Showing papers on "Wavelet published in 1989"
TL;DR: In this paper, it is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2 /sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.
Abstract: Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions. In L/sup 2/(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. >
20,028 citations
TL;DR: In this paper, the authors study the properties of multiresolution approximation and prove that it is characterized by a 2π periodic function, which is further described in terms of wavelet orthonormal bases.
Abstract: A multiresolution approximation is a sequence of embedded vector spaces V j jmember Z for approximating L 2 (R) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 2π periodic function which is further described. From any multiresolution approximation, we can derive a function ψ(x) called a wavelet such that √ 2 j ψ(2 j x −k) (k ,j)member Z 2 is an orthonormal basis of L 2 (R). This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space H s .
2,286 citations
TL;DR: The author describes the mathematical properties of such decompositions and introduces the wavelet transform, which relates to the decomposition of an image into a wavelet orthonormal basis.
Abstract: The author reviews recent multichannel models developed in psychophysiology, computer vision, and image processing. In psychophysiology, multichannel models have been particularly successful in explaining some low-level processing in the visual cortex. The expansion of a function into several frequency channels provides a representation which is intermediate between a spatial and a Fourier representation. The author describes the mathematical properties of such decompositions and introduces the wavelet transform. He reviews the classical multiresolution pyramidal transforms developed in computer vision and shows how they relate to the decomposition of an image into a wavelet orthonormal basis. He discusses the properties of the zero crossings of multifrequency channels. Zero-crossing representations are particularly well adapted for pattern recognition in computer vision. >
2,109 citations
TL;DR: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states, focusing on Weyl–Heisenberg coherent states and affine coherent states.
Abstract: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called ’wavelets,’ which arise as translations and dilations of a single function. In each case it is shown how to represent any function in $L^2 ({\bf R})$ as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included.
1,121 citations
01 Jan 1989
TL;DR: One of the aims of wavelet transforms is to provide an easily interpretable visual representation of signals that is a prerequisite for applications such as selective modifications of signals or pattern recognition.
Abstract: One of the aims of wavelet transforms is to provide an easily interpretable visual representation of signals This is a prerequisite for applications such as selective modifications of signals or pattern recognition
356 citations
01 Jan 1989
TL;DR: It will be shown that the computation load grows with the scale factor of the analysis, which leads to a prohibitive computation time, so a more effective computation procedure is needed.
Abstract: The computation of the wavelet transform involves the computation of the convolution product of the signal to be analysed by the analysing wavelet. It will be shown that the computation load grows with the scale factor of the analysis. We are interested in musical sounds lasting a few seconds. Using a straightforward algorithm leads to a prohibitive computation time, so we need a more effective computation procedure.
261 citations
TL;DR: A method for texture discrimination and image segmentation using local features based on the Gabor approach is introduced and the results show the insensitivity of the discrimination to relatively high noise levels, comparable to the performances of the human observer.
Abstract: Recent studies of cortical simple cell function suggest that the primitives of image representation in vision have a wavelet form similar to Gabor elementary functions (EFs). It is shown that textures and fully textured images can be practically decomposed into, and synthesized from, a finite set of EFs. Textured-images can be synthesized from a set of EFs using an image coefficient library. Alternatively, texturing of contoured (cartoonlike) images is analogous to adding chromaticity information to contoured images. A method for texture discrimination and image segmentation using local features based on the Gabor approach is introduced. Features related to the EF's parameters provide efficient means for texture discrimination and classification. This method is invariant under rotation and translation. The performance of the classification appears to be robust with respect to noisy conditions. The results show the insensitivity of the discrimination to relatively high noise levels, comparable to the performances of the human observer. >
207 citations
01 Jan 1989
TL;DR: The motivations of the models developed in psychophysiology, computer vision, and image processing are described and how they relate to the wavelet transform is described.
