Showing papers on "Harmonic wavelet transform published in 1992"
TL;DR: It is proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents.
Abstract: The mathematical characterization of singularities with Lipschitz exponents is reviewed. Theorems that estimate local Lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed. It is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately. The local frequency of such oscillations is measured from the wavelet transform modulus maxima. It has been shown numerically that one- and two-dimensional signals can be reconstructed, with a good approximation, from the local maxima of their wavelet transform modulus. As an application, an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales. In two dimensions, the wavelet transform maxima indicate the location of edges in images. >
4,064 citations
TL;DR: A scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed and it is shown that the wavelet transform is particularly well adapted to progressive transmission.
Abstract: A scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed. This method involves two steps. First, a wavelet transform used in order to obtain a set of biorthogonal subclasses of images: the original image is decomposed at different scales using a pyramidal algorithm architecture. The decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image. Second, according to Shannon's rate distortion theory, the wavelet coefficients are vector quantized using a multiresolution codebook. To encode the wavelet coefficients, a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed. In order to allow the receiver to recognize a picture as quickly as possible at minimum cost, a progressive transmission scheme is presented. It is shown that the wavelet transform is particularly well adapted to progressive transmission. >
3,925 citations
TL;DR: Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allows one to unfold a signal, or a field, into both space and scale, and possibly directions.
Abstract: Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allows one to unfold a signal, or a field, into both space and scale, and possibly directions. They use analyzing functions, called wavelets, which are localized in space. The scale decomposition is obtained by dilating or contracting the chosen analyzing wavelet before convolving it with the signal. The limited spatial support of wavelets is important because then the behavior of the signal at infinity does not play any role. Therefore the wavelet analysis or syn thesis can be performed locally on the signal, as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions. Wavelet transforms have been applied mostly to signal processing, image coding, and numerical analysis, and they are still evolving. So far there are only two complete presentations of this topic, both written in French, one for engineers (Gasquet & Witomski 1 990) and the other for mathematicians (Meyer 1 990a), and two conference proceedings, the first in English (Combes et al 1 989), the second in French (Lemarie 1 990a). In preparation are a textbook (Holschneider 199 1 ), a course (Dau bee hies 1 99 1), three conference procecdings (Mcyer & Paul 199 1 , Beylkin et al 199 1b, Farge et al 1 99 1), and a special issue of IEEE Transactions
2,770 citations
TL;DR: It is shown 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.
Abstract: Two separately motivated implementations of the wavelet transform are brought together. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by the choice of filters. In fact, the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, it is shown 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 computes the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Relevant signal processing parameters are examined, and it is observed that orthonormality is balanced by restrictions on resolution. >
1,856 citations
TL;DR: A tutorial review of both linear and quadratic representations is given, and examples of the application of these representations to typical problems encountered in time-varying signal processing are provided.
Abstract: A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided. >
1,587 citations
TL;DR: The 2-D orthogonal wavelet transform decomposes images into both spatial and spectrally local coefficients and gave high to acceptable quality reconstruction at compression ratios for the Miss America and Lena monochrome images.
Abstract: The 2-D orthogonal wavelet transform decomposes images into both spatial and spectrally local coefficients. The transformed coefficients were coded hierarchically and individually quantized in accordance with the local estimated noise sensitivity of the human visual system (HVS). The algorithm can be mapped easily onto VLSI. For the Miss America and Lena monochrome images, the technique gave high to acceptable quality reconstruction at compression ratios of 0.3-0.2 and 0.64-0.43 bits per pixel (bpp), respectively. >
857 citations
CNET1
TL;DR: The goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts.
Abstract: Several algorithms are reviewed for computing various types of wavelet transforms: the Mallat algorithm (1989), the 'a trous' algorithm, and their generalizations by Shensa. The goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts. Most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform. Although this general method is already efficient, it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques, such as the FFT (fast Fourier transform), in a suitable manner. The modified algorithms are termed 'fast' because of their ability to reduce the computational complexity per computed coefficient from L to log L (within a small constant factor) for large filter lengths L. For short filters, smaller gains are obtained: 'fast running FIR (finite impulse response) filtering' techniques allow one to achieve typically 30% savings in computations. >
639 citations
Book•
30 Sep 1992
TL;DR: Wavelet Theory Extentions and Ambiguity Functions, Linear Systems Modelling with Wavelet Theory, and Practical Resolution, Gain, and Processing Structures.
Abstract: Foreword. Preface. 1. Introduction/Background 2. The Wavelet Transform. 3. Practical Resolution, Gain, and Processing Structures. 4. Wavelet Theory Extentions and Ambiguity Functions. 5. Linear Systems Modelling with Wavelet Theory. 6. Wideband Scattering and Environmental Imaging. Related Research. References. Subject Index.
