Showing papers on "Continuous wavelet transform published in 1999"

Efficient time series matching by wavelets

Abstract: Time series stored as feature vectors can be indexed by multidimensional index trees like R-Trees for fast retrieval. Due to the dimensionality curse problem, transformations are applied to time series to reduce the number of dimensions of the feature vectors. Different transformations like Discrete Fourier Transform (DFT) Discrete Wavelet Transform (DWT), Karhunen-Loeve (KL) transform or Singular Value Decomposition (SVD) can be applied. While the use of DFT and K-L transform or SVD have been studied on the literature, to our knowledge, there is no in-depth study on the application of DWT. In this paper we propose to use Haar Wavelet Transform for time series indexing. The major contributions are: (1) we show that Euclidean distance is preserved in the Haar transformed domain and no false dismissal will occur, (2) we show that Haar transform can outperform DFT through experiments, (3) a new similarity model is suggested to accommodate vertical shift of time series, and (4) a two-phase method is proposed for efficient n-nearest neighbor query in time series databases.

Image processing with complex wavelets

Abstract: We first review how wavelets may be used for multi–resolution image processing, describing the filter–bank implementation of the discrete wavelet transform (DWT) and how it may be extended via separable filtering for processing images and other multi–dimensional signals. We then show that the condition for inversion of the DWT (perfect reconstruction) forces many commonly used wavelets to be similar in shape, and that this shape produces severe shift dependence (variation of DWT coefficient energy at any given scale with shift of the input signal). It is also shown that separable filtering with the DWT prevents the transform from providing directionally selective filters for diagonal image features. Complex wavelets can provide both shift invariance and good directional selectivity, with only modest increases in signal redundancy and computation load. However, development of a complex wavelet transform (CWT) with perfect reconstruction and good filter characteristics has proved difficult until recently. We now propose the dual–tree CWT as a solution to this problem, yielding a transform with attractive properties for a range of signal and image processing applications, including motion estimation, denoising, texture analysis and synthesis, and object segmentation.

Multiridge detection and time-frequency reconstruction

Abstract: The ridges of the wavelet transform, the Gabor transform, or any time-frequency representation of a signal contain crucial information on the characteristics of the signal. Indeed, they mark the regions of the time-frequency plane where the signal concentrates most of its energy. We introduce a new algorithm to detect and identify these ridges. The procedure is based on an original form of Markov chain Monte Carlo algorithm especially adapted to the present situation. We show that this detection algorithm is especially useful for noisy signals with multiridge transforms. It is a common practice among practitioners to reconstruct a signal from the skeleton of a transform of the signal (i.e., the restriction of the transform to the ridges). After reviewing several known procedures, we introduce a new reconstruction algorithm, and we illustrate its efficiency on speech signals and its robustness and stability on chirps perturbed by synthetic noise at different SNRs.

Determination of interferometer phase distributions by use of wavelets.

Abstract: A new technique for directly extracting phase gradients from two-dimensional (2-D) interferometer fringe data is presented. One finds the gradients by tracking the maximum modulus of the continuous wavelet transform of the fringe data and the phase distribution that is obtained, with a small error, by integration. Problems associated with phase unwrapping are thereby avoided. The technique is compared with standard methods, and excellent agreement is found. In common with Fourier-transform methods, the technique is capable of extracting the full 2-D phase distribution from a single image.

An introduction to wavelets through linear algebra

Abstract: Preface Acknowledgments Prologue: Compression of the FBI Fingerprint Files 1 Background: Complex Numbers and Linear Algebra 1.1 Real Numbers and Complex Numbers 1.2 Complex Series, Euler's Formula, and the Roots of Unity 1.3 Vector Spaces and Bases 1.4 Linear Transformations, Matrices, and Change of Basis 1.5 Diagonalization of Linear Transformations and Matrices 1.6 Inner Products, Orthonormal Bases, and Unitary Matrices 2 The Discrete Fourier Transform 2.1 Basic Properties of the Discrete Fourier Transform 2.2 Translation-Invariant Linear Transformations 2.3 The Fast Fourier Transform 3 Wavelets on $bZ_N$ 3.1 Construction of Wavelets on $bZ_N$: The First Stage 3.2 Construction of Wavelets on $bZ_N$: The Iteration Step 3.3 Examples and Applications 4 Wavelets on $bZ$ 4.1 $\ell ^2(bZ)$ 4.2 Complete Orthonormal Sets in Hilbert Spaces 4.3 $L^2([-\pi ,\pi ))$ and Fourier Series 4.4 The Fourier Transform and Convolution on $\ell ^2(bZ)$ 4.5 First-Stage Wavelets on $bZ$ 4.6 The Iteration Step for Wavelets on $bZ$ 4.7 Implementation and Examples 5 Wavelets on $bR$ 5.1 $L^2(bR)$ and Approximate Identities 5.2 The Fourier Transform on $bR$ 5.3 Multiresolution Analysis and Wavelets 5.4 Construction of Multiresolution Analyses 5.5 Wavelets with Compact Support and Their Computation 6 Wavelets and Differential Equations 6.1 The Condition Number of a Matrix 6.2 Finite Difference Methods for Differential Equations 6.3 Wavelet-Galerkin Methods for Differential Equations Bibliography Index

