Abstract: 1 Wavelets.- 1.1 What can wavelets offer?.- 1.2 General remarks.- 1.3 Data compression.- 1.4 Local adaptivity.- 1.5 Nonlinear smoothing properties.- 1.6 Synopsis.- 2 The Haar basis wavelet system.- 3 The idea of multiresolution analysis.- 3.1 Multiresolution analysis.- 3.2 Wavelet system construction.- 3.3 An example.- 4 Some facts from Fourier analysis.- 5 Basic relations of wavelet theory.- 5.1 When do we have a wavelet expansion?.- 5.2 How to construct mothers from a father.- 5.3 Additional remarks.- 6 Construction of wavelet bases.- 6.1 Construction starting from Riesz bases.- 6.2 Construction starting from m0.- 7 Compactly supported wavelets.- 7.1 Daubechies' construction.- 7.2 Coiflets.- 7.3 Symmlets.- 8 Wavelets and Approximation.- 8.1 Introduction.- 8.2 Sobolev Spaces.- 8.3 Approximation kernels.- 8.4 Approximation theorem in Sobolev spaces.- 8.5 Periodic kernels and projection operators.- 8.6 Moment condition for projection kernels.- 8.7 Moment condition in the wavelet case.- 9 Wavelets and Besov Spaces.- 9.1 Introduction.- 9.2 Besov spaces.- 9.3 Littlewood-Paley decomposition.- 9.4 Approximation theorem in Besov spaces.- 9.5 Wavelets and approximation in Besov spaces.- 10 Statistical estimation using wavelets.- 10.1 Introduction.- 10.2 Linear wavelet density estimation.- 10.3 Soft and hard thresholding.- 10.4 Linear versus nonlinear wavelet density estimation.- 10.5 Asymptotic properties of wavelet thresholding estimates.- 10.6 Some real data examples.- 10.7 Comparison with kernel estimates.- 10.8 Regression estimation.- 10.9 Other statistical models.- 11 Wavelet thresholding and adaptation.- 11.1 Introduction.- 11.2 Different forms of wavelet thresholding.- 11.3 Adaptivity properties of wavelet estimates.- 11.4 Thresholding in sequence space.- 11.5 Adaptive thresholding and Stein's principle.- 11.6 Oracle inequalities.- 11.7 Bibliographic remarks.- 12 Computational aspects and software.- 12.1 Introduction.- 12.2 The cascade algorithm.- 12.3 Discrete wavelet transform.- 12.4 Statistical implementation of the DWT.- 12.5 Translation invariant wavelet estimation.- 12.6 Main wavelet commands in XploRe.- A Tables.- A.1 Wavelet Coefficients.- A.2.- B Software Availability.- C Bernstein and Rosenthal inequalities.- D A Lemma on the Riesz basis.- Author Index.

Abstract: In this paper, we introduce a new multiresolution watermarking method for digital images. The method is based on the discrete wavelet transform (DWT). Pseudo-random codes are added to the large coefficients at the high and middle frequency bands of the DWT of an image. It is shown that this method is more robust to proposed methods to some common image distortions, such as the wavelet transform based image compression, image rescaling/stretching and image halftoning. Moreover, the method is hierarchical.

Abstract: This paper describes a new motion estimation algorithm that is potentially useful for both computer vision and video compression applications. It is hierarchical in structure, using a separable two-dimensional (2-D) discrete wavelet transform (DWT) on each frame to efficiently construct a multiresolution pyramid of subimages. The DWT is based on a complex-valued pair of four-tap FIR filters with Gabor-like characteristics. The resulting complex DWT (CDWT) effectively implements an analysis by an ensemble of Gabor-like filters with a variety of orientations and scales. The phase difference between the subband coefficients of each frame at a given subpel bears a predictable relation to a local translation in the region of the reference frame subtended by that subpel. That relation is used to estimate the displacement field at the coarsest scale of the multiresolution pyramid. Each estimate is accompanied by a directional confidence measure in the form of the parameters of a quadratic matching surface. The initial estimate field is progressively refined by a coarse-to fine strategy in which finer scale information is appropriately incorporated at each stage. The accuracy, efficiency, and robustness of the new algorithm are demonstrated in comparison testing against hierarchical implementations of intensity gradient-based and fractional-precision block matching motion estimators.

