Showing papers on "Wavelet published in 1993"

Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition

Abstract: We describe a recursive algorithm to compute representations of functions with respect to nonorthogonal and possibly overcomplete dictionaries of elementary building blocks e.g. affine (wavelet) frames. We propose a modification to the matching pursuit algorithm of Mallat and Zhang (1992) that maintains full backward orthogonality of the residual (error) at every step and thereby leads to improved convergence. We refer to this modified algorithm as orthogonal matching pursuit (OMP). It is shown that all additional computation required for the OMP algorithm may be performed recursively. >

Wavelets and Operators

Abstract: Over the last two years, wavelet methods have shown themselves to be of considerable use to harmonic analysts and, in particular, advances have been made concerning their applications. The strength of wavelet methods lies in their ability to describe local phenomena more accurately than a traditional expansion in sines and cosines can. Thus, wavelets are ideal in many fields where an approach to transient behaviour is needed, for example, in considering acoustic or seismic signals, or in image processing. Yves Meyer stands the theory of wavelets firmly upon solid ground by basing his book on the fundamental work of Calderon, Zygmund and their collaborators. For anyone who would like an introduction to wavelets, this book will prove to be a necessary purchase.

Texture analysis and classification with tree-structured wavelet transform

Abstract: A multiresolution approach based on a modified wavelet transform called the tree-structured wavelet transform or wavelet packets is proposed. The development of this transform is motivated by the observation that a large class of natural textures can be modeled as quasi-periodic signals whose dominant frequencies are located in the middle frequency channels. With the transform, it is possible to zoom into any desired frequency channels for further decomposition. In contrast, the conventional pyramid-structured wavelet transform performs further decomposition in low-frequency channels. A progressive texture classification algorithm which is not only computationally attractive but also has excellent performance is developed. The performance of the present method is compared with that of several other methods. >

An Introduction to Random Vibrations, Spectral and Wavelet Analysis

Abstract: 1. Introduction To Probability Distributions And Averages. 2. Joint Probability Distributions, Ensemble Averages. 3. Correlation. 4. Fourier Analysis. 5. Spectral Density. 6. Excitation - Response Relations For Linear Systems. 7. Transmission Of Random Vibration. 8. Statistics Of Narrow Band Processes. 9. Accuracy Of Measurements. 10. Digital Spectral Analysis I: Discrete Fourier Transforms. 11. Digital Spectral Analysis II: Windows And Smoothing. 12. The Fast Fourier Transform. 14. Application Notes. 15. Multi-Dimensional Spectral Analysis. 16. Response Of Continuous Linear Systems To Stationary Random Excitation. 17. Discrete Wavelet Analysis.

Texture classification by wavelet packet signatures

Abstract: This correspondence introduces a new approach to characterize textures at multiple scales. The performance of wavelet packet spaces are measured in terms of sensitivity and selectivity for the classification of twenty-five natural textures. Both energy and entropy metrics were computed for each wavelet packet and incorporated into distinct scale space representations, where each wavelet packet (channel) reflected a specific scale and orientation sensitivity. Wavelet packet representations for twenty-five natural textures were classified without error by a simple two-layer network classifier. An analyzing function of large regularity (D/sub 20/) was shown to be slightly more efficient in representation and discrimination than a similar function with fewer vanishing moments (D/sub 6/) In addition, energy representations computed from the standard wavelet decomposition alone (17 features) provided classification without error for the twenty-five textures included in our study. The reliability exhibited by texture signatures based on wavelet packets analysis suggest that the multiresolution properties of such transforms are beneficial for accomplishing segmentation, classification and subtle discrimination of texture. >

Best wavelet packet bases in a rate-distortion sense

Abstract: A fast rate-distortion (R-D) optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented. The scheme guarantees operation on the convex hull of the operational R-D curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity. Applications for this coding technique include R. Coefman et al.'s (Yale Univ., 1990) generalized multiresolution wavelet packet decomposition, iterative subband coders, and quadtree structures. Applications to image processing involving wavelet packets as well as discrete cosine transform (DCT) quadtrees are presented. >

An introduction to wavelet analysis in oceanography and meteorology - With application to the dispersion of Yanai waves

Abstract: Wavelet analysis is a relatively new technique that is an important addition to standard signal analysis methods. Unlike Fourier analysis that yields an average amplitude and phase for each harmonic in a dataset, the wavelet transform produces an instantaneous estimate or local value for the amplitude and phase of each harmonic. This allows detailed study of nonstationary spatial or time-dependent signal characteristics. The wavelet transform is discussed, examples are given, and some methods for preprocessing data for wavelet analysis are compared. By studying the dispersion of Yanai waves in a reduced gravity equatorial model, the usefulness of the transform is demonstrated. The group velocity is measured directly over a finite range of wavenumbers by examining the time evolution of the transform. The results agree well with linear theory at higher wavenumber but the measured group velocity is reduced at lower wavenumbers, possibly due to interaction with the basin boundaries.

Analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations

Abstract: A representation of a class of feedforward neural networks in terms of discrete affine wavelet transforms is developed. It is shown that by appropriate grouping of terms, feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings. It is shown that the wavelet transform formalism provides a mathematical framework within which it is possible to perform both analysis and synthesis of feedforward networks. For the purpose of analysis, the wavelet formulation characterizes a class of mappings which can be implemented by feedforward networks as well as reveals an exact implementation of a given mapping in this class. Spatio-spectral localization properties of wavelets can be exploited in synthesizing a feedforward network to perform a given approximation task. Two synthesis procedures based on spatio-spectral localization that reduce the training problem to one of convex optimization are outlined. >

VLSI architectures for discrete wavelet transforms

Abstract: A folded architecture and a digit-serial architecture are proposed for implementation of one- and two-dimensional discrete wavelet transforms. In the one-dimensional folded architecture, the computations of all wavelet levels are folded to the same low-pass and high-pass filters. The number of registers in the folded architecture is minimized by the use of a generalized life time analysis. The converter units are synthesized with a minimum number of registers using forward-backward allocation. The advantage of the folded architecture is low latency and its drawbacks are increased hardware area, less than 100% hardware utilization, and the complex routing and interconnection required by the converters used. These drawbacks are eliminated in the alternate digit-serial architecture at the expense of an increase in the system latency and some constraints on the wordlength. In latency-critical applications, the use of the folded architecture is suggested. If latency is not so critical, the digit-serial architecture should be used. The use of a combined folded and digit-serial architecture is proposed for implementation of two-dimensional discrete wavelet transforms. >

Wavelet transforms and atmopsheric turbulence.

Abstract: Wavelet cross spectra and cross scalograms are used to analyze the time-scale structure of bivariate turbulence data from the boundary layer over the ocean. The cross scalogram for the streamwise and vertical turbulent velocity components shows a highly intermittent pattern with significant contributions of opposite signs appearing at two specific scales, ∼60 m and ∼2 km, believed to be related to small-scale turbulent mixing and large-scale secondary flow in the boundary layer

Singularity spectrum of fractal signals from wavelet analysis: Exact results

Abstract: The multifractal formalism for singular measures is revisited using the wave transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimension D(h) of the set of singularities of Hoelder exponent h can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of order n, provided one uses an analyzing wavelet that possesses at least N > n vanishing moments. However, it is shown that a C[infinity] behavior generally induces a phase transition in the D(h) singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the number N of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numericalmore » examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.« less

Theory of regular M-band wavelet bases

Abstract: Orthonormal M-band wavelet bases have been constructed and applied by several authors. This paper makes three main contributions. First, it generalizes the minimal length K-regular 2-band wavelets of Daubechies (1988) to the M-band case by deriving explicit formulas for K-regular M-band scaling filters. Several equivalent characterizations of K-regularity are given and their significance explained. Second, two approaches to the construction of the (M-1) wavelet filters and associated wavelet bases are described; one relies on a state-space characterization with a novel technique to obtain the unitary wavelet filters; the other uses a factorization approach. Third, this paper gives a set of necessary and sufficient condition on the M-band scaling filter for it to generate an orthonormal wavelet basis. The conditions are very similar to those obtained by Cohen (1990) and Lawton (1990) for 2-band wavelets. >

Wavelet transforms versus Fourier transforms

Abstract: This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The «wavelet transform» maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them - always indirectly and recursively

Wavelet-based representations for the 1/f family of fractal processes : Fractals in electrical engineering

Abstract: The 1/f family of fractal random processes model a truly extraordinary range of natural and man-made phenomena, many of which arise in a variety of signal processing scenarios. Yet despite their apparent importance, the lack of convenient representations for 1/f processes has, at least until recently, strongly limited their popularity. In this paper, we demonstrate that 1/f processes are, in a broad sense, optimally represented in terms of orthonormal wavelet bases. Specifically, via a useful frequency domain characterization for 1/f processes, we develop the wavelet expansion's role as a Karhunen-Loeve-type expansion for 1/f processes. As an illustration of potential, we show that wavelet based representations naturally lead to highly efficient solutions to some fundamental detection and estimation problems involving 1/f processes

