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Showing papers on "Wavelet published in 1990"


Journal ArticleDOI
TL;DR: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied and the notion of time-frequency localization is made precise, within this framework, by two localization theorems.
Abstract: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems. >

6,180 citations



Book
01 Feb 1990
TL;DR: The Heisenberg boxes must be congruent, which minimizes entropy over all bases corresponding to disjoint dyadic covers of the segment.
Abstract: Best Level: this forces all of the Heisenberg boxes to have the same time scale. In particular, they must be congruent. There are (log N) such bases for a segment of length N, and the one displayed has minimum entropy. Best Basis: this minimizes entropy over all bases corresponding to disjoint dyadic covers of the segment. There are more than 2 N such bases for a segment of length N. Printing the Window The entire contents of the key window may be printed at full scale with the \Print" menu item. Une base orthonormale de L 2 (R), dont les el ements sont bien localis es dans l'espace de phase et leurs supports adapt es a toute partition sym etrique de l'espace des fr equences, s erie I, C. R.

511 citations


Journal ArticleDOI
TL;DR: This analysis shows that forward running wavelets dominate during both the acceleration and deceleration phases of blood flow in the aorta, and is a time domain analysis which can be applied to nonperiodic or transient flow.
Abstract: The one-dimensional equations of flow in the elastic arteries are hyperbolic and admit nonlinear, wavelike solutions for the mean velocity, U, and the pressure, P. Neglecting dissipation, the solutions can be written in terms of wavelets defined as differences of the Riemann invariants across characteristics. This analysis shows that the product, dUdP, is positive definite for forward running wavelets and negative definite for backward running wavelets allowing the determination of the net magnitude and direction of propagating wavelets from pressure and velocity measured at a point in the artery. With the linearizing assumption that intersecting wavelets are additive, the forward and backward running wavelets can be separately calculated. This analysis, applied to measurements made in the ascending aorta of man, shows that forward running wavelets dominate during both the acceleration and deceleration phases of blood flow in the aorta. The forward and backward running waves calculated using the linearized analysis are similar to the results of an impedance analysis of the data. Unlike the impedance analysis, however, this is a time domain analysis which can be applied to nonperiodic or transient flow.

453 citations


Journal ArticleDOI
TL;DR: Orthonormal wavelet bases are used to provide a new construction for nearly 1/f processes from a set of uncorrelated random variables.
Abstract: While so-called 1/f or scaling processes emerge regularly in modeling a wide range of natural phenomena, as yet no entirely satisfactory framework has been described for the analysis of such processes. Orthonormal wavelet bases are used to provide a new construction for nearly 1/f processes from a set of uncorrelated random variables. >

290 citations


Journal ArticleDOI
TL;DR: The discrete wavelet transform can be implemented in VLSI more efficiently than the FFT, and a single chip implementation is described.
Abstract: The wavelet transform is a very effective signal analysis tool for many problems for which Fourier based methods have been inapplicable, expensive for real-time applications, or can only be applied with difficulty. The discrete wavelet transform can be implemented in VLSI more efficiently than the FFT. A single chip implementation is described.

198 citations


Proceedings ArticleDOI
03 Apr 1990
TL;DR: A two-step scheme for image compression that takes into account psychovisual features in space and frequency domains is proposed, and a progressive transmission scheme is presented, particularly well adapted to progressive transmission.
Abstract: A two-step scheme for image compression that takes into account psychovisual features in space and frequency domains is proposed. A wavelet transform is first used in order to obtain a set of orthonormal 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 the number of pixels required to describe the image at a constant. 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. The wavelet transform is particularly well adapted to progressive transmission. >

160 citations


Proceedings ArticleDOI
03 Apr 1990
TL;DR: The discrete version of the wavelet transform, which has recently emerged as a powerful tool for nonstationary signal analysis is closely related to filter banks, which have been studied in digital signal processing.
Abstract: The discrete version of the wavelet transform, which has recently emerged as a powerful tool for nonstationary signal analysis is closely related to filter banks, which have been studied in digital signal processing. Also, multiresolution signal analysis has been used in image processing. The relationship between these techniques is indicated. It is shown how to construct biorthogonal systems with linear-phase finite impulse response (FIR) filters and with regular analysis and synthesis. Some examples of practical interest are given. The complexity of the discrete wavelet transform is also discussed. >