346 citations
TL;DR: The theory of a new general class of signal energy representations depending on time and scale is developed, and specific choices allow recovery of known definitions, and provide a continuous transition from Wigner-Ville to either spectrograms or scalograms (squared modulus of the WT).
Abstract: The theory of a new general class of signal energy representations depending on time and scale is developed Time-scale analysis has been introduced recently as a powerful tool through linear representations called (continuous) wavelet transforms (WTs), a concept for which an exhaustive bilinear generalization is given Although time scale is presented as an alternative method to time frequency, strong links relating the two are emphasized, thus combining both descriptions into a unified perspective The authors provide a full characterization of the new class: the result is expressed as an affine smoothing of the Wigner-Ville distribution, on which interesting properties may be further imposed through proper choices of the smoothing function parameters Not only do specific choices allow recovery of known definitions, but they also provide, via separable smoothing, a continuous transition from Wigner-Ville to either spectrograms or scalograms (squared modulus of the WT) This property makes time-scale representations a very flexible tool for nonstationary signal analysis >
326 citations
TL;DR: The wavelet transform is described, which is particularly useful in those cases in which the shape of the mother wavelet is approximately known a priori and the bank of the VanderLugt matched filters is considered.
Abstract: The wavelet transform is a powerful tool for the analysis of short transient signals. We detail the advantages of the wavelet transform over the Fourier transform and the windowed Fourier transform and consider the wavelet as a bank of the VanderLugt matched filters. This methodology is particularly useful in those cases in which the shape of the mother wavelet is approximately known a priori. A two-dimensional optical correlator with a bank of the wavelet filters is implemented to yield the time-frequency joint representation of the wavelet transform of one-dimensional signals.
145 citations
TL;DR: A wavelet transform specifically designed for Fourier analysis at multiple scales is described and shown to be capable of providing a local representation which is particularly well suited to segmentation problems.
Abstract: A wavelet transform specifically designed for Fourier analysis at multiple scales is described and shown to be capable of providing a local representation which is particularly well suited to segmentation problems. It is shown that, by an appropriate choice of analysis window and sampling intervals, it is possible to obtain a Fourier representation which can be computed efficiently and overcomes the limitations of using a fixed scale of window, yet by virtue of its symmetry properties allows simple estimation of such fundamental signal parameters as instantaneous frequency and onset time/position. The transform is applied to the segmentation of both image and audio signals, demonstrating its power to deal with signal events which are localized in either time/space or frequency. Feature extraction and segmentation are performed through the introduction of a class of multiresolution Markov models, whose parameters represent the signal events underlying the segmentation. >
TL;DR: In this paper, the wavelet transform is implemented using an optical multichannel correlator with a bank of wavelet filter filters, which provide a shift-invariant wavelet transformation with continuous translation and discrete dilation.
Abstract: The wavelet transform is implemented using an optical multichannel correlator with a bank of wavelet transform filters. This approach provides a shift-invariant wavelet transform with continuous translation and discrete dilation parameters. The wavelet transform filters can be in many cases simply optical transmittance masks. Experimental results show detection of the frequency transition of the input signal by the optical wavelet transform.
Journal Article•
TL;DR: It is shown that the best transforms for transform image coding, namely, the scrambled real discrete Fourier transform, the discrete cosine transform, and the discrete Cosine-III transform are also the best for image enhancement.
Abstract: Blockwise transform image enhancement techniques are discussed. Previously, transform image enhancement has usually been based on the discrete Fourier transform (DFT) applied to the whole image. Two major drawbacks with the DFT are high complexity of implementation involving complex multiplications and additions, with intermediate results being complex numbers, and the creation of severe block effects if image enhancement is done blockwise. In addition, the quality of enhancement is not very satisfactory. It is shown that the best transforms for transform image coding, namely, the scrambled real discrete Fourier transform, the discrete cosine transform, and the discrete cosine-III transform, are also the best for image enhancement. Three techniques of enhancement discussed in detail are alpha-rooting, modified unsharp masking, and filtering motivated by the human visual system response (HVS). With proper modifications, it is observed that unsharp masking and HVS-motivated filtering without nonlinearities are basically equivalent. Block effects are completely removed by using an overlap-save technique in addition to the best transform.
01 Jan 1992
TL;DR: In this paper, it was shown that the cross terms that exist in the energy distribution of the wavelet transform are comparable with those found in the Wigner dis- tribution (WD), a quadratic time-frequency representation, and the short time Fourier transform (STFT), of closely spaced signals.
