Number Of Wavelet Coefficients Research Articles

Combination of Continuous Wavelet Transform and Successive Projection Algorithm for the Estimation of Winter Wheat Plant Nitrogen Concentration

Plant nitrogen concentration (PNC) is a traditional standard index to measure the nitrogen nutritional status of winter wheat. Rapid and accurate diagnosis of PNC performs an important role in mastering the growth status of winter wheat and guiding field precision fertilization. In this study, the in situ hyperspectral reflectance data were measured by handheld SVC HR−1024I (SVC) passive field spectroradiometer and PNC were determined by the modified Kjeldahl digestion method. Continuous wavelet transform (CWT), successive projection algorithm (SPA) and partial least square (PLS) regression were combined to construct an efficient method for estimating winter wheat PNC. The main objectives of this study were to (1) use CWT to extract various wavelet coefficients under different decomposition scales, (2) use SPA to screen sensitive wavelet coefficients as independent variables and combine with PLS regression to establish winter wheat PNC estimation models, respectively, and (3) compare the precision of PLS regression models to find a reliable model for estimating winter wheat PNC during the growing season. The results of this paper showed that properly increasing the decomposition scale of CWT could weaken the impact of high-frequency noise on the prediction model. The number of wavelet coefficients has been significantly reduced after screened by SPA. The PNC estimation model (CWT–Scale6–SPA–PLS) based on the wavelet coefficients of the sixth decomposition scale most accurately predicted the PNC (the determination coefficient of the calibration set (Rc2) was 0.85. Root mean square error of the calibration set (RMSEc) was 0.27. The determination coefficient of the validation set (Rv2) was 0.84. Root mean square error of the validation set (RMSEv) was 0.28 and relative prediction deviation (RPD) was 2.47). CWT-Scale6-SPA-PLS can be used to predict PNC. The optimal winter wheat PNC prediction model based on CWT proposed in this study is a reliable method for rapid and nondestructive monitoring of PNC and provides a new technical method for precision nitrogen management.

Differential protection scheme for power transformers using matched wavelets

A new approach for differential protection based on the mother wavelets which match in shape with inrush and fault current waveforms is proposed in this work. It is demonstrated that if the matched wavelet is used for processing inrush and fault waveforms of transformer then its detection capability improves. In case of false tripping of transformer, the equipment is to be opened for inspection for faults. This leads to equipment outage because there is no mechanism to ascertain that there was no fault. It is needed that inrush and fault be detected independently not only for the purpose of faster and better detection but also for studying mal-operations and post-analysis. For a faster and better detection, a new protection scheme is also proposed making use of independent detection of fault and inrush waveforms. To find matched inrush and fault wavelets, differential evolution algorithm is used. Out of various possible solutions for different number of wavelet coefficients, a choice is made by further testing on various switching angles. The effectiveness of the proposed method is tested through PSCAD simulations on transformers of different ratings.

Averaged Probability of the Error in Calculating Wavelet Coefficients for the Random Sample Size

Signal denoising methods based on the threshold processing of wavelet coefficients are widely used in various application areas. When applying these methods, it is usually assumed that the number of wavelet coefficients is fixed, and the noise distribution is Gaussian. Such a model has been well studied in the literature, and optimal threshold values have been calculated for different signal classes and loss functions. However, in some situations the sample size is not known in advance and is modeled by a random variable. In this paper, we consider a model with a random number of observations contaminated by a Gaussian noise, and study the behavior of the loss function based on the probabilities of errors in calculating wavelet coefficients for a growing sample size.

Extraction of coherent clusters and grid adaptation in particle-laden turbulence using wavelet filters

In this study, a wavelet-based method for extraction of clusters of inertial particles in turbulent flows is presented that is based on decomposing Eulerian particle-number-density fields into the sum of coherent (organized) and incoherent (disorganized) components. The coherent component is associated with the clusters and is extracted by filtering the wavelet-transformed particle-number-density field based on an energy threshold. The method is applied to direct numerical simulations of homogeneous-isotropic turbulence laden with small Lagrangian particles. The analysis shows that in regimes where the preferential concentration is important, the coherent component representing the clusters can be described by just 1.6% of the total number ofwavelet coefficients, thereby illustrating the sparsity of the particle-number-density field. On the other hand, the incoherent portion is visually structureless and much less correlated than the coherent one. An application of the method, motivated by particle-laden radiative-heat-transfer simulations, is illustrated in the form of a grid-adaptation algorithm that results in nonuniform meshes with fine and coarse elements near and away from particle clusters, respectively. In regimes where preferential concentration in clusters is important, the grid adaptation leads to a significant reduction of the number of control volumes by one to two orders of magnitude.