130 citations


01 Dec 1990
TL;DR: An adaptative version of the algorithm exists that allows one to reduce in a significant way the number of degrees of freedom required for a good computation of the solution of the Burgers equation.
Abstract: The Burgers equation with a small viscosity term, initial and periodic boundary conditions is resolved using a spatial approximation constructed from an orthonormal basis of wavelets. The algorithm is directly derived from the notions of multiresolution analysis and tree algorithms. Before the numerical algorithm is described these notions are first recalled. The method uses extensively the localization properties of the wavelets in the physical and Fourier spaces. Moreover, the authors take advantage of the fact that the involved linear operators have constant coefficients. Finally, the algorithm can be considered as a time marching version of the tree algorithm. The most important point is that an adaptive version of the algorithm exists: it allows one to reduce in a significant way the number of degrees of freedom required for a good computation of the solution. Numerical results and description of the different elements of the algorithm are provided in combination with different mathematical comments on the method and some comparison with more classical numerical algorithms.

114 citations


Patent
31 May 1990
TL;DR: In this paper, a method for coding an image or other two-dimensional data array to provide a sequence of images having differing spatial frequency content is disclosed, which generates the output images by taking weighted sums of the pixels in the input image.
Abstract: A method for coding an image or other two-dimensional data array to provide a sequence of images having differing spatial frequency content is disclosed The method generates the output images by taking weighted sums of the pixels in the input image The weights are sets of two-dimensional irreducible scaling and wavelet coefficients

111 citations


Patent
Wayne M. Lawton1
24 Jul 1990
TL;DR: In this paper, a modular digital signal processor system for calculating wavelet-analysis transformations and wavelet synthesis transformations of one-dimensional numerical data and multidimensional numerical data for solving speech processing and other problems is presented.
Abstract: A modular digital signal processor system for calculating "wavelet-analysis transformations" and "wavelet-synthesis transformations" of one-dimensional numerical data and multi-dimensional numerical data for solving speech processing and other problems. The system includes one or more "dual-convolver" components, "analyzer-adjunct" components, "synthesizer-adjunct" components, "de-interleaver components" and "interleaver components", and specific configurations of these components for implementing specific functions. Each dual-convolver is capable of loading a finite number of numerical values into its coefficient registers and subsequently performing two convolution operations on input sequences of numerical values to produce an output sequence of numerical values. Two dual-convolvers are configured with an analyzer adjunct or synthesizer adjunct to respectively build a single stage analyzer or synthesizer. Analyzers and synthesizers are configured in conjunction with interleavers and de-interleaver components to build wavelet sub-band processors capable of decomposing one-dimensional sequences of numerical data or multi-dimensional arrays of numerical data into constituent wavelets and to synthesize the original sequences or arrays from their constituent wavelets. Synthesizers are configured with interleavers to build function generators capable of calculating functions including wavelet functions to within any specified degree of detail.

Journal ArticleDOI
TL;DR: This letter has found that using the wavelet transform in time and space, combined with a multiresolution approach, leads to an efficient and effective method of compression.
Abstract: This letter present results on using wavelet transforms in both space and time for compression of real time digital video data. The advantages of the wavelet transform for static image analysis are well known.2 We have found that using the wavelet transform in time and space, combined with a multiresolution approach, leads to an efficient and effective method of compression. In addition, the computational requirements are considerably less than for other compression methods, and are more suited to VLSI implementation. Some preliminary results of compression on a sample video will be presented.