Abstract: In this paper we review recent multichannel models de- veloped in psychophysiology, computer vision, and image processing. In psychophysiology, multichannel models have been particularly suc- cessful in explaining some low-level processing in the visual cortex. The expansion of a function into several frequency channels provides a rep- resentation which is intermediate between a spatial and a Fourier rep- resentation. We describe the mathematical properties of such decom- positions and introduce the wavelet transform. We review the classical multiresolution pyramidal transforms developed in computer vision and show how they relate to the decomposition of an image into a wavelet orthonormal basis. In the last section we discuss the properties of the zero crossings of multifrequency channels. Zero-crossings represen- tations are particularly well adapted for pattern recognition in com- puter vision. I. INTRODUCTION ITHIN the last 10 years, multifrequency channel W decompositions have found many applications in image processing. In the psychophysiology of human vi- sion, multichannel models have also been particularly successful in explaining some low-level biological pro- cesses. The expansion of a function into several fre- quency channels provides a representation which is inter- mediate between a spatial and a Fourier representation. In harmonic analysis, this kind of transform appeared in the work of Littlewood and Payley in the 1930's. More re- search has recently been focused on this domain with the modeling of a new decomposition called the wavelet transform. In this paper we review the recent multichan- nel models developed in psychophysiology, computer vi- sion, and image processing. We describe the motivations of the models within each of these disciplines and show how they relate to the wavelet transform. In psychophysics and the physiology of human vision, evidence has been gathered showing that the retinal image is decomposed into several spatially oriented frequency channels. In the first section of this paper, we describe the experimental motivations for this model. Biological studies of human vision have always been a source of ideas for computer vision and image processing research. Indeed, the human visual system is generally considered to be an optimal image processor. The goal is not to im- itate the processings implemented in the human brain, but rather to understand the motivations of such processings
114 citations
Patent•
11 Sep 1989TL;DR: In this paper, a system for compressing images is described, which utilizes a transformation which is equivalent to expanding the image using a system of wavelets having finite support, and it is shown that the system can be used to compress images.
Abstract: A system for compressing images is disclosed. The system utilizes a transformation which is equivalent to expanding the image using a system of wavelets having finite support.
84 citations
AT&T1
TL;DR: This work focuses on orthonormal bases of wavelets, in particular bases ofwavelets with finite support, and defines wavelets and the wavelet transform.
Abstract: We define wavelets and the wavelet transform. After discussing their basic properties, we focus on orthonormal bases of wavelets, in particular bases of wavelets with finite support.
Book•
01 Jan 1989
TL;DR: A real-time algorithm for Signal Analysis with the help of the Wavelet Transform is described in this paper. But it is not suitable for the use of the wavelet transform in computer vision applications.
Abstract: I Introduction to Wavelet Transforms.- Reading and Understanding Continuous Wavelet Transforms.- Orthonormal Wavelets.- Orthonormal Bases of Wavelets with Finite Support - Connection with Discrete Filters.- II Some Topics in Signal Analysis.- Some Aspects of Non-Stationary Signal Processing with Emphasis on Time-Frequency and Time-Scale Methods.- Detection of Abrupt Changes in Signal Processing.- The Computer, Music, and Sound Models.- III Wavelets and Signal Processing.- Wavelets and Seismic Interpretation.- Wavelet Transformations in Signal Detection.- Use of Wavelet Transforms in the Study of Propagation of Transient Acoustic Signals Across a Plane Interface Between Two Homogeneous Media.- Time-Frequency Analysis of Signals Related to Scattering Problems in Acoustics Part I: Wigner-Ville Analysis of Echoes Scattered by a Spherical Shell.- Coherence and Projectors in Acoustics.- Wavelets and Granular Analysis of Speech.- Time-Frequency Representations of Broad-Band Signals.- Operator Groups and Ambiguity Functions in Signal Processing.- IV Mathematics and Mathematical Physics.- Wavelet Transform Analysis of Invariant Measures of Some Dynamical Systems.- Holomorphic Integral Representations for the Solutions of the Helmholtz Equation.- Wavelets and Path Integrals.- Mean Value Theorems and Concentration Operators in Bargmann and Bergman Space.- Besov-Sobolev Algebras of Symbols.- Poincare Coherent States and Relativistic Phase Space Analysis.- A Relativistic Wigner Function Affiliated with the Weyl-Poincare Group.- Wavelet Transforms Associated to the n-Dimensional Euclidean Group with Dilations: Signal in More Than One Dimension.- Construction of Wavelets on Open Sets.- Wavelets on Chord-Arc Curves.- Multiresolution Analysis in Non-Homogeneous Media.- About Wavelets and Elliptic Operators.- Towards a Method for Solving Partial Differential Equations Using Wavelet Bases.- V Implementations.- A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform.- An Implementation of the "algorithme a trous" to Compute the Wavelet Transform.- An Algorithm for Fast Imaging of Wavelet Transforms.- Multiresolution Approach to Wavelets in Computer Vision.- Index of Contributors.
