Design Of Wavelet Research Articles

The biorthogonal wavelets that are redundant-free and nearly shift-insensitive

It is well known that discrete wavelet transform (DWT) is sensitive to shift, which means a slight shift of feature in the original signal may cause unpredictable changes in the analysis subbands. Some modified versions of DWT can reduce the shift sensitivity, however, they are all redundant. In this article, we shows the shift sensitivity is caused by the aliasing terms formed in the downsampling operation during analysis process. A novel scheme for the design of wavelet is proposed to reduce the effect of aliasing terms as much as possible in the general framework of DWT. A few of biorthogonal wavelets have been designed and applied in the simulation examples. The results of examples demonstrate the efficiency of the designed wavelets in the term of shift insensitivity and nonredundancy.

Corrections to ‘Area- and Power-Efficient Design of Daubechies Wavelet Transforms Using Folded AIQ Mapping’

Islam and Wahid proposed an area- and power-efficient design of Daubechies wavelet transforms. However, it was found that the matrix decompositions of the folded algebraic-integer-quantization scheme for DAUB4 and DAUB6 are incorrect and do not correspond with their architecture designs. We propose modifications not only to the algorithm but also to its implementation. Compared with the original method, the proposed design requires fewer adders and maintains the same critical path.

Real-time defect detection of steel wire rods using wavelet filters optimized by univariate dynamic encoding algorithm for searches

We propose a new defect detection algorithm for scale-covered steel wire rods. The algorithm incorporates an adaptive wavelet filter that is designed on the basis of lattice parameterization of orthogonal wavelet bases. This approach offers the opportunity to design orthogonal wavelet filters via optimization methods. To improve the performance and the flexibility of wavelet design, we propose the use of the undecimated discrete wavelet transform, and separate design of column and row wavelet filters but with a common cost function. The coefficients of the wavelet filters are optimized by the so-called univariate dynamic encoding algorithm for searches (uDEAS), which searches the minimum value of a cost function designed to maximize the energy difference between defects and background noise. Moreover, for improved detection accuracy, we propose an enhanced double-threshold method. Experimental results for steel wire rod surface images obtained from actual steel production lines show that the proposed algorithm is effective.

Design of a Novel Wavelet Based Transient Detection Unit for In-Vehicle Fault Determination and Hybrid Energy Storage Utilization

This paper addresses the performance and reliability issues as encountered by the present EV technology. The research work presented in this paper is based on an ongoing project which has a twofold research motivation: 1) use of wavelet analysis to determine transients in electric vehicles; and 2) application the information obtained during transient detection to address the performance and reliability concerns. The novel wavelet based transient detection unit designed and developed as part of this project is discussed in this paper. The prototype is developed on a low-cost embedded system in order to provide an economical solution. Significance of the developed transient detection unit is also illustrated in this paper by developing applications that ensure a robust and reliable drivetrain with enhanced performance. Firstly, the information gathered from the transient detection unit is used to determine in-vehicle electrical faults. Secondly, this transient detector is used to facilitate the optimization of hybrid energy storage system (battery/ultra-capacitor combination) and to ensure smooth battery operation.

Bearing and Gear Fault Diagnosis Using Adaptive Wavelet Transform of Vibration Signals

The continuous wavelet transform (CWT) is widely used for analyzing non-stationary vibration signals from rotating machines. It is a well known fact that, if the shape of the wavelet function matches adequately with the shape of the burst present in signal at a specific scale and location, a large transform value is obtained, and a low transform value is obtained if the signal and wavelet do not correlate well. It is also worth noticing that each rotating system and its corresponding defect signature are unique in nature. Therefore, the challenge remains in arriving with a robust wavelet function for the wavelet transform technique to solve any specific class of problem in hand. This challenge has been addressed in the present work and in order to understand the defect signature appropriately, the concept of adaptive wavelet design and its implementation for defect identification are discussed. In presence of seeded defects, a random burst from the vibration signal is used for designing the corresponding adaptive wavelet. Similarly, the designed adaptive wavelet is used to compute the CWT coefficients. The CWT coefficients so generated are compared with the standard wavelet based scalograms. The adaptive wavelet transform is shown to be apposite in analyzing the vibration signals and also is corroborated with the acquired experimental data.

Uncertainty relations for sets of self adjoint operators

AbstractUncertainty relations quantify the joint localisation in time respectively space domain and Fourier domain. For that reason they are an important tool in the design of wavelets. Uncertainty relations in higher dimension are most often realized as the tensor product of one dimensional uncertainty relations, whence these uncertainty relations are not invariant under rotations. The property of invariance with respect to rotation is however very desirable for image processing. We will give a new uncertainty relation for sets of self‐adjoint operators that yields the desired invariance in many examples. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solvability of the Zero-Pinning Technique to Orthonormal Wavelet Design

A zero-pinning technique for orthonormal wavelet design proposed by Tay results in a system of linear equations. We first prove the existence and uniqueness to the solution of the linear system. For orthonormal wavelet filters, non-negativity is known to be a necessary condition. However, it is not sufficient. A tau-cycle condition is cited as one in verifying a wavelet filter being orthonormal. Finally, we show that the amplitude of ripples between two successive zeros of the parametric Bernstein polynomials decreases as the distance between these two zeros decreases.