Fourier and Wavelet Analysis

Abstract: 1 Metrie and Normed Spaces.- 1.1 Metrie Spaces.- 1.2 Normed Spaces.- 1.3 Inner Product Spaces.- 1.4 Orthogonality.- 1.5 Linear Isometry.- 1.6 Holder and Minkowski Inequalities Lpand lpSpaces..- 2 Analysis.- 2.1 Balls.- 2.2 Convergence and Continuity.- 2.3 Bounded Sets.- 2.4 Closure and Closed Sets.- 2.5 Open Sets.- 2.6 Completeness.- 2.7 Uniform Continuity.- 2.8 Compactness.- 2.9 Equivalent Norms.- 2.10 Direct Sums.- 3 Bases.- 3.1 Best Approximation.- 3.2 Orthogonal Complements and the Projection Theorem.- 3.3 Orthonormal Sequences.- 3.4 Orthonormal Bases.- 3.5 The Haar Basis.- 3.6 Unconditional Convergence.- 3.7 Orthogonal Direct Sums.- 3.8 Continuous Linear Maps.- 3.9 Dual Spaces.- 3.10 Adjoints.- 4 Fourier Series.- 4.1 Warmup.- 4.2 Fourier Sine Series and Cosine Series.- 4.3 Smoothness.- 4.4 The Riemann-Lebesgue Lemma.- 4.5 The Dirichlet and Fourier Kernels.- 4.6 Point wise Convergence of Fourier Series.- 4.7 Uniform Convergence.- 4.8 The Gibbs Phenomenon.- 4.9 - Divergent Fourier Series.- 4.10 Termwise Integration.- 4.11 Trigonometric vs. Fourier Series.- 4.12 Termwise Differentiation.- 4.13 Dido's Dilemma.- 4.14 Other Kinds of Summability.- 4.15 Fejer Theory.- 4.16 The Smoothing Effect of (C, 1) Summation.- 4.17 Weierstrass's Approximation Theorem.- 4.18 Lebesgue's Pointwise Convergence Theorem.- 4.19 Higher Dimensions.- 4.20 Convergence of Multiple Series.- 5 The Fourier Transform.- 5.1 The Finite Fourier Transform.- 5.2 Convolution on T.- 5.3 The Exponential Form of Lebesgue's Theorem.- 5.4 Motivation and Definition.- 5.5 Basics/Examplesv.- 5.6 The Fourier Transform and Residues.- 5.7 The Fourier Map.- 5.8 Convolution on R.- 5.9 Inversion, Exponential Form.- 5.10 Inversion, Trigonometric Form.- 5.11 (C, 1) Summability for Integrals.- 5.12 The Fejer-Lebesgue Inversion Theorem.- 5.13 Convergence Assistance.- 5.14 Approximate Identity.- 5.15 Transforms of Derivatives and Integrals.- 5.16 Fourier Sine and Cosine Transforms.- 5.17 Parseval's Identities.- 5.18 The L2Theory.- 5.19 The Plancherel Theorem.- 5.20 Point wise Inversion and Summability.- 5.21 - Sampling Theorem.- 5.22 The Mellin Transform.- 5.23 Variations.- 6 The Discrete and Fast Fourier Transforms.- 6.1 The Discrete Fourier Transform.- 6.2 The Inversion Theorem for the DFT.- 6.3 Cyclic Convolution.- 6.4 Fast Fourier Transform for N=2k.- 6.5 The Fast Fourier Transform for N=RC.- 7 Wavelets.- 7.1 Orthonormal Basis from One Function.- 7.2 Multiresolution Analysis.- 7.3 Mother Wavelets Yield Wavelet Bases.- 7.4 From MRA to Mother Wavelet.- 7.5 Construction of - Scaling Function with Compact Support.- 7.6 Shannon Wavelets.- 7.7 Riesz Bases and MRAs.- 7.8 Franklin Wavelets.- 7.9 Frames.- 7.10 Splines.- 7.11 The Continuous Wavelet Transform.