Proceedings ArticleDOI
01 Jun 1990
TL;DR: In this paper, the use of orthonormal bases of compactly supported wavelets to represent a discrete signal in 2 dimensions yields a localized representation of coefficient energy, and subsequent coding of the multiresolution representation is achieved through techniques such as scalar/vector quantization, hierarchical quantization and entropy coding to achieve compression.
Abstract: Multilevel unitary wavelet transform methods for image compression are described. The sub-band decomposition preserves geometric image structure within each sub-band or level. This yields a multilevel image representation. The use of orthonormal bases of compactly supported wavelets to represent a discrete signal in 2 dimensions yields a localized representation of coefficient energy. Subsequent coding of the multiresolution representation is achieved through techniques such as scalar/vector quantization, hierarchical quantization, entropy coding, and non-linear prediction to achieve compression. Performance advantages over the Discrete Cosine Transform are discussed. These include reduction of errors and artifacts typical of Fourier-based spectral methods, such as frequency-domain quantization noise and the Gibbs phenomenon. The wavelet method also eliminates distortion arising from data blocking. The paper includes a quick review of past/present compression techniques, with special attention paid to the Haar transfOrm, the simplest wavelet transform, and conventional Fourier-based subband coding. Computational results are presented.

Journal ArticleDOI
TL;DR: In this paper, the wavelet transform is applied to two-dimensional dye concentration data in turbulent jets at moderate Reynolds numbers, revealing the nature and limitations of scale similarity of the inner structure of the scalar, and the stringiness associated with small scales.

Journal Article
TL;DR: The wavelet transform appears to be a natural alternative to the decompositions commonly used in fluid dynamics and turbulence (mainly the Fourier decomposition) as mentioned in this paper, and the most attractive properties of wavelets are reviewed and explained using the classical language of turbulence.
Abstract: The basic definitions and the most attractive properties of the wavelet transform are reviewed and explained using the classical language of turbulence The wavelet transform appears to be a natural alternative to the decompositions commonly used in fluid dynamics and turbulence (mainly the Fourier decomposition)

Journal ArticleDOI
TL;DR: In this paper, a wave theoretical wavelet estimation method is derived for estimating the total wavelet, including the phase and source array pattern, when the source is completely unknown (discrete and/or continuously distributed) the method predicts the wavefield due to this source.
Abstract: A new and general wave theoretical wavelet estimation method is derived. Knowing the seismic wavelet is important both for processing seismic data and for modeling the seismic response. To obtain the wavelet, both statistical (e.g., Wiener-Levinson) and deterministic (matching surface seismic to well-log data) methods are generally used. In the marine case, a far-field signature is often obtained with a deep-towed hydrophone. The statistical methods do not allow obtaining the phase of the wavelet, whereas the deterministic method obviously requires data from a well. The deep-towed hydrophone requires that the water be deep enough for the hydrophone to be in the far field and in addition that the reflections from the water bottom and structure do not corrupt the measured wavelet. None of the methods address the source array pattern, which is important for amplitude-versus-offset (AVO) studies.This paper presents a method of calculating the total wavelet, including the phase and source-array pattern. When the source locations are specified, the method predicts the source spectrum. When the source is completely unknown (discrete and/or continuously distributed) the method predicts the wavefield due to this source. The method is in principle exact and yet no information about the properties of the earth is required. In addition, the theory allows either an acoustic wavelet (marine) or an elastic wavelet (land), so the wavelet is consistent with the earth model to be used in processing the data. To accomplish this, the method requires a new data collection procedure. It requires that the field and its normal derivative be measured on a surface. The procedure allows the multidimensional earth properties to be arbitrary and acts like a filter to eliminate the scattered energy from the wavelet calculation. The elastic wavelet estimation theory applied in this method may allow a true land wavelet to be obtained. Along with the derivation of the procedure, we present analytic and synthetic examples.

Journal ArticleDOI
TL;DR: In this paper, the fractal Hausdorff dimension of the data space consisting of the centroid velocities over the x-y plane is noted to be definable as 2.35 + or - 0.01.
Abstract: In the present wavelet analysis of the C-13O spectral data from L1551's outflow region, the fractal Hausdorff dimension of the data space consisting of the centroid velocities over the x-y plane is noted to be definable as 2.35 + or - 0.01. An analysis of surface density yields a dimensional value close to 3, rather than Prasad and Sreenivasan's (1990) 2.36 + or - 0.05 for the Kolmogorov range of the turbulence associated with a water jet in quiet surroundings; this suggests that there is gas between the emissivity peaks, as supported by the larger dominant structure found for the surface density. 13 refs.