Abstract: The wavelet transform (WT), a time-scale repre- sentation, is linear by definition. However, the nonlinear en- ergy distribution of this transform is often used to represent the signal; it contains ''cross terms" which could cause prob- lems while analyzing multicomponent signals. In this paper, we show that the cross terms that exist in the energy distribution of the WT are comparable with those found in the Wigner dis- tribution (WD), a quadratic time-frequency representation, and the energy distribution of the short time Fourier transform (STFT), of closely spaced signals. The cross terms of the WT and the STFT energy distributions occur at the intersection of their respective WT and STFT spaces, while for the WD they occur midtime and midfrequency. The parameters of the cross terms are a function of the difference in center frequencies and center times of the perpended signals. The amplitude of these cross terms can be as large as twice the product of the magni- tudes of the transforms of the two signals in question in all three cases. In this paper, we consider the significance of the effect of the cross terms on the analysis of a multicomponent signal in each of these three representations. We also compare the advantages and disadvantages of all of these methods in appli- cations to signal processing.
TL;DR: It is shown that cross terms comparable to those found in the Wigner distribution exist for the energy distributions of the wavelet transform and the short-time Fourier transform, and that the presence of these cross terms could lead to problems in analyzing a multicomponent signal.
Abstract: It is shown that cross terms comparable to those found in the Wigner distribution (WD) exist for the energy distributions of the wavelet transform (WT) and the short-time Fourier transform (STFT). The geometry of the cross terms is described by deriving mathematical expressions for the energy distributions of the STFT and the WT of a multicomponent signal. From those mathematical expressions it is inferred that the STFT and the WT cross terms: (1) occur at the intersection of the respective transforms of the two signals under consideration, whereas the WD cross terms occur at mid-time-frequency of the two signals; (2) are oscillatory in nature, as are the WD cross terms, and are modulated by a cosine whose argument is a function of the difference in center times and center frequencies of the signals under consideration; and (3) can have a maximum amplitude as large as twice the product of the magnitude of the transforms of the two signals in question, like WD cross terms. It is shown that the presence of these cross terms could lead to problems in analyzing a multicomponent signal. The consequences of this effect with respect to speech applications are discussed. >
Book•
01 Jan 1992
TL;DR: In this paper, the authors discuss signal analysis discrete-time signal processing, wavelets for Quincunx pyramid, transform maxima and multiscale edges, among other topics; numerical analysis; other applications the optical wave transform, continuous wavelet transform, quantum mechanics; and theoretical develop
Abstract: Contributions discuss signal analysis discrete-time signal processing, wavelets for Quincunx pyramid, transform maxima and multiscale edges, among other topics; numerical analysis; other applications the optical wave transform, continuous wavelet transform, quantum mechanics; and theoretical develop
TL;DR: A preliminary investigation into the wavelet transform application to the study of both ECG and heart rate variability data is described, suggesting that it is well suited to this task.
Journal Article•
TL;DR: The adaptive chirplet transform as discussed by the authors is a special case of the wavelet transform, which can be used to select an optimal set of bases without the need to manually intervene.
Abstract: The "chirplet" transform unifies manyof the disparate signal representation methods. In particular, the wide ra nge of time-frequency (TF) methods such as the Fourier transform, spectrogram, Wigner distribution, ambiguity function, wideband ambiguity function, and wavelet transform may each be shown to be a special case of the chirplet transform. The above-mentioned TF methods as well as many new ones may be derived by selecting appropriate 2-D manifolds from within the 8-D "chirplet space" (with appropriate smoothing kernel). Furthermore, the chirplet transform is a framework for deriving new signal representations. The chirplet transform is a mapping from a 1-D domain to an 8-D range (in contrast to the wavelet, for example, which is a 1-D to 2-D mapping). Display of the 8-D space is at best difficult. (Although it may be displayed by moving a mesh around in a 3-D virtual world, the whole space cannot be statically displayed in its entirety.) Computation of the 8-D range is also difficult. The adaptive chirplet transform attempts to alleviate some of these problems by selecting an optimal set of bases without the need to manually intervene. The adaptive chirplet, based on expectation maximization, may also form the basis for a classifier (such as a radial basis function neural network) in TF space.
TL;DR: Close-form expressions are derived for the worst-case detection performance for all possible mismatch signals of a given energy, which make it possible to evaluate and compare the performance of various transient detection algorithms, for both single-channel and multichannel data.
