Showing papers on "Wavelet published in 1991"

Wavelets and signal processing

Abstract: A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes. The discussion includes nonstationary signal analysis, scale versus frequency, wavelet analysis and synthesis, scalograms, wavelet frames and orthonormal bases, the discrete-time case, and applications of wavelets in signal processing. The main definitions and properties of wavelet transforms are covered, and connections among the various fields where results have been developed are shown. >

Fast wavelet transforms and numerical algorithms I

Abstract: A class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors. Previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices. In contrast, the methods presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators. The algorithms of this paper require order O(N) or O(N log N) operations to apply an N × N matrix to a vector (depending on the particular operator and the version of the algorithm being used), and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here.

Zero-crossings of a wavelet transform

Abstract: The completeness, stability, and application to pattern recognition of a multiscale representation based on zero-crossings is discussed. An alternative projection algorithm is described that reconstructs a signal from a zero-crossing representation, which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero-crossings. The reconstruction algorithm has a fast convergence and each iteration requires O(N log/sup 2/ (N)) computation for a signal of N samples. The zero-crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems. As an example, the implementation and results of a coarse-to-fine stereo-matching algorithm are described. >

Wavelets and multifractal formalism for singular signals: Application to turbulence data.

Abstract: The wavelet decomposition is used to generalize the multifractal formalism to singular signals The singularity spectrum is directly determined from the scaling behavior of partition functions that are defined from the wavelet transform modulus maxima Illustrations on fractal signals with a recursive structure, eg, devil's staircases, are shown Applications to fully developed turbulence data and Brownian signals are reported

Describing functions: Atomic decompositions versus frames

Abstract: The theory of frames and non-orthogonal series expansions with respect to coherent states is extended to a general class of spaces, the so-called coorbit spaces. Special cases include wavelet expansions for the Besov-Triebel-Lizorkin spaces, Gabortype expansions for modulation spaces, and sampling theorems for wavelet and Gabor transforms.

Analysis of turbulence in the orthonormal wavelet representation

Abstract: The usefulness of the wavelet transform for the analysis of turbulent flow fields is explored by examining the wavelet transform properties of a decomposition of turbulent velocity fields into modes that exhibit the localization in a wavenumber and physical space. The calculations are performed on 3D fields from direct numerical simulations of isotropic flow and homogeneous shear flow, and from measurements in two laboratory wind-tunnel experimental velocity signals (boundary layer and wake behind a circular cylinder). The analysis confirmed that there is strong spatial intermittency in nonlinear quantities; their mean spectral behavior results from a delicate balance between large positive and negative excursions. The wavelet analysis is a way to quantify these observations in a standardized fashion by using 'flow-independent eddies' to decompose the velocity field.

The determination of the seismic quality factor q from vsp data: a comparison of different computational methods1

Abstract: Ten methods for the computation of attenuation have been investigated, namely: amplitude decay, analytical signal, wavelet modelling, phase modelling, frequency modelling, rise-time, pulse amplitude, matching technique, spectral modelling and spectral ratio. In particular, we have studied the reliability of each of these methods in estimating correct values of Q using three synthetic VSP seismograms for plane P-waves with different noise contents. The investigations proved that no single method is generally superior. Rather, some methods are more suitable than others in specific situations depending on recording, noise or geology. The analytical signal method has been demonstrated to be superior if true amplitude recordings are available. Otherwise spectral modelling or, in the ‘ noise-free’ case the spectral ratio method, is optimal. Finally, two field VSPs in sediments are investigated. Only in the case of the highest quality VSP can significant information be deduced from the computed attenuation.

Using the refinement equations for the construction of Pre-Wavelets II: powers and two

Abstract: We study basic questions of wavelet decompositions associated with multiresolution analysis. A rather complete analysis of multiresolution associated with the solution of a refinement equation is presented. The notion of extensibility of a finite set of Laurent polynomials is shown to be central in the construction of wavelets by decomposition of spaces. Two examples of extensibility, first over the torus and then in complex space minus the coordinate axes are discussed. In each case we are led to a decomposition of the fine space in a multiresolution analysis as a sum of the adjacent coarse space plus an additional space spanned by the multiinteger translates of a finite number of pre-wavelets. Several examples are provided throughout to illustrate the general theory.