On the use of wavelet expansions in the method of moments (EM scattering)

Abstract: An approach which incorporates the theory of wavelet transforms in method-of-moments solutions for electromagnetic wave interaction problems is presented. The unknown field or response is expressed as a twofold summation of shifted and dilated forms of a properly chosen basis function, which is often referred to as the mother wavelet. The wavelet expansion can adaptively fit itself to the various length scales associated with the scatterer by distributing the localized functions near the discontinuities and the more spatially diffused ones over the smooth expanses of the scatterer. The approach is thus best suited for the analysis of scatterers which contain a broad spectrum of length scales ranging from a subwavelength to several wavelengths. Using a Galerkin method and subsequently applying a threshold procedure, the moment-method matrix is rendered sparsely populated. The structure of the matrix reveals the localized scale-fitting distribution long before the matrix equation is solved. The performance of the proposed discretization scheme is illustrated by a numerical study of electromagnetic coupling through a double-slot aperture. >

Wavelet transforms versus Fourier transforms

Abstract: This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The ``wavelet transform'' maps each $f(x)$ to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them --- always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including high-definition television). So far the Fourier Transform --- or its 8 by 8 windowed version, the Discrete Cosine Transform --- is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory.

Wavelet-based representations for the 1/f family of fractal processes

Abstract: It is demonstrated that 1/f fractal processes are, in a broad sense, optimally represented in terms of orthonormal wavelet bases Specifically, via a useful frequency-domain characterization for 1/f processes, the wavelet expansion's role as a Karhunen-Loeve-type expansion for 1/f processes is developed As an illustration of potential, it is shown that wavelet-based representations naturally lead to highly efficient solutions to some fundamental detection and estimation problems involving 1/f processes >

Harmonic wavelet analysis

Abstract: A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4$\pi $x)-exp(i2$\pi $x)}/i2$\pi $x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.

Non-separable bidimensional wavelet bases

Abstract: We build orthonormal and biorthogonal wavelet bases of L2(R2) with dilation matrices of determinant 2. As for the one dimensional case, our construction uses a scaling function which solves a two-scale difference equation associated to a FIR filter. Our wavelets are generated from a single compactly supported mother function. However, the regularity of these functions cannot be derived by the same approach as in the one dimensional case. We review existing techniques to evaluate the regularity of wavelets, and we introduce new methods which allow to estimate the smoothness of non-separable wavelets and scaling functions in the most general situations. We illustrate these with several examples

On the Construction of Multivariate (pre) Wavelets

Abstract: A new approach for the construction of wavelets and prewavelets on R d from multiresolution is presented. The method uses only properties of shift- invariant spaces and orthogonal projectors from L2(R d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multi- resolution. We present a new approach for the construction of wavelets and prewavelets on R a from multiresolution. Our method, which is based on our earlier work (BDR), (BDR1), uses only properties of shift-invariant spaces and orthogonal projectors from L2(R a) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows us to derive in a simple way previous constructions of wavelets, as well as new constructions, and to settle completely certain basic questions about multiresolution. A univariate function ~ E L2(R ) is called an orthogonal wavelet if its normalized, translated dilates ~kj.k:= 2k/2r j, k rZ, form an orthonormal basis for L~(R). In other words, this system is complete and satisfies the orthogonality conditions

Wavelet radiosity

Abstract: Radiosity methods have been shown to be an effective means to solve the global illumination problem in Lambertian diffuse environments. These methods approximate the radiosity integral equation by projecting the unknown radiosity function into a set of basis functions with limited support resulting in a set of n linear equations where n is the number of discrete elements in the scene. Classical radiosity methods required the evaluation of n interaction coefficients. Efforts to reduce the number of required coefficients without compromising error bounds have focused on raising the order of the basis functions, meshing, accounting for discontinuities, and on developing hierarchical approaches, which have been shown to reduce the required interactions to O(n). In this paper we show that the hierarchical radiosity formulation is an instance of a more general set of methods based on wavelet theory. This general framework offers a unified view of both higher order element approaches to radiosity and the hierarchical radiosity methods. After a discussion of the relevant theory, we discuss a new set of linear time hierarchical algorithms based on wavelets such as the multiwavelet family and a flatlet basis which we introduce. Initial results of experimentation with these basis sets are demonstrated and discussed. CR