Abstract: The problem of detecting short-duration nonstationary signals, which are commonly referred to as transients, is addressed. Transients are characterized by a signal model containing some unknown parameters, and by a 'model mismatch' representing the difference between the model and the actual signal. Both linear and nonlinear signal models are considered. The transients are assumed to undergo a noninvertible linear transformation prior to the application of the detection algorithm. Examples of such transforms include the short-time Fourier transform, the Gabor transform, and the wavelet transform. Closed-form expressions are derived for the worst-case detection performance for all possible mismatch signals of a given energy. These expressions make it possible to evaluate and compare the performance of various transient detection algorithms, for both single-channel and multichannel data. Numerical examples comparing the performance of detectors based on the wavelet transform and the short-time Fourier transform are presented. >
26 May 1992
Patent•
NEC1
TL;DR: In a wavelet transform apparatus including a plurality of series of delay units and at least one convolution calculating circuit, the two-band analysis filter circuit is operated in multiplicity as mentioned in this paper.
Abstract: In a wavelet transform apparatus including a plurality of series of delay units and at least one convolution calculating circuit to form a two-band analysis filter circuit, the two-band analysis filter circuit, i.e., the convolution calculating circuit is operated in multiplicity.
TL;DR: The wavelet transform is an expansion onto a basis of functions which are affine in the physical domain (e.g. time) as discussed by the authors, which is an expanison of an arbitrary function onto a localised basis of multiscale chirps (swept frequency wave packets).
Abstract: A novel transform is proposed, which is an expanison of an arbitrary function onto a localised basis of multiscale chirps (swept frequency wave packets) for which the term ‘chirplets’ has been used. The wavelet transform is an expansion onto a basis of functions which are affine in the physical domain (e.g. time). In other words they are translates and dilates of one mother wavelet. The proposed basis is an extension of affinity, from the physical (time) domain, to the time-frequency domain. The basis includes both the wavelet and the short-time Fourier transform (STFT) as special cases (the degree of freedom modulation is simply attained through a translation in frequency). Furthermore, the basis include shear in time, and shear in frequency, leading to a broader class of chirping bases. Numerous practical applications of the chirplet have been found, such as in Doppler radar signal processing.
TL;DR: It is well known that the infinite Kramers-Kronig transform is equivalent to the infinite Hilbert transform, which is the equivalent of the allied Fourier integrals as discussed by the authors.
Abstract: It is well known that the infinite Kramers–Kronig transform is equivalent to the infinite Hilbert transform, which is equivalent to the allied Fourier integrals. The Hilbert transform can thus be i...
TL;DR: Computational results and theoretical analysis indicate that image reconstruction from the localized phase representation requires fewer computer operations and yields an improved rate of convergence compared to reconstructing from the global phase representation.
Abstract: The authors present a novel approach to image representation using partial information defined by the localized phase. The scheme is implemented using the short-time (short-distance) Fourier transform. This is a generalization of the Gabor scheme which is well-established with regard to biological representation of visual information at the level of the visual cortex. Similar to processing in vision, the DC component is first extracted from the signal and treated separately. Computational results and theoretical analysis indicate that image reconstruction from the localized phase representation requires fewer computer operations and yields an improved rate of convergence compared to reconstruction from the global phase representation. It is also implementable with fast algorithms using highly parallel architecture. >
TL;DR: A joint transform correlation system based on wavelet transforms is introduced that enables this optical correlator to identify the specific features and distinguish similar characters in images.
Abstract: A joint transform correlation system based on wavelet transforms is introduced. The selection of wavelets and the optical wavelet transform of images enables this optical correlator to identify the specific features and distinguish similar characters. Preliminary experimental results are given.
TL;DR: A novel scheme for optical realization of wavelet transform for a one-dimensional signal is described and some preliminary experimental results are demonstrated.
Abstract: A novel scheme for optical realization of wavelet transform for a one-dimensional signal is described. Using commercially available components, the proposed system can perform wavelet transform in real time. Some preliminary experimental results are demonstrated.
23 Mar 1992
TL;DR: The authors have developed an expansion they call the chirplet transform, which has been successfully applied to a wide variety of signal processing applications, including radar and image processing.
Abstract: The authors have developed an expansion they call the chirplet transform. It has been successfully applied to a wide variety of signal processing applications, including radar and image processing. There has been a recent debate as to the relative merits of an affine-in-time (wavelet) transform and the classical short-time Fourier transform (STFT) for the analysis of nonstationary phenomena. Chirplet filters embody both the wavelet and STFT as special cases by decoupling the filter bandwidths and center frequencies. Chirplets, by their embodiment of affine geometry in the time-frequency (TF) plane, may also include shears in time and frequency (chirps) and even time-bandwidth product variation (noise bursts) if desired. The most general chirplets may be derived from one or more basic ('mother') chirplets by the transformations or perspective geometry in the TF plane. >