Abstract: A method based on the discrete wavelet transform was applied to the regional-residual separation of potential fields and to the filtering of local anomalies. A specific space-scale wavelet analysis, called multiresolution analysis, allowed decomposition of the signal with respect to a vast range of scales. Different analysing wavelets were applied to anomalies in both synthetic and real cases, but the more appropriate basis needed to be chosen by requiring the maximum compactness for the multiresolution analysis. Moreover, since such analysis was found not to be shift-invariant, the same criterion was applied to choosing the best signal shift. Application of the technique to both synthetic and real cases produced an optimal space-scale representation of the fields and a consistent regional-residual separation. Furthermore, the space localization allowed a variety of filtered signals to be obtained, each one with a specific scale and local area content. Fourier and wavelet analyses were both applied to the filtering out of the intense Etna anomaly from the aeromagnetic field of Sicily. The wavelet method was more powerful, suppressing only the Etna volcano anomaly and leaving the rest of the map practically unchanged.

Abstract: Wavelet analysis is a relatively new mathematical discipline which has generated much interest in both theoretical and applied mathematics over the past decade. Crucial to wavelets are their ability to analyze different parts of a function at different scales and the fact that they can represent polynomials up to a certain order exactly. As a consequence, functions with fast oscillations, or even discontinuities, in localized regions may be approximated well by a linear combination of relatively few wavelets. In comparison, a Fourier expansion must use many basis functions to approximate such a function well. These properties of wavelets have lead to some very successful applications within the field of signal processing. This dissertation revolves around the role of wavelets in scientific computing and it falls into three parts: Part I gives an exposition of the theory of orthogonal, compactly supported wavelets in the context of multiresolution analysis. These wavelets are particularly attractive because they lead to a stable and very efficient algorithm, namely the fast wavelet transform(FWT). We give estimates for the approximation characteristics of wavelets and demonstrate how and why the FWT can be used as a front-end for efficient image compression schemes. Part II deals with vector-parallel implementations of several variants of the Fast Wavelet Transform. We develop an efficient and scalable parallel algorithm for the FWT and derive a model for its performance. Part III is an investigation of the potential for using the special properties of wavelets for solving partial differential equations numerically. Several approaches are identified and two of them are described in detail. The algorithms developed are applied to the nonlinear Schrodinger equation and Burgers' equation. Numerical results reveal that good performance can be achieved provided that problems are large, solutions are highly localized, and numerical parameters are chosen appropriately, depending on the problem in question.

Abstract: Automated signal classification systems are finding increasing use in many applications for the analysis and interpretation of large volumes of signals. Such systems show consistency of response and help reduce the effect of variabilities associated with human interpretation. This paper deals with the analysis of ultrasonic NDE signals obtained during weld inspection of piping in boiling water reactors. The overall approach consists of three major steps, namely, frequency invariance, multiresolution analysis, and neural network classification. The data are first preprocessed whereby signals obtained using different transducer center frequencies are transformed to an equivalent reference frequency signal. Discriminatory features are then extracted using a multiresolution analysis technique, namely, the discrete wavelet transform (DWT). The compact feature vector obtained using wavelet analysis is classified using a multilayer perceptron neural network. Two different databases containing weld inspection signals have been used to test the performance of the neural network. Initial results obtained using this approach demonstrate the effectiveness of the frequency invariance processing technique and the DWT analysis method employed for feature extraction.

Abstract: A three-dimensional (3-D) multiresolution analysis procedure similar to the finite-difference time-domain (FDTD) method is derived using a complete set of three-dimensional orthonormal bases of Haar scaling and wavelet functions. The expansion of the electric and the magnetic fields in these basis functions leads to the time iterative difference approximation of Maxwell's equations that is similar to the FDTD method. This technique effectively models realistic microwave passive components by virtue of its multiresolution property; the computational time is reduced approximately by half compared to the FDTD method. The proposed technique is validated by analyzing several 3-D rectangular resonators with inhomogeneous dielectric loading. It is also applied to the analyses of microwave passive devices with open boundaries such as microstrip low-pass filters and spiral inductors to extract their S-parameters and field distributions. The results of the proposed technique agree well with those of the traditional FDTD method.