Influences of the signal border extension in the discrete wavelet transform in EEG spike detection

Introduction The discrete wavelet transform is used in many studies as signal preprocessor for EEG spike detection. An inherent process of this mathematical tool is the recursive wavelet convolution over the signal that is decomposed into detail and approximation coefficients. To perform these convolutions, firstly it is necessary to extend signal borders. The selection of an unsuitable border extension algorithm may increase the false positive rate of an EEG spike detector. Methods In this study we analyzed nine different border extensions used for convolution and 19 mother wavelets commonly seen in other EEG spike detectors in the literature. Results The border extension may degrade an EEG spike detector up to 44.11%. Furthermore, results behave differently for distinct number of wavelet coefficients. Conclusion There is not a best border extension to be used with any EEG spike detector based on the discrete wavelet transform, but the selection of the most adequate border extension is related to the number of coefficients of a mother wavelet.

Tsunami source uncertainty estimation: The 2011 Japan tsunami

AbstractThis paper studies the initial sea surface displacement and its uncertainty after an earthquake based on tsunami waveforms. The spatial distribution is inferred with a Bayesian approach that provides probabilities that are interpreted as uncertainties of the displaced sea surface. The parameterization is nonlinear and treats apparent rupture velocity as unknown but assumes rise time to be fixed at 30 s. Importantly, the spatial complexity of the source is constrained by observations using a transdimensional algorithm based on a wavelet decomposition of the displacement field. In this approach, the number of wavelet coefficients is an unknown random variable that is also estimated as part of the inversion. The resulting parameterization is parsimonious in that it can adapt to the spatially varying source complexity while being consistent with the information in the tsunami waveforms. In this way, the resolution of displacement varies across the source region with more parameters introduced for parts of the source that are resolved well by the data and/or have significant complexity. The noise level (standard deviation) at each gauge is initially treated as unknown to estimate data covariance matrices. These matrices are applied in subsequent inversion and include unknown scaling which eliminates the requirement to assume station weights and accounts for temporally correlated waveform noise. The method is applied to waveforms recorded during the 2011 Japan Tsunami and results show high resolution (low uncertainty) in most parts of the source region and a previously unreported level of source detail. In particular, the main peak of the source is elongated trench parallel and shows a well‐resolved bimodal finger‐like feature in the northern source region that closely follows the trench.

A constrained two-layer compression technique for ECG waves.

This paper proposes a constrained two-layer compression technique for electrocardiogram (ECG) waves, of which encoded parameters can be directly used for the diagnosis of arrhythmia. In the first layer, a single ECG beat is represented by one of the registered templates in the codebook. Since the required coding parameter in this layer is only the codebook index of the selected template, its compression ratio (CR) is very high. Note that the distribution of registered templates is also related to the characteristics of ECG waves, thus it can be used as a metric to detect various types of arrhythmias. The residual error between the input and the selected template is encoded by a wavelet-based transform coding in the second layer. The number of wavelet coefficients is constrained by pre-defined maximum distortion to be allowed. The MIT-BIH arrhythmia database is used to evaluate the performance of the proposed algorithm. The proposed algorithm shows around 7.18 CR when the reference value of percentage root mean square difference (PRD) is set to ten.

Improved Threshold Function for Power Quality Disturbance Signal Denoising

Mutation point feature of power quality (PQ) disturbance signals are very conducive to the wavelet-based detection and localization of PQ events, but the PQ signals are often disturbed by noise. In order to suppress noise and keep mutation points, an improved threshold function was proposed. According to the fact that the wavelet coefficients of signal and noise distributed on different scale, the threshold σj2lnk amended by to calculate threshold value for each scale adaptively. (k is the number of wavelet coefficients at level j). In simulation, four type of PQ signals and three degrading degrees are testing; meanwhile, four existing algorithm with wavelet shrinkage are performed for comparison. Results reveal that the proposed scheme not only suppresses noise of PQ signal well but also keeps the mutation points nicely.