TL;DR: In this paper, a simple mathematical expression for the wavelet properties of a synthetic source wavelet is given, and the mathematical expression should be such that with a minimum number of parameters, a large flexibility in the form of wavelet can be obtained.
Abstract: Seismologists make frequent use of wavelets (also referred to as signals, signatures, or pulses), particularly in such fields as seismic filtering, wavelet processing, wave‐propagation modeling, and trace inversion. Whenever possible, the actual seismic wavelet of the real source should be considered (Hosken, 1988). However, frequently, particularly in wave‐propagation modeling, one must consider a synthetic source wavelet. This should, if possible, be given by a simple mathematical formula and possess an easy description for its most important spectral properties (e.g., amplitude and phase spectrum, main frequency, Hilbert transform, etc.). Moreover, the mathematical expression should be such that with a minimum number of parameters a large flexibility in the form of the wavelet can be obtained.
TL;DR: A selection of well established deconvolution techniques are assessed for operation in the ultrasonic environment and those algorithms which are best suited to particular ultrasonic applications and conditions are identified via a simulation approach, based around a linear systems model.
01 Jan 1989
TL;DR: Most of the Signal Processing methods which have been proposed in this direction are reviewed, with emphasis on time-frequency representations and on their time-scale versions which implicitly make use of “wavelet” concepts.
Abstract: The analysis and the processing of nonstationary signals call for specific tools which go beyond Fourier analysis. This paper is intended to review most of the Signal Processing methods which have been proposed in this direction. Emphasis is put on time-frequency representations and on their time-scale versions which implicitly make use of “wavelet” concepts. Relationships between Gabor expansion, wavelet transform and ambiguity functions are detailed by considering signal decomposition as a detection-estimation problem. This permits one to make more precise some of the links which exist between time-frequency and time-scale.
09 Nov 1989
TL;DR: In this paper, the wavelet transform was used as a detection tool for ventricular late potentials and showed that the detection is obvious from 16 beats and needs no high amplification or preliminary filtering.
Abstract: The wavelet transform, leading to a time-scale analysis, was used as a detection tool for ventricular late potentials. Preliminary results show that the detection is obvious from 16 beats and needs no high amplification or preliminary filtering. The experimental results are validated by averaging and comparing highly amplified filtered signals obtained from both a healthy person and a patient with a ventricular tachycardia diagnosis. >
01 Jan 1989
01 Jan 1989
TL;DR: The wavelet transform is presented as a mathematical microscope which is well suited for studying the local scaling properties of fractal measures and is applied to probability measures on self-similar Cantor sets, to the 2∞ cycle of period-doubling and to the golden-mean trajectories on two-tori at the onset of chaos.
Abstract: We present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal measures. We apply this technique, recently introduced in signal analysis, to probability measures on self-similar Cantor sets, to the 2∞ cycle of period-doubling and to the golden-mean trajectories on two-tori at the onset of chaos. We emphasize the wide range of application of the wavelet transform which turns out to be a natural tool for characterizing the structural properties of fractal objects arising in a variety of physical situations.
TL;DR: In this article, the authors established a rigorous relation between the wavelet transform of a measure and its local scaling exponents, and showed that the relation can be extended to the local scaling exponent.
Abstract: We establish a rigorous relation between the wavelet transform of a measure and its local scaling exponents.