Abstract: Variations of Windowed Fourier Transform and Applications: M. An, A. Brozdzik, I. Gertner, and R. Tolimieri, Weyl-Heisenberg Systems and the Finite Zak Transform. M.J. Bastiaans, Gabors Expansion and the Zak Transform for Continuous-Time and Discrete-Time Signals. W. Schempp, Non-Commutative Affine Geometry and Symbol Calculus: Fourier Transform Magnetic Resonance Imaging and Wavelets. M. Zibulski and Y.Y. Zeevi, The Generalized Gabor Scheme and Its Application In Signal and Image Representation. Construction for Special Waveforms for Specific Tasks: J.S. Byrnes, A Low Complexity Energy Spreading Transform Coder. A. Cohen and N. Dyn, Nonstationary Subdivision Schemes, Multiresolution Analysis, and Wavelet Packets. J. Prestin and K. Selig, Interpolatory and Orthonormal Trigonometric Waves. Redundant Waveform Representations for Signal Processing and Image Analysis: J.J. Benedetto, Noise Reduction in Termsof the Theory of Frames. F. Bergeaud and S. Mallat, Matching Pursuit of Images. Z. Cvetkovi( and M. Vetterli, Overcomplete Expansions and Robustness. Numerical Compression and Applications: A. Averbuch, G. Beylkin, R. Coifman, and M.Israeli, Multiscale Inversion of Elliptic Operators. A. Harten, Multiresolution Representation of Cell-Averaged Data: A Promotional Review. Analysis of Waveform Representations: C.K. Chui and C. Li, Characterizations of Smoothness viaFunctional Wavelet Transforms. M.A. Kon and L.A. Raphael, Characterizing Convergence Rates for Multiresolution. B. Rubin, On Calderon's Reproducing Formula. B. Rubin, Continuous Wavelet Transforms on a Sphere. V.A. Zheludev, Periodic Splines, Harmonic Analysis, and Wavelets. Filter Banks and Image Coding: A.J.E.M. Janssen, A Density Theorem for Time-Continuous Filter Banks. V.E. Katsnelson, Sampling and Interpolation for Functions with Multi-Band Spectrum: The Mean Periodic Continuation Method. R. Lenz and J. Svanberg, Group Theoretical Transforms, Statistical Properties of Image Spaces and Image Coding. Subject Index.

Abstract: Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in the case of band-limited wavelets.

Abstract: We explore joint source-channel coding (JSCC) for time-varying channels using a multiresolution framework for both source coding and transmission via novel multiresolution modulation constellations. We consider the problem of still image transmission over time-varying channels with the channel state information (CSI) available at (1) receiver only and (2) both transmitter and receiver being informed about the state of the channel, and we quantify the effect of CSI availability on the performance. Our source model is based on the wavelet image decomposition, which generates a collection of subbands modeled by the family of generalized Gaussian distributions. We describe an algorithm that jointly optimizes the design of the multiresolution source codebook, the multiresolution constellation, and the decoding strategy of optimally matching the source resolution and signal constellation resolution "trees" in accordance with the time-varying channel and show how this leads to improved performance over existing methods. The real-time operation needs only table lookups. Our results based on a wavelet image representation show that our multiresolution-based optimized system attains gains on the order of 2 dB in the reconstructed image quality over single-resolution systems using channel optimized source coding.

Abstract: Features in satellite images of the oceans often have weak edges. These images also have a significant amount of noise, which is either due to the clouds or atmospheric humidity. The presence of noise compounds the problems associated with the detection of features, as the use of any traditional noise removal technique will also result in the removal of weak edges. Recently, there have been rapid advances in image processing as a result of the development of the mathematical theory of wavelet transforms. This theory led to multifrequency channel decomposition of images, which further led to the evolution of important algorithms for the reconstruction of images at various resolutions from the decompositions. The possibility of analyzing images at various resolutions can be useful not only in the suppression of noise, but also in the detection of fine features and their classification. This paper presents a new computational scheme based on multiresolution decomposition for extracting the features of interest from the oceanographic images by suppressing the noise. The multiresolution analysis from the median presented by Starck-Murtagh-Bijaoui (1994) is used for the noise suppression.