A Consistent and Stable Approach to Generalized Sampling

We consider the problem of generalized sampling, in which one seeks to obtain reconstructions in arbitrary finite dimensional spaces from a finite number of samples taken with respect to an arbitrary orthonormal basis. Typical approaches to this problem consider solutions obtained via the consistent reconstruction technique or as solutions of an overcomplete linear systems. However, the consistent reconstruction technique is known to be non-convergent and ill-conditioned in important cases, such as the recovery of wavelet coefficients from Fourier samples, and whilst the latter approach presents solutions which are convergent and well-conditioned when the system is sufficiently overcomplete, the solution becomes inconsistent with the original measurements. In this paper, we consider generalized sampling via a non-linear minimization problem and prove that the minimizers present solutions which are convergent, stable and consistent with the original measurements. We also provide analysis in the case of recovering wavelets coefficients from Fourier samples. We show that for compactly supported wavelets of sufficient smoothness, there is a linear relationship between the number of wavelet coefficients which can be accurately recovered and the number of Fourier samples available.

A Novel Methodology for Memory Reduction in Distributed Arithmetic Based Discrete Wavelet Transform

Discrete Wavelet Transform (DWT) is widely used in image compression standards such as JPEG 2000. DWT can be implemented on FPGA using parallel Distributed Arithmetic (DA) architecture, which is suitable for low power implementation. However, the size of the memory in DA increases with the number of wavelet coefficients. In this paper, we propose a novel methodology to reduce the size of the Look-Up Tables (LUTs) used in DA for DWT. The table entries are sorted using Burrows-Wheeler Transform (BWT) and then compressed. The compressed table is stored in memory. During DWT/IDWT computation, without reconstructing the entire table we can recover only the required table entry. A comparative study of this methodology among different wavelets is performed. We demonstrate that the method is very effective for reducing the memory of DA architectures. A compression ratio of around 2.3:1 is achieved for the look-up table which stores the inner product of high-pass filter coefficients of Daubechies-4 (Db4) wavelet which is used in JPEG2000.

Quantisation-based video watermarking in the wavelet domain with spatial and temporal redundancy

In this article we introduce a new public digital watermarking technique for video copyright protection working in the discrete wavelet transform domain. The scheme uses binary images as watermarks. These are embedded in the detail wavelet coefficients of the middle wavelet sub-bands. The method is a combination of spread spectrum and quantisation-based watermarking. Every bit of the watermark is spread over a number of wavelet coefficients with the use of a secret key. The resilience of the watermarking algorithm was tested against a series of eight different attacks using different videos. To improve the resilience of the algorithm we use error correction codes and embed the watermark with spatial and temporal redundancy. The proposed method achieves a very good perceptual quality with mean peak signal-to-noise ratio values of the watermarked videos of more than 40 dB and high resistance to a large spectrum of attacks.

Power quality evaluation in smart grids considering modern distortion in electric power systems

High power quality level is required in smart grids especially for non-stationary situations due to increased use of nonlinear loads and PQ disturbances such as dips, swells, transients and interruptions. Many power quality indices (PQIs) are available. In this paper a new fuzzy-wavelet packet transform-based power quality Index (FWPTPQI) is developed to amalgamate existing power quality indices as the output of a fuzzy based module based on fuzzy inference systems, knowledge base and existing PQI as input. Fuzzy systems allow handling the uncertainties associated with the electric power quality evaluation. The proposed approach has been applied to two case studies; stationary balanced and non-stationary unbalanced three-phase systems. The results are compatible with prevalent situations. The new index gives significant sense of the quality of transmitted electrical power. A comparative study of using different wavelet basis functions is considered and results indicate that Daubechies 10 and Daubechies 15 could be considered as the overall best wavelet basis functions. Since the new index represents an amalgamation of the PQ indices with less number of wavelet coefficients, it helps reduce the size of data processed which is required in smart grid applications.

