Adaptive Wavelet Basis Research Articles

Compressive sensing with an adaptive wavelet basis for structural system response and reliability analysis under missing data

The challenge of determining response and reliability statistics of large-scale structural systems under earthquake induced stochastic excitations is considered where the source load data records are incomplete. To this aim, a compressive sensing based framework in conjunction with an adaptive wavelet basis is presented for reconstructing the samples with missing data and estimating the underlying process EPS. In this regard, novel insights are provided whereas certain conceptual, numerical, and practical implementation aspects of the technique are presented in detail. A numerical example pertaining to the stochastic response and reliability analysis of an eight floor reinforced concrete building structural system demonstrates the effectiveness of the proposed methodology.

A wavelet-based Projector Augmented-Wave (PAW) method: Reaching frozen-core all-electron precision with a systematic, adaptive and localized wavelet basis set

We present a Projector Augmented-Wave (PAW) method based on a wavelet basis set. We implemented our wavelet-PAW method as a PAW library in the ABINIT package [http://www.abinit.org] and into BigDFT [http://www.bigdft.org]. We test our implementation in prototypical systems to illustrate the potential usage of our code. By using the wavelet-PAW method, we can simulate charged and special boundary condition systems with frozen-core all-electron precision. Furthermore, our work paves the way to large-scale and potentially order-N simulations within a PAW method.

A new compound faults detection method for rolling bearings based on empirical wavelet transform and chaotic oscillator

The rolling bearings often suffer from compound faults in practice. The concurrence of different faults increases the fault detection difficulty and the decoupling detection of compound faults is attracting considerable attentions. Recent publications report the application of the multiwavelets and empirical mode decomposition (EMD) for compound faults decoupling. However, due to limited adaptability they would induce mode mixing or/and overestimation problems in the signal processing. Particularly, the mode mixing would greatly degrade their performance on compound faults detection. To address this issue, this work presents a new method based on the empirical wavelet transform-duffing oscillator (EWTDO) for compound faults decoupling diagnosis of rolling bearings. The empirical wavelet transform (EWT) is able to extract intrinsic modes of a signal by fully adaptive wavelet basis. Hence, the mode mixing and overestimation can be resolved in decoupling processing and the compound faults can be correctly decomposed into different single faults in the form of empirical modes. Then, each single fault frequency was incorporated into a duffing oscillator to establish its corresponding fault isolator. By directly observing the chaotic motion from the Poincar mapping of the isolator outputs the single faults were identified one by one from the empirical modes. Experimental tests were carried out on a rolling bearing fault tester to examine the efficacy of the proposed EWTDO method on compound faults detection. The analysis results show attractive performance with respect to existing decoupling approaches based on the multiwavelets and EMD. In particular, our proposed method is much more reliable in decoupling the compound faults. Hence, the proposed method has practical importance in compound faults decoupling diagnosis for rolling bears.

On the optimal choice of wavelet function for multiscale honed surface characterization

Multiscale surface topography characterization is mostly suited than standard approaches because it is more adapted to the multi-stage process generation. Wavelet transform represents a power tool to perform the multiscale decomposition of the surface topography in a wide range of wavelength. However, characterization results depend closely on the topography data acquisition instrument (resolution, height accuracy, sensitivity…) and also on the wavelet analysis method (discrete or continuous transform). In particular, the choice of wavelet function can have significant effect on the analysis results. In this paper, we present experimental work on a number of popular wavelets functions with the aim of finding wavelets that exhibit optimal description of honed surface features when continuous wavelet transform is used. We demonstrate that the regularity property of wavelet function has a significant influence on the characterization performances. This comparative study shows also that the Morlet wavelet is the more adapted wavelet basis function for multiscale characterization of honed surfaces using continuous wavelet transform.

Wavelet kernel learning

This paper addresses the problem of optimal feature extraction from a wavelet representation. Our work aims at building features by selecting wavelet coefficients resulting from signal or image decomposition on an adapted wavelet basis. For this purpose, we jointly learn in a kernelized large-margin context the wavelet shape as well as the appropriate scale and translation of the wavelets, hence the name “wavelet kernel learning”. This problem is posed as a multiple kernel learning problem, where the number of kernels can be very large. For solving such a problem, we introduce a novel multiple kernel learning algorithm based on active constraints methods. We furthermore propose some variants of this algorithm that can produce approximate solutions more efficiently. Empirical analysis show that our active constraint MKL algorithm achieves state-of-the art efficiency. When used for wavelet kernel learning, our experimental results show that the approaches we propose are competitive with respect to the state-of-the-art on brain–computer interface and Brodatz texture datasets.

Modelling cardiovascular physiological signals using adaptive Hermite and wavelet basis functions

This study presented a unified perspective of adaptive basis functions to compare Hermite decomposition and wavelet transform for the analysis of cardiovascular physiological signals. Three different algorithms were presented to carry out physiological signal modelling with adaptive Hermite basis functions (HBFs), orthonormal wavelet basis functions (OWBFs) and adaptive wavelet basis functions (AWBFs). The modelling with OWBFs is computationally efficient. However, the concomitant restrictions in mathematics make OWBFs not optimal for compact modelling. In contrast, the optimised AWBFs can model cardiovascular physiological signals compactly with the cost of losing orthonormality. It not only sacrifices the fast implementation but also degrades AWBFs in discriminant analysis. In summary, merely HBFs achieve a balanced performance in compact modelling and discriminant analysis.