Inverse Q filtering by Fourier transform

Abstract: Although some attention has been paid to the idea that seismic migration is equivalent to a type of deconvolution (of the spatial wavelet), less thought has been given to the opposite perspective: that deconvolution (of the earth Q filter) might itself be equivalent to a form of migration. The key point raised in this paper is that a dispersive 1-D backward propagation can form the basis of a number of different algorithms for inverse Q filtering, each of which is akin to a particular migration algorithm. An especially efficient algorithm can be derived by means of a coordinate transformation equivalent to that in the Stolt frequency-wavenumber migration.This fast algorithm, valid for Q constant with depth, can be extended to accommodate depth-variable Q by cascading a series of constant Q compensations, as in cascaded migration. By combining a cascaded phase compensation with a windowed approach to amplitude compensation, we obtain an algorithm that is sufficiently efficient to be used routinely for prestack data processing. Data examples compare the results of conventional processing with the more stable phase treatment that can be obtained by including prestack inverse Q filtering in the processing.

Pointwise smoothness, two-microlocalization and wavelet coefficients

Abstract: In this paper we shall compare three notions of pointwise smoothness: the usual definition, J.M. Bony's two-microlocal spaces Cx0s,s', and the corresponding definition on the wavelet coefficients. The purpose is mainly to show that these two-microlocal spaces provide "good substitutes" for the pointwise Holder regularity condition; they can be very precisely compared with this condition, they have more functional properties, and can be characterized by conditions on the wavelet coefficients. We also give applications of these properties. In Part 2 some results on the microlocal spaces contained in [B2] will be recalled. Theorems 3 and 4 are also essentially contained in [B2]. The starting point for this paper was a note ([J1]) the author had written on a comparison between the Holder criterion of regularity at a given point x0 and a corresponding property defined on the wavelet coefficients. Some easy proofs are omitted or abridged and can be found in [J2].

Necessary and sufficient conditions for constructing orthonormal wavelet bases

Abstract: This paper proves a previous conjecture of the author characterizing sequences h∈l2(Z) that yield orthonormal wavelet bases of L2(R) in terms of the multiplicity of the eigenvalue 1 of an operator associated to h. The proof utilizes a result of Cohen characterizing these sequences in terms of the real zeros of their Fourier transforms. The mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2(Z) into L2(R). Related conjectures are discussed.

VLSI architecture for 2-D Daubechies wavelet transform without multipliers

Abstract: The 2-D orthogonal wavelet tranform is proving to be a highly effective tool for image analysis. In particular, the four-coefficient Daubechies wavelet transform has excellent spatial and spectral locality, properties which make it very useful in image compression. A VLSI architecture suitable for 2D orthogonal wavelet tranforms is presented, which for the Daubechies wavelet implements the forward and inverse tranforms without multipliers. A sample implementation is described.

Spline-wavelet signal analyzers and methods for processing signals

Abstract: A processor (10) is disclosed which uses a B-spline interpolator (14) to produce a plurality of zero-level spline coefficients c 0 (n). This set of coefficients may be fed to a B-spline generator (16) to produce an approximation of the input signal, and/or may be multiplied by a set of coefficients Bn to produce a set of first-level wavelet coefficients d -1 (n). The zero-level spline coefficients are also used to create first-level spline coefficients c -1 (n). The first-level spline and wavelet coefficient c -1 (n) and d -1 (n) may be submitted to a respective B-spline generator (22) or B-wavelet generator (24) to produce a first-level spline signal components and a first-level wavelet signal component for extraction of data from the original signal. The signal may in a similar fashion be decomposed to any level of resolution desired. The signal components may then be processed, and an improved signal then reassembled from the last-level spline and the processed wavelet signals. Novel spline and wavelet generators are also disclosed.

