used as the initial value for u and v n to extract the second highest frequency component. We continue this procedure until all components or intrinsic mode functions are separated.…”

Section: Simplified Models and Computational Algorithmsmentioning

confidence: 99%

“…These modes are called intrinsic mode functions (IMFs). 32, 34, 70 Time series or signals are decomposed into a sum of IMFs which have zero mean value and equal (or different by one) number of extrema and zero crossings. Each IMF contains information of instantaneous frequency defined by Hilbert-Huang transforms.…”

Section: Introductionmentioning

confidence: 99%

“…77 The increase in the order of the PDE leads to better frequency localization, which is desirable for certain applications. Additionally, the MoDEEs need to regenerate the initial value in order to systematically extract all IMFs, an algorithm previously used in our mode decomposition via iterative filtering, 42, 70 which is equivalent to the use of high order PDE high-pass filters. The MoDEE yields band-frequency components (or modes) by recursively extracting high frequency signals using high order PDEs.…”

Section: Introductionmentioning

confidence: 99%

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Mode Decomposition Evolution Equations

2011

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Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Europhys. Lett., 59(6): 814, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett., 6(7): 165, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be easily used for secondary processing. Various simplifications of the proposed MoDEEs, including a linearized version, and an algebraic version, are discussed for computational convenience. The Fourier pseudospectral method, which is unconditionally stable for linearized the high order MoDEEs, is utilized in our computation. Validation is carried out to mode separation of high frequency adjacent modes. Applications are considered to signal and image denoising, image edge detection, feature extraction, enhancement etc. It is hoped that this work enhances the understanding of high order PDEs and yields robust and useful tools for image and signal analysis.

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Section: Discussionmentioning

confidence: 99%

Section: Simplified Models and Computational Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mode Decomposition Evolution Equations

2011

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“…Functional modes are mode components which share the same band of frequency as well as same category, i.e., trend, edge, texture, noise etc. A major motivation in this development is the empirical mode decomposition (EMD), which enables the processing of non-uniform and non-stationary signals [32, 57]. However, the PDE transform is typically more robust than the Hilbert-Huang transform [32] and can be easly applied to multi-dimensional data.…”

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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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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Iterative Filtering Decomposition Based on Local Spectral Evolution Kernel

Cited by 34 publications

References 76 publications

Mode Decomposition Evolution Equations

Mode Decomposition Evolution Equations

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

Partial differential equation transform—Variational formulation and Fourier analysis

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