Low bit rate transparent audio compression using adapted wavelets

Abstract: Describes a novel wavelet based audio synthesis and coding method. The method uses optimal adaptive wavelet selection and wavelet coefficients quantization procedures together with a dynamic dictionary approach. The adaptive wavelet transform selection and transform coefficient bit allocation procedures are designed to take advantage of the masking effect in human hearing. They minimize the number of bits required to represent each frame of audio material at a fixed distortion level. The dynamic dictionary greatly reduces statistical redundancies in the audio source. Experiments indicate that the proposed adaptive wavelet selection procedure by itself can achieve almost transparent coding of monophonic compact disk (CD) quality signals (sampled at 44.1 kHz) at bit rates of 64-70 kilobits per second (kb/s). The combined adaptive wavelet selection and dynamic dictionary coding procedures achieve almost transparent coding of monophonic CD quality signals at bit rates of 48-66 kb/s. >

Using the refinement equation for evaluating integrals of wavelets

Abstract: The Wavelet Galerkin Method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets This paper studies the problem from the point of view of stationary subdivision schemes One of the main results is to identify these integrals as components of the unique solution of a certain eigenvector-moment problem associated with the coefficients of the refinement equation Asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

Detection of turbulent coherent motions in a forest canopy part I: Wavelet analysis

Abstract: Turbulence measurements performed at high frequencies yield data revealing intermittent and multi-scale processes Analysing time series of turbulent variables thus requires extensive numerical treatment capable, for instance, of performing pattern recognition This is particularly important in the case of the atmospheric surface layer and specifically in the vicinity of plant canopies, where largescale coherent motions play a major role in the dynamics of turbulent transport processes In this paper, we examine the ability of the recently developedwavelet transform to extract information on turbulence structure from time series of wind velocities and scalars It is introduced as a local transform performing a time-frequency representation of a given signal by a specific wavelet function; unlike the Fourier transform, it is well adapted to studying non-stationary signals After the principles and the most relevant mathematical properties of wavelet functions and transform are given, we present various applications of relevance for our purpose: determination of time-scales, data reconstruction and filtering, and jump detection Several wavelet functions are inter-compared, using simple artificially generated data presenting large-scale features similar to those observed over plant canopies Their respective behaviour in the time-frequency domain leads us to assign a specific range of applications for each

Volume data and wavelet transforms

Abstract: The application of 3D orthogonal wavelet transforms to real volume data is discussed. Examples of the wavelet transform and the reconstruction of 1D functions are presented. The application of the 3D wavelet transform to real volume data generated from a series of 115 slices of magnetic resonance (MR) images is described. >

AM-FM energy detection and separation in noise using multiband energy operators

Abstract: This paper develops a multiband or wavelet approach for capturing the AM-FM components of modulated signals immersed in noise. The technique utilizes the recently-popularized nonlinear energy operator Psi (s)=(s)/sup 2/-ss to isolate the AM-FM energy, and an energy separation algorithm (ESA) to extract the instantaneous amplitudes and frequencies. It is demonstrated that the performance of the energy operator/ESA approach is vastly improved if the signal is first filtered through a bank of bandpass filters, and at each instant analyzed (via Psi and the ESA) using the dominant local channel response. Moreover, it is found that uniform (worst-case) performance across the frequency spectrum is attained by using a constant-Q, or multiscale wavelet-like filter bank. The elementary stochastic properties of Psi and of the ESA are developed first. The performance of Psi and the ESA when applied to bandpass filtered versions of an AM-FM signal-plus-noise combination is then analyzed. The predicted performance is greatly improved by filtering, if the local signal frequencies occur in-band. These observations motivate the multiband energy operator and ESA approach, ensuring the in-band analysis of local AM-PM energy. In particular, the multi-bands must have the constant-Q or wavelet scaling property to ensure uniform performance across bands. The theoretical predictions and the simulation results indicate that improved practical strategies are feasible for tracking and identifying AM-FM components in signals possessing pattern coherencies manifested as local concentrations of frequencies. >

Time-varying system identification and model validation using wavelets

Abstract: Parametric identification of time-varying (TV) systems is possible if each TV coefficient can be expanded onto a finite set of basis sequences. The problem then becomes time invariant with respect to the parameters of the expansion. The authors address the question of selecting this set of basis sequences. They advocate the use of a wavelet basis because of its flexibility in capturing the signal's characteristics at different scales, and discuss how to choose the optimal wavelet basis for a given system trajectory. They also develop statistical tests to keep only the basis sequences that significantly contribute to the description of the system's time-variation. By formulating the problem as a regressor selection problem, they apply an P-test and an AIC based approach for multiresolution analysis of TV systems. The resulting algorithm can estimate TV AR or ARMAX models and determine their orders. They apply this algorithm to both synthetic and real speech data and compare it with the Kalman filtering TV parameter estimator. >