Using the refinement equation for the construction of pre-wavelets

Abstract: A variety of methods have been proposed for the construction of wavelets. Among others, notable contributions have been made by Battle, Daubechies, Lemarie, Mallat, Meyer, and Stromberg. This effort has led to the attractive mathematical setting of multiresolution analysis as the most appropriate framework for wavelet construction. The full power of multiresolution analysis led Daubechies to the construction ofcompactly supported orthonormal wavelets with arbitrarily high smoothness. On the other hand, at first sight, it seems some of the other proposed methods are tied to special constructions using cardinal spline functions of Schoenberg. Specifically, we mention that Battle raises some doubt that his block spin method “can produce only the Lemarie Ondelettes”. A major point of this paper is to extend the idea of Battle to the generality of multiresolution analysis setup and address the easier job of constructingpre-wavelets from multiresolution.

Method and apparatus for encoding and decoding using wavelet-packets

Abstract: The disclosure involves the use of a library of modulated wavelet-packets which are effective in providing both precise frequency localization and space localization. An aspect of the disclosure involves feature extraction by determination of the correlations of a library of waveforms with the signal being processed, while maintaining, orthogonality of the set of waveforms selected (i.e. a selected advantageous basis). In a disclosed embodiment, a method is provided for encoding and decoding an input signal, comprising the following steps: applying combinations of dilations and translations of a wavelet to the input signal to obtain processed values; computing the information costs of the processed values; selecting, as encoded signals, an orthogonal group of processed values, the selection being dependent on the computed information costs; and decoding the encoded signals to obtain an output signal. The wavelet preferably has a plurality of vanishing moments. In the disclosed embodiment, the step of applying combinations of dilations and translations of the wavelet to the input signal to obtain processed values comprises correlating said combinations of dilations and translations of the wavelet with the input signal. The combinations of dilations and translations of the wavelet are designated as wavelet-packets.

An identification of Energy Cascade in Turbulence by Orthonormal Wavelet Analysis

Abstract: Orthonormal wavelet expansion method is applied to an analysis of atmospheric turbulence data which shows more than two decades of the inertial subrange spectrum. The result of the orthonor­ mal wavelet analysis of the turbulence data is discussed in comparison with those of an artificial random noise. The local wavelet spectra of turbulence show a characteristic structure. which is absent in the artificial random noise and is identified with the trace of the energy cascade process. The higher· order structure function of velocity. obtained by the wavelet analysis. shows the intermit· tent structure of the flow field. In 1941 Kolmogorov proposed a universal theory of fluid turbulence/) in which every statistical quantity concerning the inertial subrange of flow field is assumed to be determined only by the energy dissiation rate. According to this theory, the average of the n-th order of velocity difference between two points separated by spatial distance r is proportional to r"13. This prediction has been repeatedly examined in accurate experiments in high Reynolds number flows, and it is now widely accepted that as far as lower order of velocity difference is concerned, the r-dependence agrees well with the Kolmogorov theory. In particular, experimental forms of the energy spectrum of the velocity field, which corresponds to n=2, coincide with the Kolmogorov form of k- 5/3 • However, it has been repeatedly confirmed that higher order of the velocity difference has a statistical property different from Kolmogorov's prediction, so that the n-th order structure function of the velocity field, when normalized by the second order of the velocity difference, shows a clear r-dependence. This fact implies that the energy cascade process in the inertial subrange has an unnegligible deviation from the Kolmogorov picture. The deviation is reflected in the shape of the probability distribution function of the velocity difference, which has longer tail for smaller distance r. This deviation is often called intermittency, and its characterization is regarded as one of the central problems of fluid turbulence. The fact that intermittency becomes more prominent at smaller scales indicates that the intermittency is an essential part of the energy cascade process. However, this cascade process itself, sometimes called Richardson cascade, has been only a matter of theoretical consideration, and its characteristic structure has not yet been clearly captured in experiments or in numerical simula­ tions.

Wavelet construction using Lagrange halfband filters

Abstract: Using the approach described by M.J.T. Smith and T.P. Barnwell (1986) for obtaining exact-reconstruction filter banks, the authors present conjugate-quadrature and linear-phase solutions for two-channel filter banks using Lagrange halfband filters. It is shown that the wavelet solutions obtained by I. Daubechies (1988) under certain regularity conditions are the same as the conjugate-quadrature solutions derived from Lagrange halfband filters using the above approach. The linear-phase solution that is described provides filters with simple coefficients. >

On-signal decomposition techniques

Abstract: Well-known block transforms and perfect reconstruction orthonormal filter banks are evaluated based on their frequency behavior and energy compaction. The filter banks outperform the block transforms for the signal sources considered. Although the latter are simpler to implement and already the choice of the existing video coding standards, filter banks with simple algorithms may well become the signal decomposition technique for the next generation video codecs, which require a multiresolution signal representation.