DissertationDOI
01 Jan 1990
TL;DR: In this article, Benedetto and Heil showed that the Wiener transform is an invertible mapping of the spectrum of infinite-energy signals, and that the higher-dimensional variation spaces are complete by using Masani's helices; this generalizes a one-dimensional result of Lau and Chen.
Abstract: Title of Dissertation: WIENER AMALGAM SPACES IN GENERALIZED HARMONIC ANALYSIS AND WAVELET THEORY Christopher Edward Heil, Doctor of Philosophy, 1990 Dissertation directed by: Professor John J. Benedetto, Department of Mathematics This thesis is divided into four parts. Part I, Introduction and Notation, describes the results contained in the thesis and their background. Part II, Wiener Amalgam Spaces, is an expository introduction to Feichtinger's general amalgam space theory, which is used in the remainder of the thesis to formulate and prove results. Part III, Generalized Harmonic Analysis, presents new results in that area. Part IV, Wavelet Theory, contains exposition and miscellaneous results on Gabor ( also known as Weyl-Heisenberg) wavelets. Amalgam, or mixed-norm, spaces are Banach spaces of functions determined by a norm which distinguishes between local and global properties of functions. Specific cases were introduced by Wiener. Feichtinger has developed a far-reaching generalization of amalgam spaces, which allows general function spaces norms as local or global components. We use Feichtinger's amalgam theory, on d-dimensional Euclidean space under componentwise multiplication, to prove that the Wiener transform (introduced by Wiener to analyze the spectra of infinite-energy signals) is an invertible mapping of -_______ _. . ........ _.,... .....,.~~~~the amalgam space with local L2 and global LtJ. components onto an appropriate space defined in terms of the variation of functions, for each q between one and infinity. As corollaries, we obtain results of Beurling on the Fourier transform and results of Lau and Chen on the Wiener transform. Moreover, our results are carried out in higher dimensions. In addition, we prove that the higher-dimensional variation spaces are complete by using Masani's helices; this generalizes a one-dimensional result of Lau and Chen. In wavelet theory, we present a survey of frames in Hilbert and Banach spaces and the use of the Zak transform in analyzing Gabor wavelets. Frames are an alternative to unconditional bases in these spaces; like bases, they Provide representations of each element of the space in terms of the frame elements, and do so in a way in which the scalars in the representation are explicitly known. However, unlike bases, the representations need not be unique. We then discuss the specific case of Gabor frames in the space of square-integrable functions, concentrating on the role of the Zak transform in the analysis of such frames. J ,, ~ ~ :I (. • l . '

Patent
20 Feb 1990
TL;DR: In this paper, the correlation is done both on the signals coming from the correlation at the previous step by the scale Φ O signal and on the signal coming from correlation at preceding step by signal Φ o.
Abstract: To analyze short-duration signals by means of an analysis using MARRAT's algorithm with the wavelets of DAUBECHIES, at each step the correlation is done both on the signals coming from the correlation at the previous step by the scale Φ O signal and on the signals coming from the correlation at the preceding step by the signal Φ O . Thus, at the stage p, there is obtained a homogeneous signal consisting of 2 p ×N p points of analysis. This makes it possible, in the method of analysis by wavelets, to obtain a depiction of the results having a homogeneous form making it easier to interpret them.

Proceedings ArticleDOI
03 Apr 1990
TL;DR: It is shown, in particular, that Gaussian kernels provide a continuous transition between spectrograms and scalograms by means of the Wigner-Ville distribution, which makes it a versatile tool for the analysis of nonstationary signals.
Abstract: A formalism of signal energy representations depending on time and scale is presented. Precise links between time-frequency and time-scale energy distributions are provided. It is known that a full description of the former is given by Cohen's class, which can be described as a generalization of the spectrogram appropriately parameterized by a smoothing function acting on the Wigner-Ville distribution. A full description of the latter is given, resulting in a class of representations in which the smoothing of the Wigner-Ville distribution is scale-dependent. Through proper choice of the smoothing function, interesting properties may be imposed on the representation, which makes it a versatile tool for the analysis of nonstationary signals. Also, specific choices allow known definitions to be recovered (including the Bertrands' and the energetic version of the wavelet transform, referred to as the scalogram). Another very flexible choice uses separable smoothing functions. It is shown, in particular, that Gaussian kernels provide a continuous transition between spectrograms and scalograms by means of the Wigner-Ville distribution. >