Abstract: The reflection seismic signal observed at the surface is the convolution of a wavelet with a reflection sequence representing the geology. Deconvolution of the observations without prior knowledge of the wavelet can be done by making assumptions about the statistics of the reflection sequence. In particular, the widely used prediction error filter is obtained by assuming that the power spectra of reflection sequences are white. However, evidence from well logs suggests that the power spectra are in fact proportional to a power of the frequency f, that is, to f α, with α equal approximately to 1.
Patent•
30 May 1989
TL;DR: In this paper, the authors proposed a method for interpolation of severely aliased events in which composite wavelets of a section of seismic data traces are separated by wave field decomposition into their component wavelets.
Abstract: Methods for interpolation of severely aliased events in which composite wavelets of a section of seismic data traces are separated by wave field decomposition into their component wavelets. The spatially aliased component wavelets are identified to form a first class of wavelets. A second class of wavelets is comprised of non-aliased component wavelets. The wavelets in the second class or non-aliased component wavelets are interpolated in accordance with known techniques, such for example, as sinc interpolation whereas the spatially aliased component wavelets are interpolated in accordance with dip guided interpolation to provide the same number of traces as those comprising the second class of wavelets. The interpolated first and second class of wavelets are summed to provide a section of seismic data substantially free of spatial aliasing.
TL;DR: In this article, a simple closed-form expression relates one-dimensional output cumulant statistics with the parameters of a known-order moving-average wavelet, and based on this relationship, the author obtains unique parameter and phase estimates of autoregressive moving average seismic wavelets.
Abstract: A simple, closed-form expression relates one-dimensional output cumulant statistics with the parameters of a known-order moving-average wavelet. Based on this relationship the author obtains unique parameter and phase estimates of autoregressive moving-average seismic wavelets. The input reflectivity sequence is assumed to be non-Gaussian, independent, and identically distributed. The wavelet is not assumed to be minimum phase and is allowed to include all-pass factors. The seismogram is contaminated by additive-colored Gaussian noise. Simulations demonstrate that the algorithm works well for moderate-size data records with a relatively low signal-to-noise ratio. >
01 Jan 1989
TL;DR: In this article, an algorithm for computing the expansion coefficients of a signal in terms of wavelet bases has been found, the structure of which is that of a pruned-tree quadrature mirror multirate filter bank.
Abstract: Summary form only given. A new class of orthogonal basis functions that can be relevant to signal processing has recently been introduced. These bases are constructed from a single smooth bandpass function psi (t), the wavelet, by considering its translates and dilates on a dyadic grid 2/sup n/, 2/sup n/m of points, psi /sub n,m/(t)=2/sup -n/2/ psi (2/sup -n/t-m). It is required that psi (t) be well localized in both the time and frequency domain, without violating the uncertainty principle. Any one-dimensional signal can be represented by the bidimensional set of its expansion coefficients. Multidimensional signals can also be expanded in terms of wavelet bases. An algorithm for computing the expansion coefficients of a signal in terms of wavelet bases has been found, the structure of which is that of a pruned-tree quadrature mirror multirate filter bank. The construction of wavelet bases and their relation to filter banks, together with several design techniques for wavelet generating quadrature mirror filters and examples, are reviewed. >
03 Oct 1989
TL;DR: A generalized SSP based on a wavelet decomposition, in which the received signal is decomposed over a basis of elementary wavelets translated in frequency and dilated in time, and followed as in the conventional SSP by an optimization algorithm is introduced.
Abstract: The split-spectrum processing (SSP) technique is known to improve flaw detection in materials in which the coarse microstructure produces broadband noise of large amplitude, which masks useful signals It is shown that the spectral decomposition used in the SSP is equivalent to the time-frequency Gabor decomposition A generalized SSP based on a wavelet decomposition, in which the received signal is decomposed over a basis of elementary wavelets translated in frequency and dilated in time, and followed as in the conventional SSP by an optimization algorithm is introduced The flexibility in the choice of the wavelet basis allows implementation of efficient signal decomposition Three wavelet bases are presented: Gaussian-shaped wavelets, binary wavelets, and autoregressive wavelets Applications of the SSP with wavelet decomposition to noise reduction and deconvolution are illustrated with experimental and simulated data >
01 Jan 1989
TL;DR: Wavelets present good properties of global approximation (good frequency localization) and their spatial localization allows precise approximation of discontinuities, without producing spurious fluctuations all over the domain.