Abstract: Wavelet-based methods have proven their efficiency for visualization at different levels of detail, progressive transmission, and compression of large data sets. The required core of all wavelet-based methods is a hierarchy of meshes that satisfies subdivision-connectivity. This hierarchy has to be the result of a subdivision process starting from a base mesh. Examples include quadtree uniform 2D meshes, octree uniform 3D meshes, or 4-to-1 split triangular meshes. In particular, the necessity of subdivision-connectivity prevents the application of wavelet-based methods on irregular triangular meshes. In this paper, a "wavelet-like" decomposition is introduced that works on piecewise constant data sets over irregular triangular surface meshes. The decomposition/reconstruction algorithms are based on an extension of wavelet-theory allowing hierarchical meshes without property. Among others, this approach has the following features: it allows exact reconstruction of the data set, even for nonregular triangulations, and it extends previous results on Haar-wavelets over 4-to-1 split triangulations.

Abstract: Subdivision is a powerful paradigm for the generaton of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is the construction of curves that interpolate a griven set of points and minimize a fairness functional (variational design). In the context of subdivision, fairing leads to special schemes requiring the solution of a banded linear system at every subdivision step. We present several examples of such schemes including one that reproduces nonuniform interpolating cubic splines. Expressing the construction in terms of certain elementary operations we are able to embed variational subdivision in the lifting framework, a powerful technique to construct wavelet filter banks given a subdivision scheme. This allows us to extend the traditional lifting scheme for FIR filters to a certain class of IIR filters. Consquently, we how how to build variationally optimal curves and associated, stable wavelets in a straightforward fashion. The algorithms to perform the corresponding decomposition and reconstruction transformations are easy to implement and efficient enough for interactive applications.

Abstract: The ARSIS concept is designed to increase the spatial resolution of an image without modification of its spectral contents, by merging structures extracted from a higher resolution image of the same scene, but in a different spectral band. It makes use of wavelet transforms and multiresolution analysis. It is currently applied in an operational way with dyadic wavelet transforms that limit the merging of images whose ratio of their resolution is a power of 2. Rational discrete wavelet transforms can be approximated numerically by rational filter banks which would enable a more general merging. Indeed, in theory, the ratio of the resolution of the images to merge is a power of a certain family of rational numbers. The aim of this paper is to examine whether the use of those approximations of rational wavelet transforms are efficient within the ARSIS concept. This work relies on a particular case: the merging of a 10 m SPOT Panchromatic image and a 30 m Landsat Thematic Mapper multispectral image to synthesize 10m multispectral image TM-HR.

Abstract: We undertake the study of EEG-signals by recourse to a wavelet based multiresolution analysis as adapted to an information-measure-scenario. Different information measures are employed. It is shown that non-extensive ones seem to be of particular usefulness.

Abstract: This paper proposes a unified approach to digital watermarking of images and video based on the 2D and 3D discrete wavelet transforms. The hierarchical nature of the wavelet representation allows multiresolutional detection of the digital watermark, which is a Gaussian distributed random vector added to all the high pass bands in the wavelet domain. We show that, when subjected to distortion from compression or image halftoning, the corresponding watermark can still be correctly identified at each resolution (excluding the lowest one) in the wavelet domain. Computational saving from such a multiresolution watermarking framework is obvious, especially for the video case.

Abstract: This paper reports about a quantitative evaluation of pyramid- based schemes performing a feature-based fusion of data from multispectral and panchromatic imaging sensors having different ground resolutions. A critical point is performances evaluation of image data fusion. A set of quantitative parameters has been recently proposed. Both visual quality, regarded as contrast, presence of fine details, and absence of impairments and artifacts (e.g., blur, ringing), and spectral fidelity (i.e., preservation of spectral signatures) are concerned and embodied in the measurements. The aim of the present work is to provide a comprehensive performance comparison on SPOT data among three feature-based schemes for image fusion, as well as on a specific case study on which multisensor observations were available. Out of the three methods compared, respectively based on high-pass filtering (HPF), wavelet transform (WT), and generalized Laplacian pyramid (GLP), the latter two are far more efficient than the former, thus establishing the advantages for data fusion of a formally multiresolution analysis.