Proceedings ArticleDOI
04 Dec 1990
TL;DR: An efficient architecture is presented to synthesize filters of arbitrary orientations from linear combinations of basis filters, allowing one to adaptively 'steer' a filter to any orientation, and to determine analytically the filter output as a function of orientation.
Abstract: An efficient architecture is presented to synthesize filters of arbitrary orientations from linear combinations of basis filters, allowing one to adaptively 'steer' a filter to any orientation, and to determine analytically the filter output as a function of orientation. The authors show how to design and steer filters, and present examples of their use in several tasks: the analysis of orientation and phase, angularly adaptive filtering, edge detection, and shape-from-shading. It is also possible to build a self-similar steerable pyramid representation which may be considered to be a steerable wavelet transform. The same concepts can be generalized to the design of 3-D steerable filters, which should be useful in the analysis of image sequences and volumetric data. >

Journal ArticleDOI
TL;DR: In this article, an orthonormal wavelet analysis with conditional sampling is applied to data of wind turbulence, yielding Kolmogorov's spectrum and the dissipation correlation with the intermittency exponent,11"0.2.
Abstract: Orthonormal wavelet expansion is applied to experimental data of turbulence. A direct relation is found between the wavelet spectrum and the Fourier spectrum. The orthonormal wavelet analysis with conditional sampling is applied to data of wind turbulence, yielding Kolmogorov's spectrum and the dissipation correlation with the intermittency exponent ,11"'0.2. Fourier transform method is a fundamental and indispensable tool in data analysis since it enables us to decompose data into components with different scales. Many fundamental properties of physical systems have been described in terms of Fourier spectrum, that is, the amplitude of Fourier coefficients. However, since Fourier spectrum totally ignores the phase of each Fourier coefficient, it lacks infor­ mation about positions of local events which underlie the characteristics of the spectrum. The Fourier spectrum analysis therefore encounters difficulty in ana)yzing data in temporal or spatial intervals which include different kinds of local events. The method of scale analysis applicable even to such complicated situations should enable us to identify the origin of characteristics of the spectrum with local events occurring in physical space. This requirement would be satisfied at least partially by an expansion in terms of basis functions which are local both in physical and Fourier space, although the locality is limited in its extent by the uncertainty principle. In this paper, we adopt an expansion method' in terms of orthonormal wavelets (discrete wavelet analysis) as one of such types of expansion method. The orthonormal wavelet expansion is a discrete version of continuous wavelet analysis.!) The latter is an integral transform method with kernel functions obtained by translating and dilating a l~calized function (analyzing wavelet). The continuous wavelet transform of a square integrable function is an isometric transform between a Hilbert space (V space on Rn) and V space on a locally compact topological group (a group of translation and dilation) with its Haar measure: 2H ) The continuous wavelet is a useful tool especially for studying a singularity or a fractal structure of a given function. 5H ) In particular, the energy cascade process in fully-developed turbulence has been captured remarkably in such an analysis. 7 ) However, it is not very advantageous if one is interested in the energetic aspect because the kernel functions are not mutually orthogonal and no physically immediate meaning can be associated with the expansion coefficients.

Patent
23 Aug 1990
TL;DR: In this paper, a system for compressing images using wavelets having finite support is described, which is equivalent to expanding the image using a system of wavelets with finite support.
Abstract: A system (10) compressing images is disclosed. The system utilizes a transformation (12) which is equivalent to expanding the image using a system of wavelets having finite support.

Journal ArticleDOI
David F. Aldridge1
TL;DR: The purpose of this note is to quantitatively describe the characteristics of a relatively unfamiliar wavelets, called the Berlage wavelet, that is appropriate for seismic modeling studies.
Abstract: Symmetric wavelets are commonly used in seismic modeling studies. However, accurate simulation of many physical wave propagation phenomena requires a causal waveform possessing a certain degree of differentiability. The purpose of this note is to quantitatively describe the characteristics of a relatively unfamiliar wavelet, called the Berlage wavelet, that is appropriate for such studies.