Wavelet-transform-based algorithm for harmonic analysis of power system waveforms

Abstract: This paper develops an approach based on wavelet transform for the evaluation of harmonic contents of power system waveforms. The new algorithm can simultaneously identify all harmonics including integer, noninteger and subharmonics. In the first step of the approach, the frequency spectrum of the waveform is decomposed into subbands using discrete wavelet packet transform filter banks. In the second step, continuous wavelet transform is applied to nonzero subbands to evaluate the harmonic contents. In the first step, there is the problem of images due to down samplings and up samplings of the waveform. A method for alleviating this image problem is developed in the paper. Methods are developed to accurately quantify harmonics frequency amplitude and phase. The approach is validated by its application to synthesised waveforms and to power system waveforms measured in the Western Australia system. It is found to be powerful and suitable for practical use.

New signal processing tools applied to power quality analysis

Abstract: This paper deals with the comparison of new signal processing tools for power quality analysis. Three new signal processing techniques are considered: the continuous wavelet transform, the multiresolution analysis and the quadratic transform. Their theoretical behaviours are investigated using the basic theory of the Fourier transform. Then, examples of the four most frequent disturbances met in the power system are chosen. Finally, each kind of electrical disturbance is analyzed with example representing each tool. A qualitative comparison of results shows the advantages and drawbacks of each new signal processing technique applied to voltage disturbance analysis. The continuous wavelet transform appears to be a reliable method for detecting and measuring voltage sags, flicker and transients in power quality analysis.

Image compression using wavelets

Abstract: The discrete wavelet transform (DWT) represents images as a sum of wavelet functions (wavelets) on different resolution levels. The basis for the wavelet transform can be composed of any function that satisfies requirements of multiresolution analysis. It means that there exists a large selection of wavelet families depending on the choice of wavelet function. The choice of wavelet family depends on the application. In image compression application this choice depends on image content. This paper provides fundamentals of wavelet based image compression. The options for wavelet image representations are tested. The results of image quality measurements for different wavelet functions, image contents, compression ratios and resolutions are given.

Identification of sources of potential fields with the continuous wavelet transform: Basic theory

Abstract: The continuous wavelet transform is used to analyze potential fields and to locate their causative sources. A particular class of wavelets is introduced which remains invariant under the action of the upward continuation operator in potential field theory. These wavelets make the corresponding wavelet transforms easy to analyze and the sources' parameters (horizontal location, depth, multipolar nature, and strength) simple to estimate. Practical issues (effects of noise, choice of the analyzing wavelet, etc.) are addressed. A field data example corresponding to a near-surface magnetic survey is discussed. Applications to the high-resolution aeromagnetic survey of French Guyana will be discussed in the next paper of the series.

Multiple wavelet threshold estimation by generalized cross validation for images with correlated noise

Abstract: Denoising algorithms based on wavelet thresholding replace small wavelet coefficients by zero and keep or shrink the coefficients with absolute value above the threshold. The optimal threshold minimizes the error of the result as compared to the unknown, exact data. To estimate this optimal threshold, we use generalized cross validation. This procedure does not require an estimation for the noise energy. Originally, this method assumes uncorrelated noise. In this paper, we describe how we can extend it to images with correlated noise.

Speed estimation in wireless systems using wavelets

Abstract: A new technique is described for estimating the speed of a mobile station in a wireless system. The mobile speed maps the characteristic spatial scale of the received signal into a characteristic temporal scale. The continuous wavelet transform tracks changes in the temporal scale to estimate the mobile speed as a function of time. This technique requires neither knowledge of the average received power of the nonstationary signal nor adaptation of a temporal observation window, in contrast to other speed estimators given in the literature. Simulations show the tracking of a variable speed profile.

The use of the wavelet transform to describe embolic signals.