Adaptive Multidimensional Filtering

Abstract: This thesis contains a presentation and an analysis of adaptive filtering strategies for multidimensional data. The size, shape and orientation of the flter are signal controlled and thus adapted locally to each neighbourhood according to a predefined model. The filter is constructed as a linear weighting of fixed oriented bandpass filters having the same shape but different orientations. The adaptive filtering methods have been tested on both real data and synthesized test data in 2D, e.g. still images, 3D, e.g. image sequences or volumes, with good results. In 4D, e.g. volume sequences, the algorithm is given in its mathematical form. The weighting coefficients are given by the inner products of a tensor representing the local structure of the data and the tensors representing the orientation of the filters. The procedure and lter design in estimating the representation tensor are described. In 2D, the tensor contains information about the local energy, the optimal orientation and a certainty of the orientation. In 3D, the information in the tensor is the energy, the normal to the best ftting local plane and the tangent to the best fitting line, and certainties of these orientations. In the case of time sequences, a quantitative comparison of the proposed method and other (optical flow) algorithms is presented. The estimation of control information is made in different scales. There are two main reasons for this. A single filter has a particular limited pass band which may or may not be tuned to the different sized objects to describe. Second, size or scale is a descriptive feature in its own right. All of this requires the integration of measurements from different scales. The increasing interest in wavelet theory supports the idea that a multiresolution approach is necessary. Hence the resulting adaptive filter will adapt also in size and to different orientations in different scales.

Processing via spectral modeling

Abstract: The widespread occurrence of subtle trap accumulations offshore Brazil has led to the need for the development of a high resolution processing scheme that helps the delineation of these features. The process consists of three stages, the first of which is deterministic and stochastic deconvolution. The second stage is the deconvolution of the residual wavelet by means of spectral modeling. The last stage consists of the correction of the color of the reflectivity function using a model developed for the area. An important conclusion that is drawn from the model is that the acoustic impedance is not white. Rather it is as red as the corresponding reflectivity is blue. Successful results from the application of the proposed technique to real data indicate that the color compensation is of second order importance as compared with the first two stages of the proposed scheme.

Gabor Representations of Spatiotemporal Visual Images

Abstract: We review Gabor''s Uncertainty Principle and the limits it places on the representation of any signal. Representations in terms of Gabor elementary functions (Gaussian-modulated sinusoids), which are optimal in terms of this uncertainty principle, are compared with Fourier and wavelet representations. We also review Daugman''s evidence for representations based on two-dimensional Gabor functions in mammalian visual cortex. We suggest three-dimensional Gabor elementary functions as a model for motion selectivity in complex and hypercomplex cells in visual cortex. This model also suggests a computational role for low frequency oscillations (such as the alpha rhythm) in visual cortex.

Image coding using lattice vector quantization of wavelet coefficients

Abstract: An image coding scheme has been introduced by the authors (see IEEE ICASSP, p.2297, 1990). This scheme involves two steps. A biorthogonal wavelet transform is applied to the original image, and wavelet coefficients are then vector quantized using the LBG (Linde, Buzo and Gray, 1980) method. The purpose of this work is to propose a new scheme for vector quantization of wavelet coefficients. The proposed method is based on lattice vector quantization. The application of the D/sub 4/, E/sub 8/ and Barnes-Wall Lambda /sub 16/ lattices is investigated. These lattices are used to encode wavelet coefficients whose PDFs are close to Laplacian. A variable-length coding method is applied and the trade-off between distortion and optimal rate is investigated. Experimental results on the Lena image using the Lambda /sub 16/ lattice leads to a peak signal-to-noise ratio (PSNR) of 31.14 dB at 0.08 bpp. This result outperforms, to the authors knowledge, all other methods. Edges which are most of interest for image analysis are particularly sharp without any smoothing artefacts. >

Wavelet analysis of fully developed turbulence data and measurement of scaling exponents

Abstract: Wavelet analysis can be used to measure directly the scaling exponents characterizing the local multifractal behaviour of the velocity field at inertial-range scales, without recourse to dissipation-type quantities. Preliminary results using data from the Modane S1 wind tunnel indicate that the most frequent exponents are close to the Kolmogorov 1941 value of 1/3. Violent rare events are however found with negative exponents of −0.1 or less.