01 Dec 1990
TL;DR: In this paper, the authors performed an analysis of planar cuts through three-dimensional turbulent fields (planar channel flow and mixing layer) using the 2D continuous wavelet transform.
Abstract: We perform an analysis of planar cuts through three dimensional turbulent fields (planar channel flow and mixing layer) using the 2D continuous wavelet transform We propose two new diagnostics: (1) a measure of intermittency I(r, vector x), which is the ratio of local energy and average energy at a given scale r; and (2) a local Reynolds number, defined on the local velocity contribution at a given scale, computed from the wavelet transform of the three velocity components, the scale of the transform, and molecular viscosity; this gives a representation of the local non-linearity of the flow viewed in both space and scale We find, for the analyzed flows, strong small-scale intermittency located in the ejection regions for the channel flow and in the vortex core of the mixing layer

Proceedings ArticleDOI
16 Jun 1990
TL;DR: An algorithm that reconstructs one-dimensional signals and images from their sharper variation points at dyadic scales is described and it is proved that the evolution across scales of the wavelet maxima characterizes the local shape of the sharp variations of the signal.
Abstract: An algorithm that reconstructs one-dimensional signals and images from their sharper variation points at dyadic scales is described. This algorithm exactly reconstructs images from their multiscale edges. It is proved that the evolution across scales of the wavelet maxima characterizes the local shape of the sharp variations of the signal. One can thus not only detect edges but also classify them. The wavelet maxima representation is a new reorganization of the image information that makes it possible to develop algorithms uniquely based on edges for solving image processing problems. >

Journal ArticleDOI
TL;DR: In this article, a wavelet analysis over the circle is presented, in which the spaces of infinitely times differentiable functions, tempered distributions, and square integrable functions are analyzed by means of the wavelet transform.
Abstract: The construction of a wavelet analysis over the circle is presented. The spaces of infinitely times differentiable functions, tempered distributions, and square integrable functions over the circle are analyzed by means of the wavelet transform.

Patent
02 Oct 1990
TL;DR: In this article, an improved method and apparatus for conducting seismic prospecting is provided wherein more accurate estimates of the source wavelet can be achieved, where the pulses are converted to digital signals in a signal processor (212) and stored on a recorder (210).
Abstract: An improved method and apparatus for conducting seismic prospecting is provided wherein more accurate estimates of the source wavelet can be achieved. Receivers (202) detect pulses at vertically offset locations. The pulses are converted to digital signals in a signal processor (212) and stored on a recorder (210). A processor (108) operates on the stored signals to produce an output wavelet A'(107 ,θ) in accord with the equation ∫(P.G n -G.P n )dx g .

Proceedings ArticleDOI
04 Dec 1990
TL;DR: A coding algorithm is described that selects the most important image edges in order to obtain a compact representation and gives a precise characterization of the edge type which can be used for pattern recognition.
Abstract: The edges of an image can be detected at different scales from the local maxima of its wavelet transform. an algorithm is described that reconstructs images from their edges at dyadic scales. The wavelet maxima representation is a novel reorganization of the image information that makes it possible to develop algorithms uniquely based on edges for solving image processing and computer vision problems. The evolution of the wavelet maxima across scales gives a precise characterization of the edge type which can be used for pattern recognition. A coding algorithm is described that selects the most important image edges in order to obtain a compact representation. >

Book ChapterDOI
01 Jan 1990
TL;DR: The goal of computer vision is to imitate the human ability to interpret the information content of images and it is shown that multiresolution approaches to images provide efficient strategies for computer vision algorithms.
Abstract: The goal of computer vision is to imitate the human ability to interpret the information content of images. An image is acquired by a video camera and a digitizer provides as output an array of 512 by 512 points called pixels. Each pixel gives the value of the local light intensity in the image. In computer vision we are developing numerical algorithms for understanding these images. For example, one would like to build a computer program which is able to recognize that image 8(a) is the portrait of a woman with a hat. Since the work of Rosenfeld and Thurston [1] several researchers have shown that multiresolution approaches to images provide efficient strategies for computer vision algorithms. An image can be interpreted as a sum of details which appear at different resolutions. Such a multiresolution decomposition is meaningful because to each resolution corresponds a different type of structure in the image. At a coarse resolution these details will correspond to borders of large structures like the hat of image 8(a) whereas at a finer resolution these details will rather provide texture information like in the hairs of the woman.