Abstract: Wavelets present good properties of global approximation (good frequency localization) and their spatial localization allows precise approximation of discontinuities, without producing spurious fluctuations all over the domain.
01 Jan 1989
TL;DR: In this paper, the authors explain some possible uses of the wavelet transform in Seismic Interpretation (SEI) in the context of seismic data acquisition, where the source signal used during seismic acquisition is approximated by a wavelet form.
Abstract: The concept of the wavelet is not new in the petroleum industry, and particularly in geophysics, but this term is used here with a different meaning. For a geophysicist the source signal used during seismic acquisition is often approximated by a wavelet form. The mathematical approach of Wavelet Transform (W.T) permits to go further in the exploitation of their interesting properties. In this article, we explain some possible uses in Seismic Interpretation.
06 Sep 1989
TL;DR: In this paper, a family of wavelets is generated by dilating and translating a single function of one variable: h/sub a,b/(x)= mod a mod /sup -1/2/h (x-b/a).
Abstract: Summary form only given. In the simplest case, a family wavelets is generated by dilating and translating a single function of one variable: h/sub a,b/(x)= mod a mod /sup -1/2/h (x-b/a). The parameters a and b may vary continuously, or be restricted to a discrete lattice of values a=a/sub 0//sup m/, b=na/sub 0//sup m/B/sub 0/. If the dilation and translation steps a/sub 0/ and b/sub 0/ are not too large, then any L/sup 2/-function can be completely characterized by its inner products with the elements of such a discrete lattice of wavelets. Moreover, one can construct numerically stable algorithms for the reconstruction of a function from these inner products (the wavelet coefficients). For special choices of the wavelet h decomposition and reconstruction can be done very fast, via a tree algorithm. The wavelet coefficients of a function give a time-frequency decomposition of the function, with higher time resolution for high-frequency than for low-frequency components. >
06 Sep 1989
TL;DR: In this article, the scale is allowed to change at various points in space, as well as the analyzing wavelets, and various versions of wavelet analysis valid in a non-translation-invariant setting are described.
Abstract: Summary form only given, as follows. Various versions of wavelet analysis valid in a non-translation-invariant setting are described. Here, the scale is allowed to change at various points in space, as well as the analyzing wavelets. Such a generalized time (space) frequency analysis could find uses in a variety of signal and image processing contexts, as well as in the study of partial differential operators with variable coefficients arising in a nonhomogeneous medium, and gives rise to fast numerical algorithms. This multiresolution analysis, say for an edge detection problem, takes into account the variable geometry and sensitivity of the receptors. >
01 Jan 1989
TL;DR: In this paper, the authors proposed wavelets, a new method for dealing with transient signals referred to as wavelets and employing time compression (or dilation) rather than a variation of frequency of the modulated sinusoid.
Abstract: In the analysis of transient signals such as those encountered in speech, or in certain kinds of image processing, standard Fourier analysis is often not satisfactory. This is because the basis functions of Fourier analysis (sines, cosines, complex exponentials) extend over infinite time whereas the signals to be analysed are short-time transients A new method for dealing with transient signals has recently appeared in the literature [1-6]. The basis functions are referred to as wavelets, and they employ time compression (or dilation) rather than a variation of frequency of the modulated sinusoid. Hence all the wavelets have the same number of cycles . The analyzing wavelets must satisfy a few simple conditions, but are not otherwise specified. There is therefore a wide latitude in the choice of these functions and they can be taylored to specific applications. The wavelets are well founded on rigorous mathematical theory, and the expansions are robust. We have applied them to detect ventricular delayed potentials (VLP) in the electrocardiogram