Instantaneous frequency and amplitude at the envelope peak of a constant-phase wavelet

Abstract: Robertson and Nogami (1984) have shown that the instantaneous frequency at the peak of a zero‐phase Ricker wavelet is exactly equal to that wavelet’s average Fourier spectral frequency weighted by its amplitude spectrum. Bodine (1986) gave an example which shows this is also true for constant‐phase bandpass wavelets. Here I prove that this holds for any constant‐phase wavelet. I then develop an equation expressing this quantity as a function of propagation time through an attenuating medium. A corresponding equation is derived for the amplitude of the envelope peak. Taken together, these may aid in the analysis of seismic data as suggested by Robertson and Nogami (1984), Bodine (1986), and Robertson and Fisher (1988).

A comparison of a wavelet functions for pitch detection of speech signals

Abstract: The authors compare the relative performances of linear phase wavelets and minimum phase wavelets for the estimation of pitch periods using an event detection algorithm based upon the dyadic wavelet transform (D/sub y/WT). They apply the D/sub y/WT to detect the glottal closure, which they define as an event, and estimate the pitch period by measuring the time interval between two such events. Comparative examples are given of applying the D/sub y/WT using both linear phase and minimum phase wavelets on synthetic as well as actual speech data to evaluate their relative performance in pitch detection. The D/sub y/WT pitch detector using a spline wavelet gives the best results. >

The wavelet transform and its applications to phonocardiogram signal analysis.

Abstract: The wavelet transform, which is the decomposition of a signal into a set of independent frequency channels, is shown to be a useful diagnostic tool in the analysis of heartbeat sounds. In particular, the wavelet transform enables the experimentalist to obtain qualitative and quantitative measurements of time-frequency characteristics of phonocardiogram (PCG) signals.

The Discrete Wavelet Transform

Abstract: : In a general sense, this report represents an effort to clarify the relationship of discrete and continuous wavelet transforms. More narrowly, it focuses on bringing together two separately motivated implementations of the wavelet transform, the algorithm a trous and Mallat's multiresolution decomposition. These algorithms are special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by one's choice of filters. In fact, the a trous algorithm, originally devised as a computationally efficient implementation, is more properly viewed as a nonorthogonal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, we show that the commonly used Lagrange a trous filters are in one-to-one correspondence with the convolutional squares of the Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, and conditions are derived under which it computer the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Finally, relevant signal-processing parameters are examined, and it is remarked that orthonormality is balanced by restrictions on resolution.

Amplitude-versus-offset measurement errors in a finely layered medium

Abstract: Recently Spratt (1987) showed how amplitude-versus-offset analysis (AVO) can be sensitive to small residual velocity errors. However, even when the velocity is determined perfectly, serious AVO distortions remain due to normal-moveout stretch, differential tuning as a function of offset, spherical divergence, and source and receiver directivity patterns. I have found that all of these errors can be expanded in a Taylor series about the zero-offset event time, assuming it is much larger than the wavelet width. The first term of this series represents the residual velocity error term found by Spratt, while the second term encompasses the remaining effects mentioned. In practice, either term can be larger than the underlying amplitude variations being estimated. For example, Ricker wavelet stretch leads to a peak AVO error which is 61 percent of the peak zero-offset reflectivity, even though the velocity field is uniform and correct. This result is independent of the wavelet frequency, and the range of incidence angles used in the analysis. Positive gradients in moveout velocity amplify this error, while narrowband filtering of the data prior to AVO analysis greatly widens its temporal extent. Aligning a particular event with static shifts instead of normal-moveout correction can eliminate stretch, but not differential tuning error, in a finely layered target zone whose wavelets overlap.