Robust wavelet-based video watermarking scheme for copyright protection using the human visual system

The development of the information technology and computer networks facilitates easy duplication, manipulation, and distribution of digital data. Digital watermarking is one of the proposed solutions for effectively safeguarding the rightful ownership of digital images and video. We propose a public digital watermarking technique for video copyright protection in the discrete wavelet transform domain. The scheme uses binary images as watermarks. These are embedded in the detail wavelet coefficients of the middle wavelet subbands. The method is a combination of spread spectrum and quantization-based watermarking. Every bit of the watermark is spread over a number of wavelet coefficients with the use of a secret key by means of quantization. The selected wavelet detail coefficients from different subbands are quantized using an optimal quantization model, based on the characteristics of the human visual system (HVS). Our HVS-based scheme is compared to a non-HVS approach. The resilience of the watermarking algorithm is tested against a series of different spatial, temporal, and compression attacks. To improve the robustness of the algorithm, we use error correction codes and embed the watermark with spatial and temporal redundancy. The proposed method achieves a good perceptual quality and high resistance to a large spectrum of attacks.

A robust digital watermarking scheme for video copyright protection in the wavelet domain

This paper proposes a novel digital watermarking method for video based on a multi-resolution wavelet decomposition. The watermark used is a binary image. This watermark is embedded in the wavelet coefficients of the LH, HL and HH sub-bands of the second wavelet decomposition level by quantization. Every bit of the watermark is spread over a number of wavelet coefficients with the use of a key. The resulting watermarking scheme can be used for public watermarking applications, where the original image is not available for watermark extraction. We have tested the resilience of the watermarking algorithm against a series nine different attacks for different videos and improved the decoding BER by redundant embedding of the same watermark in different frames and by using an error correction code. Experimental results show that the embedded watermark is invisible and robust to attacks.

Uniformly sampling multi-resolution analysis for image-based relighting

Image-based relighting allows us to efficiently light a scene under complicated illumination conditions. However, the traditional cubemap based multi-resolution analysis unevenly samples the spherical surface, with a higher sampling rate near the face corners and a lower one near the face centers. The non-uniformity penalizes the efficiency of data representation. This paper presents a uniformly sampling multi-resolution analysis approach, namely the icosahedron spherical wavelets (ISW), for image-based relighting under time-varying distant environment. Since the proposed ISW approach provides a highly uniform sampling distribution over the spherical domain, we thus can efficiently handle high frequency variations locally in the illumination changes as well as reduce the number of wavelet coefficients needed in the renderings. Furthermore, visual artifacts are demonstrated to be better suppressed in the proposed ISW approach. Compared with the traditional cubemap based multi-resolution analysis approach, we show that our approach can effectively produce higher quality image sequences that are closer to the ground truth in terms of percentage square errors.

Novel power quality indices based on wavelet packet transform for non-stationary sinusoidal and non-sinusoidal disturbances

High power quality (PQ) level represents one of the main objectives towards smart grid. The currently used PQIs that are a measure of the PQ level are defined under the umbrella of the Fourier foundation that produces unrealistic results in case of non-stationary PQ disturbances. In order to accurately measure those indices, wavelet packet transform (WPT) is used in this paper to reformulate the recommended PQIs and hence benefiting from the WPT capabilities in accurately analyzing non-stationary waveforms and providing a uniform time–frequency sub-bands leading to reduced size of the data to be processed which is a necessity to facilitate the implementation of smart grid. Numerical examples’ results considering non-stationary waveforms prove the suitability of the WPT for PQIs measurement; also the results indicate that Daubechies 10 could be the best candidate wavelet basis function that could provide acceptable accuracy while requiring less number of wavelet coefficients and hence reducing the data size. Moreover, a new time–frequency overall and node crest factors are introduced in this paper. The new node crest factor is able to determine the node or the sub-band that is responsible for the largest impact which could not be achieved using traditional approaches.

Clustering MIT–BIH arrhythmias with Ant Colony Optimization using time domain and PCA compressed wavelet coefficients

In this paper, Ant Colony Optimization (ACO) based clustering analysis of ECG arrhythmias taken from the MIT-BIH Arrhythmia Database is proposed. Both time domain and discrete wavelet transform (DWT) based frequency domain features are used in the analysis. Since the number of wavelet coefficients are huge amount as compared to the time domain parameters, Principal Component Analysis (PCA) based compression is applied on them in order to decrease their number to the number of time domain features. Then, the reduced numbers of frequency parameters are combined with the time domain features, in order to get the total feature sets. Different types of feature sets are tried and the classification results are compared. These are: time domain feature set, frequency domain feature set and the mixture of them. A neural network algorithm is developed in parallel to verify and measure the ACO classifier's success. Moreover, linear discriminant analysis (LDA) is used to show the effect of clustering on the system's results. The method is tested with MIT-BIH database to classify normal beats and five different critical and having vital importance arrhythmia types. Chosen six classes are normal sinus rhythm, premature ventricular contraction (PVC), atrial premature contraction (APC), right bundle branch block (RBBB), ventricular fusion (F) and fusion (f). Comparison results indicate that the mixture feature set gave a better success for the classification.