Steerable Pyramids and Tight Wavelet Frames in $L_{2}({\BBR}^{d})$

We present a functional framework for the design of tight steerable wavelet frames in any number of dimensions. The 2-D version of the method can be viewed as a generalization of Simoncelli's steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization. The backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal. The steerable wavelets are obtained by applying a one-to-many mapping (Nth-order generalized Riesz transform) to the primal ones. The shaping of the steerable wavelets is controlled by an M×M unitary matrix (where M is the number of wavelet channels) that can be selected arbitrarily; this allows for a much wider range of solutions than the traditional equiangular configuration (steerable pyramid). We give a complete functional description of these generalized wavelet transforms and derive their steering equations. We describe some concrete examples of transforms, including some built around a Mallat-type multiresolution analysis of L(2)(R(d)), and provide a fast Fourier transform-based decomposition algorithm. We also propose a principal-component-based method for signal-adapted wavelet design. Finally, we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms. The results are in favor of an optimized wavelet design (equalized principal component analysis), which consistently performs best.

SVD properties of orthogonal two-channel filter banks

We study the Singular Value Decomposition (SVD) of the three building blocks of the matrix representation of two-channel orthogonal filter banks. We show the highly structured decomposition of these blocks and derive their mutual relations. We give examples to exploit the SVD properties in the implementation and design of orthogonal filter banks and wavelets.

Spike detection in human muscle sympathetic nerve activity using a matched wavelet approach

AbstractSympathetic nerve recordings associated with blood pressure regulation can be recorded directly using microneurography. A general characteristic of this signal is spontaneous burst activity of spikes (action potentials) separated by silent periods against a background of considerable Gaussian noise. During measurement with electrodes, the raw muscle sympathetic nerve activity (MSNA) signal is amplified, band-pass filtered, rectified and integrated. This integration process removes information regarding action potential content and their discharge properties. This paper proposes a new method for detecting action potentials from the raw MSNA signal to enable investigation of post-ganglionic neural discharge properties.The new method is based on the design of a mother wavelet that is matched to an actual mean action potential template extracted from a real raw MSNA signal. To detect action potentials, the new matched wavelet is applied to the MSNA signal using a continuous wavelet transform following a thresholding procedure and finding of a local maxima that indicates the location of action potentials. The performance of the proposed method versus two previous wavelet-based approaches was evaluated using (1) real MSNA recorded from seven healthy participants and, (2) simulated MSNA. The results show that the new matched wavelet performs better than the previous wavelet-based methods that use a non-matched wavelet in detecting action potentials in the MSNA signal.

Area- and Power-Efficient Design of Daubechies Wavelet Transforms Using Folded AIQ Mapping

In this brief, we present an efficient design of a shared architecture to compute two 8-point Daubechies wavelet transforms. The architecture is based on a two-level folded mapping technique that is developed on the factorization and decomposition of transform matrices exploiting the symmetrical structure. The chip occupies a 2.08-mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> silicon area, runs at 100 MHz, and consumes 4.51 mW of power in 0.18-m CMOS technology.

Algorithm design and implementation of finite wavelet grow tree

Based on grow tree composite model, Finite Field Wavelet Grow Tree (FW-GT) was proposed in this paper. FW-GT is a novel framework to be used in data encryption enhancing data security. It is implemented by replacement operator and wavelet operator. Forward integration and inverse decomposition of FW-GT are performed by replacement, inverse wavelets and its corresponding replacement, wavelet transforms. Replacement operator joined nonlinear factor, wavelet operator completed data transformation between lower dimensional space and higher dimensional space. FW-GT security relies on the difficulty of solving nonlinear equations over finite fields. By using FW-GT, high security of data could be obtained at the cost of low computational complexity. It proved FW-GT algorithm’s correctness in this paper. The experimental result and theory analysis shows the excellent performance of the algorithm.

Design of Regular Wavelets Using a Three-Step Lifting Scheme

We propose structural multidimensional multichannel filter banks with desirable numbers of vanishing moments for the analysis and synthesis banks. For a two-channel filter bank, we use a three-step lifting scheme as opposed to the conventional two-step lifting method in order to provide more symmetry between the analysis and synthesis filters. We show that the resulting filters have more regularity, lower frame bounds ratio, and better frequency selectivity. We also extend our design to a general multichannel framework. Three steps of lifting scheme provides us a degree of freedom which we can benefit from toward a more flexible design.

Swarm Intelligence Approach to Wavelet Design for Hyperspectral Image Classification

Wavelets are known to be a valuable tool for analyzing hyperspectral images. In this letter, we propose to further improve their performance by means of a novel classification-driven design scheme that aims at deriving a wavelet that best represents in terms of between-class discrimination capability the spectral signatures conveyed by a given hyperspectral image. This is achieved by adopting a polyphase representation of the wavelet filter bank and formulating the wavelet optimization problem within a particle-swarm-optimization (PSO) framework. Experimental results show that the proposed wavelet design method outperforms the popular Daubechies wavelets whatever the classifier type adopted in the classification process.