Adaptive wavelet modelling of a nested 3 factor experimental design in NIR chemometrics

The objective of this study was to investigate the effects of some commonly used sample preparation procedures, including overnight freezing, and the type of homogeniser on the near-infrared (NIR) spectra of red grape homogenates. Homogenates ( n = 284) of three red grape varieties were prepared using one of three types of homogenisers after one of two types of short term storage (fresh and overnight freezing) and then scanned in a FOSS NIRSystems6500 instrument (400–2500 nm). The NIR spectral data were then analysed using various discrimination techniques, namely Penalized Discriminant Analysis (PDA), Multivariate Adaptive Regression Splines discriminant analysis (MARS-DA) and Random Forests (RF) yielding correct classification rates (CCR) of 63.4%, 58.6% and 45.6%, respectively. To improve the CCR of the discrimination models, feature extraction from the NIR spectral data was performed using an adaptive discrete wavelet transformation (DWT). The DWT algorithm employs an adaptive wavelet basis function that maximizes the discrimination between the different combinations of homogenisers, storage and grape varieties. The results after adaptive DWT on the NIR spectra resulted in CCRs of 99.93%, 99.2% and 76.4% for PDA, MARS-DA and RF, respectively. Further analysis of adaptive DWT PDA via MANOVA revealed significant differences in the main and interaction effects of the three treatments, which were then associated with specific regions within the NIR spectrum.

Discriminative fabric defect detection using adaptive wavelets

We propose a new method for fabric defect detection by in- corporating the design of an adaptive wavelet-based feature extractor with the design of an Euclidean distance-based detector. The proposed method characterizes the fabric image with multiscale wavelet features by using undecimated discrete wavelet transforms. Each nonoverlapping window of the fabric image is then detected as defect or nondefect with an Euclidean distance-based detector. Instead of using the standard wavelet bases, an adaptive wavelet basis is designed for the detection of fabric defects. Minimization of the detection error is achieved by incor- porating the design of the adaptive wavelet with the design of the detec- tor parameters using a discriminative feature extraction (DFE) training method. The proposed method has been evaluated on 480 defect samples from five types of defects, and 480 nondefect samples, where a 97.5% detection rate and 0.63% false alarm rate were achieved. The evaluations were also carried out on unknown types of defects, where a 93.3% detection rate and 3.97% false alarm rate were achieved in the detection of 180 defect samples and 780 nondefect samples. © 2002

Numerical simulation of a mixing layer in an adaptive wavelet basis

This note presents an adaptive wavelet method to compute two-dimensional turbulent flows. The Navier–Stokes equations in vorticity–velocity form are discretized using a Petrov–Galerkin scheme. The vorticity field is developed into an orthogonal wavelet series where only the most significant coefficients are retained. The testfunctions are adapted to the linear part of the equation so that the resulting stiffness matrix turns out to be the identity. The nonlinear term is evaluated on a locally refined grid in physical space. This numerical scheme is applied to simulate a temporally developing mixing layer. A comparison with a classical pseudo-spectral method is used for validation of the new method. The results show that the formation of Kelvin–Helmholtz vortices is well captured and all scales of the flow are well represented.

Computation of decaying turbulence in an adaptive wavelet basis

The paper presents computations of decaying two-dimensional turbulence in an adaptive wavelet basis. At each time step the vorticity is represented by an adaptively selected set of wavelet functions which adjusts to the instantaneous distribution of vorticity. Essential features are the use of operator–adapted test functions and the adaptive evaluation of the convection term. The results of this new algorithm are compared to a classical Fourier method and a Fourier method supplemented with wavelet compression in each time step. They show that turbulent flows with a multitude of spatial scales can be computed with a reduced number of degrees of freedom. The investigation of diverse spectral and statistical criteria validates the wavelet approach.

Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis

We decompose turbulent flows into two orthogonal parts: a coherent, inhomogeneous, non-Gaussian component and an incoherent, homogeneous, Gaussian component. The two components have different probability distributions and different correlations, hence different scaling laws. This separation into coherent vortices and incoherent background flow is done for each flow realization before averaging the results and calculating the next time step. To perform this decomposition we have developed a nonlinear scheme based on an objective threshold defined in terms of the wavelet coefficients of the vorticity. Results illustrate the efficiency of this coherent vortex extraction algorithm. As an example we show that in a 2562 computation 0.7% of the modes correspond to the coherent vortices responsible for 99.2% of the energy and 94% of the enstrophy. We also present a detailed analysis of the nonlinear term, split into coherent and incoherent components, and compare it with the classical separation, e.g., used for large eddy simulation, into large scale and small scale components. We then propose a new method, called coherent vortex simulation (CVS), designed to compute and model two-dimensional turbulent flows using the previous wavelet decomposition at each time step. This method combines both deterministic and statistical approaches: (i) Since the coherent vortices are out of statistical equilibrium, they are computed deterministically in a wavelet basis which is remapped at each time step in order to follow their nonlinear motions. (ii) Since the incoherent background flow is homogeneous and in statistical equilibrium, the classical theory of homogeneous turbulence is valid there and we model statistically the effect of the incoherent background on the coherent vortices. To illustrate the CVS method we apply it to compute a two-dimensional turbulent mixing layer.

Texture characterization and defect detection using adaptive wavelets

Many textures such as woven fabrics and composites have a regular and repeating texture. This paper presents a new method to capture the texture information using adaptive wavelet bases. Wavelets are compact functions which can be used to generate a multiresolution analysis. Texture constraints are used to adapt the wavelets to better characterize specific textures. An adapted wavelet basis has very high sensitivity to the abrupt changes in the texture structure caused by de- fects. This paper demonstrates how adaptive wavelet basis can be used to locate defects in woven fabrics. © 1996 Society of Photo-Optical Instrumenta- tion Engineers.