Wavelet series based iterative learning controller design for industrial robot manipulators

In this paper, a wavelet series based learning controller has been proposed for the tracking control of robot manipulator. Wavelet series approximation is used to approximate the desired and actual trajectories of the system into finite number of wavelet coefficients. A learning controller is designed in wavelet domain which forces the wavelet coefficients of the actual output approach to the corresponding wavelet coefficients of the desired trajectory, which are known constants, such that the tracking of robot manipulator is achieved. Numerical experimentation studies show the ability of the learning controller designed in wavelet domain in the presence of uncertainties in robot manipulator system.

Multiwavelet Constructions and Volterra Kernel Identification

The Volterra series is commonly used for the modeling of nonlinear dynamical systems. In general, however, a large number of terms are needed to represent Volterra kernels, with the number of required terms increasing exponentially with the order of the kernel. Therefore, reduced-order kernel representations are needed in order to employ the Volterra series in engineering practice. This paper presents an approach whereby multiwavelets are used to obtain low-order estimates of first-, second-, and third-order Volterra kernels. A family of multiwavelets is constructed from the classical finite element basis functions using the technique of intertwining. The resulting multiwavelets are piecewise-polynomial, orthonormal, compactly-supported, and can be constructed with arbitrary approximation order. Furthermore, these multiwavelets are easily adapted to the domains of support of the Volterra kernels. In contrast, most wavelet families do not possess this characteristic. Higher-dimensional multiwavelets can easily be constructed by taking tensor products of the original one-dimensional functions. Therefore, it is straightforward to extend this approach to the representation of higher-order Volterra kernels. This kernel identification algorithm is demonstrated on a prototypical oscillator with a quadratic stiffness nonlinearity. For this system, it is shown that accurate kernel estimates can be obtained in terms of a relatively small number of wavelet coefficients. These results indicate the potential of the multiwavelet-based algorithm for obtaining reduced-order models for a large class of weakly nonlinear systems.

Efficient spatial and temporal representations of global ionosphere maps over Japan using B-spline wavelets

Wavelet expansion has been demonstrated to be suitable for the representation of spatial functions. Here we propose the so-called B-spline wavelets to represent spatial time-series of GPS-derived global ionosphere maps (GIMs) of the vertical total electron content (TEC) from the Earth’s surface to the mean altitudes of GPS satellites, over Japan. The scalar-valued B-spline wavelets can be defined in a two-dimensional, but not necessarily planar, domain. Generated by a sequence of knots, different degrees of B-splines can be implemented: degree 1 represents the Haar wavelet; degree 2, the linear B-spline wavelet, or degree 4, the cubic B-spline wavelet. A non-uniform version of these wavelets allows us to handle data on a bounded domain without any edge effects. B-splines are easily extended with great computational efficiency to domains of arbitrary dimensions, while preserving their properties. This generalization employs tensor products of B-splines, defined as linear superposition of products of univariate B-splines in different directions. The data and model may be identical at the locations of the data points if the number of wavelet coefficients is equal to the number of grid points. In addition, data compression is made efficient by eliminating the wavelet coefficients with negligible magnitudes, thereby reducing the observational noise. We applied the developed methodology to the representation of the spatial and temporal variations of GIM from an extremely dense GPS network, the GPS Earth Observation Network (GEONET) in Japan. Since the sampling of the TEC is registered regularly in time, we use a two-dimensional B-spline wavelet representation in space and a one-dimensional spline interpolation in time. Over the Japan region, the B-spline wavelet method can overcome the problem of bias for the spherical harmonic model at the boundary, caused by the non-compact support. The hierarchical decomposition not only allows an inexpensive calculation, but also separates visualisation at different levels of detail. Each level corresponds to a certain spatial frequency band, leading to a detection of structures and enhancement in the ionosphere at different resolutions.