Frequency-Domain Design of Overcomplete Rational-Dilation Wavelet Transforms

The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Q-factor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied. The new wavelet transform can attain higher Q-factors (desirable for processing oscillatory signals) or the same low Q-factor of the dyadic wavelet transform. The new wavelet transform is modestly overcomplete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible 'constant-Q' discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for L2(R). The wavelet can be made to resemble a Gabor function and can hence have good concentration in the time-frequency plane. The construction of the new wavelet transform depends on the judicious use of both the transform's redundancy and the flexibility allowed by frequency-domain filter design.

QRS COMPLEX DETECTION USING OPTIMAL DISCRETE WAVELET

Wavelet transform has emerged as a powerful tool for time frequency analysis of complex nonstationary signals such as the electrocardiogram (ECG) signal. In this paper, the design of good wavelets for cardiac signal is discussed from the perspective of orthogonal filter banks. Optimum wavelet for ECG signal is designed and evaluated based on perfect reconstruction conditions and QRS complex detection. The performance is evaluated by using the ECG records from the MIT-BIH arrhythmia database. In the first step, the filter coefficients (optimum wavelet) is designed by reparametrization of filter coefficients. In the second step, ECG signal is decomposed to three levels using the optimum wavelet and reconstructed. From the reconstructed signal, the range of error signal is calculated and it is compared with the performance of other suitable wavelets already available in the literature. The optimum wavelet gives the maximum error range as 10-14–10-11 which is better than that of other wavelets existing in the literature. In the third step, the baseline wandering is removed from the ECG signal for better detection of QRS complex. The optimum wavelet detects all R peaks of all records. That is using optimum wavelet 100% sensitivity and positive predictions are achieved. Based on the performance, it is confirmed that optimum wavelet is more suitable for ECG signal.

Towards global design of orthogonal filter banks and wavelets

This paper investigates several design issues concerning two-channel conjugate quadrature (CQ) filter banks and orthogonal wavelets. It is well known that optimal designs of CQ filters and wavelets in the least squares (LS) or minimax sense are nonconvex problems and to date only local solutions can be claimed. By virtue of recent progress in global polynomial optimization and a direct design technique for CQ filters, we in this paper present a design strategy that may be viewed as our endeavors towards global solutions for CQ filters. Two design scenarios are considered, namely the least squares designs with vanishing moment (VM) requirement, and equiripple (i.e. minimax) designs with VM requirement. Simulation studies are presented to verify our design concept for both LS and minimax designs of low-order CQ filters, and to evaluate and compare the proposed algorithms with existing design algorithms for high-order CQ filters.

Existence and design of biorthogonal matrix-valued wavelets

In this paper, we study biorthogonal matrix-valued wavelets for analyzing matrix-valued signals based on matrix multiresolution analysis. Firstly, sufficient conditions for the existence of biorthogonal matrix-valued scaling function are established in terms of two-scale matrix symbols. Then we focus on the construction of biorthogonal matrix-valued wavelet. Two designs based on factorizations of biorthogonal two-scale matrix symbol are presented. In particular, explicit constructing formulations for biorthogonal matrix-valued wavelets are given. With these formulations, highpass filters { G k } and { G ̃ k } of biorthogonal matrix-valued wavelets can be given explicitly by lowpass filters { H k } and { H ̃ k } of their corresponding biorthogonal scaling functions. Finally, according to our designs, two examples of two-scale matrix filters are given.

Design and properties of vector-valued wavelets associated with an orthogonal vector-valued scaling function

The notion of vector-valued wavelets associated with an orthogonal vector-valued scaling function with 3-scale is introduced. The existence of orthogonal vector-valued wavelets is discussed and a necessary and sufficient condition is presented by means of the theory of vector-valued multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a class of orthogonal vector-valued wavelets with compact support is proposed. The concept of vector-valued wavelet packets is introduced and their properties are characterized by virtue of operator theory, time–frequency analysis method. Moreover, it is shown how to construct various orthonormal bases of space L 2 ( R , C μ ) ( 2 ⩽ μ ∈ Z ) from these wavelet packets. Relation to some physical theories such as fractal theory and E-infinity theory is also discussed.

Practical wavelet design on the sphere

This paper considers the design of isotropic analysis functions on the sphere which are perfectly limited in the spectral domain and optimally localized in the spatial domain. This is motivated by the need of localized analysis tools in domains where the data is lying on the sphere, e.g. the science of the Cosmic Microwave Background. Our construction is derived from the localized frames introduced by F. Narcowich et al. [F. Narcowich, P. Petrushev, J. Ward, Localized tight frames on spheres, SIAM J. Math. Anal. 38 (2) (2006) 574–594]. The analysis frames are optimized for given applications and compared numerically using various criteria.