Abstract: A number of methods to detect cerebral emboli and differentiate them from artefacts using Doppler ultrasound have been described in the literature. In most, Fourier transform-based (FT) spectral analysis has been used. The FT is not ideally suited to analysis of short-duration embolic signals due to an inherent trade-off between temporal and frequency resolution. An alternative approach that might be expected to describe embolic signals well is the wavelet transform. Wavelets are ideally suited for the analysis of sudden short-duration signal changes. Therefore, we have implemented a wavelet-based analysis and compared the results of this with a conventional FFT-based analysis. The temporal resolution, as measured by the half-width maximum, was significantly better for the continuous wavelet transform (CWT), mean (SD) 8.40 (8.82) ms, compared with the 128-point FFT, 12.92 (9.70) ms, and 64-point FFT, 10.80 (5.69) ms. Time localization of the CWT for the embolic signal was also significantly better than the FFT. The wavelet transform appears well suited to the analysis of embolic signals offering superior time resolution and time localization to the FFT.

On-line detection of the breakage of small diameter drills using current signature wavelet transform

Abstract: This paper presents on-line tool breakage detection of small diameter drills by monitoring the AC servo motor current. The continuous wavelet transform was used to decompose the spindle AC servo motor current signal and the discrete wavelet transform was used to decompose the feed AC servo motor current signal in time–frequency domain. The tool breakage features were extracted from the decomposed signals. Experimental results show that the proposed monitoring system possessed an excellent on-line capability; in addition, it had a low sensitivity to change of the cutting conditions and high success rate for the detection of the breakage of small diameter drills.

Revealing a lognormal cascading process in turbulent velocity statistics with wavelet analysis

Abstract: We use the continuous wavelet transform to extract a cascading process from experimental turbulent velocity signals. We mainly investigate various statistical quantities such as the singularity spe...

Simulation of nonGaussian long-range-dependent traffic using wavelets

Abstract: In this paper, we develop a simple and powerful multiscale model for the synthesis of nonGaussian, long-range dependent (LRD) network traffic. Although wavelets effectively decorrelate LRD data, wavelet-based models have generally been restricted by a Gaussianity assumption that can be unrealistic for traffic. Using a multiplicative superstructure on top of the Haar wavelet transform, we exploit the decorrelating properties of wavelets while simultaneously capturing the positivity and “spikiness” of nonGaussian traffic. This leads to a swift O(N) algorithm for fitting and synthesizing N-point data sets. The resulting model belongs to the class of multifractal cascades, a set of processes with rich statistical properties. We elucidate our model’s ability to capture the covariance structure of real data and then fit it to real traffic traces. Queueing experiments demonstrate the accuracy of the model for matching real data. Our results indicate that the nonGaussian nature of traffic has a significant effect on queuing.

Numerical solution of the identification problem for the attenuated Radon transform

Abstract: The attenuated Radon transform serves as a mathematical tool for single-photon emission computerized tomography (SPECT). The identification problem for the attenuated Radon transform is to find the attenuation coefficient, which is a parameter of the transform, from the values of the transform alone. Previous attempts to solve this problem used range theorems for the continuous attenuated/exponential Radon transform. We consider a matrix representation of the transform and formulate the corresponding discrete consistency conditions in the form of the orthogonal projection of the data vector onto the orthogonal complement of the column space of the matrix. The singular value decomposition is applied to compute the orthogonal projector and its Frechet derivative. The numerical algorithm suggested is based on the Newton method with the Tikhonov regularization. Results of numerical experiments and inversion of the measured SPECT data are considered.

Wavelet analysis of time-series: coherent structures, chaos and noise

Abstract: The continuous and orthogonal wavelet transforms are used to analyze time-series data. The analysis involves signal decomposition into scale components using both Grossman–Morlet and Daubechies type wavelets. A number of simulated and experimental data vectors exhibiting different types of coherent structures, chaos and noise is analyzed. The study shows that wavelet analysis provides a unifying framework for the description of many phenomena in time-series.

An approach using wavelet transform for fault type identification in digital relaying

Abstract: This paper proposes the use of wavelet transform (WT) in power transmission lines for identifying fault types. WT multiresolution properties are quite adequate for detection of fast changes contained in the disturbed signal. Wavelet decomposition associated with modal components have shown to be an excellent alternative for quick identification of faulted phases. Digitized phase voltage and/or current signals are fed to wavelet filters. The coefficients obtained are then transformed by a modified modal transformation matrix. From the resulting signals, the energy is measured with short time intervals. With an appropriately chosen threshold the fault type is identified. EMTP simulations are used to test and validate the proposed methodology. The obtained results are encouraging and the proposed technique requires a low computational complexity, allowing it to be used as part of a high speed protective relay.

A survey of mixed transform techniques for speech and image coding

Abstract: The goal of transform based coding is to build a representation of a signal using the smallest number of weighted basis functions possible, while maintaining the ability to reconstruct the signal with adequate fidelity. Mixed transform techniques, which employ subsets of non-orthogonal basis functions chosen from two or more transform domains, have been shown to consistently yield more efficient signal representations than those based on one transform. This paper provides a survey of mixed transform techniques, also known as multitransforms or mixed basis representations, which have been developed for speech and image coding.

Invariant Image Retrieval Using Wavelet Maxima Moment

Abstract: Wavelets have been shown to be an effective analysis tool for image indexing due to the fact that spatial information and visual features of images could be well captured in just a few dominant wavelet coefficients. A serious problem with current wavelet-based techniques is in the handling of affine transformations in the query image. In this work, to cure the problem of translation variance with wavelet basis transform while keeping a compact representation, the wavelet transform modulus maxima is employed. To measure the similarity between wavelet maxima representations, which is required in the context of image retrieval systems, the difference of moments is used. As a result, each image is indexed by a vector in the wavelet maxima moment space. Those extracted features are shown to be robust in searching for objects independently of position, size, orientation and image background.

Spline functions and the theory of wavelets

Abstract: Spline Functions: Introduction and summary by H. Brunner Radial extensions of vertex data by L. P. Bos and D. Holland The use of splines in the numerical solutions of differential and Volterra integral equations by H. Brunner On best error bounds for deficient splines by F. Dubeau and J. Savoie Optimal error bounds for spline interpolation on a uniform partition by F. Dubeau and J. Savoie Modelization of flexible objects using constrained optimization and B-spline surfaces by J.-P. Dussault and N. Pfister New control polygons for polynomial curves by J. C. Fiorot and P. Jeannin Splines in approximation and differential operators: $(m,\ell,s)$ interpolating-spline by A. Le Mehaute and A. Bouhamidi New families of B-splines on uniform meshes of the plane by P. Sablonniere Theory of Wavelets: Introduction and summary by S. Jaffard Analysis of Hermite-interpolatory subdivision schemes by N. Dyn and D. Levin Some directional microlocal classes defined using wavelet transforms by M. Holschneider Nonseparable biorthogonal wavelet bases of $L^2(\mathbb R^n)$ by A. Karoui and R. Vaillancourt Local bases: Theory and applications by J. Kovacevic and R. Bernardini On the $L^p$-Lipschitz exponents of the scaling functions by K.-S. Lau and M.-F. Ma Robust speech and speaker recognition using instantaneous frequencies and amplitudes obtained with wavelet-derived synchrosqueezing measures by S. Maes Extensions of the Heisenberg group and wavelet analysis in the plane by E. Schulz and K. F. Taylor Wavelets in Physics: Introduction and summary by A. Arneodo Coherent states and quantization by S. Twareque Ali Wavelets in molecular and condensed-matter physics by J.-P. Antoine Wavelets in atomic physics by J.-P. Antoine, Ph. Antoine, and B. Piraux The wavelet $\epsilon$-expansion and Hausdorff dimension by G. Battle Modelling the coupling between small and large scales in the Kuramoto-Sivashinsky equation by J. Elezgaray, G. Berkooz, and P. Holmes Continuous wavelet transform analysis of one-dimensional quantum ground states by C. R. Handy and R. Murenzi Oscillating singularities and fractal functions by A. Arneodo, E. Bacry, S. Jaffard, and J. F. Muzy Splines and Wavelets in Statistics: Introduction and summary by B. Macgibbon Wavelet estimators for change-point regression models by A. Antoniadis Wavelet thresholding for non (necessarily) Guassian noise: A preliminary report by R. Averkamp and C. Houdre Deslauries-Dubuc: Ten years after by D. L. Donoho and T. P. Y. Yu Some theory for $L$-spline smoothing by J. O. Ramsay and N. Heckman Spectral representation and estimation for locally stationary wavelet processes by R. von Sachs, G. P. Nason, and G. Kroisandt.