Abstract: [PNA-R9810] Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the so-called pyramid decomposition scheme. This report proposes a general axiomatic pyramid decomposition scheme for signal analysis and synthesis. This scheme comprises the following ingredients: (i) the pyramid consists of a (finite or infinite) number of levels such that the information content decreases towards higher levels; (ii) each step towards a higher level is constituted by an (information-reducing) analysis operator, whereas each step towards a lower level is modeled by an (information-preserving) synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. In this report, several examples are described of linear as well as nonlinear (e.g., morphological) pyramid decomposition schemes. Some of these examples are known from the literature (Laplacian pyramid, morphological granulometries, skeleton decomposition) and some of them are new (morphological Haar pyramid, median pyramid). Furthermore, the report makes a distinction between single-scale and multiscale decomposition schemes (i.e. without or with sample reduction).#[PNA-R9905] In its original form, the wavelet transform is a linear tool. However, it has been increasingly recognized that nonlinear extensions are possible. A major impulse to the development of nonlinear wavelet transforms has been given by the introduction of the lifting scheme by Sweldens. The aim of this report, which is a sequel to a previous report devoted exclusively to the pyramid transform, is to present an axiomatic framework encompassing most existing linear and nonlinear wavelet decompositions. Furthermore, it introduces some, thus far unknown, wavelets based on mathematical morphology, such as the morphological Haar wavelet, both in one and two dimensions. A general and flexible approach for the construction of nonlinear (morphological) wavelets is provided by the lifting scheme. This paper discusses one example in considerable detail, the max-lifting scheme, which has the intriguing property that it preserves local maxima in a signal over a range of scales, depending on how local or global these maxima are.

Abstract: The paper deals with the problem of reconstruction of nonlinearities in a certain class of nonlinear systems of composite structure from their input-output observations when prior information about the system is poor, thus excluding the standard parametric approach to the problem. The multiresolution idea, being the fundamental concept of modern wavelet theory, is adopted, and the Haar multiresolution analysis in particular is applied to construct nonparametric identification techniques of nonlinear characteristics. The pointwise convergence properties of the proposed identification algorithms are established. Conditions for the convergence are given; and for nonlinearities satisfying a local Lipschitz condition, the rate of convergence is evaluated, With applications in mind, the problem of data-driven selection of the optimum resolution degree in the identification procedure, essential for the multiresolution analysis, is considered as well. The theory is verified by computer simulations.

Abstract: A well-documented problem in the analysis of real-world measurements is that the data may suffer from several problems, including data dropouts, an irregularly spaced sampling grid, and time-delayed sampling. These data irregularities render many traditional signal processing techniques unusable, and thus the data must be interpolated onto an even grid before scientific analysis techniques can be applied. We propose a method to perform a reconstruction of nonuniformly sampled signals using a wavelet basis fit in a multiresolutional setting. The advantage of using a wavelet basis is that we are able to not only reconstruct the signal using global information, but we are also able to take advantage of locally dense areas of sampling to reconstruct at higher resolutions.

Abstract: This paper evaluates the performance of an image compression system based on wavelet-based subband decomposition and vector quantization. The images are decomposed using wavelet filters into a set of subbands with different resolutions corresponding to different frequency bands. The resulting subbands are vector quantized using the Linde-Buzo-Gray (1980) algorithm and various fuzzy algorithms for learning vector quantization (FALVQ). These algorithms perform vector quantization by updating all prototypes of a competitive neural network through an unsupervised learning process. The quality of the multiresolution codebooks designed by these algorithms is measured on the reconstructed images belonging to the training set used for multiresolution codebook design and the reconstructed images from a testing set.

Abstract: Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the so-called pyramid decomposition scheme. This report proposes a general axiomatic pyramid decomposition scheme for signal analysis and synthesis. This scheme comprises the following ingredients: (i) the pyramid consists of a (finite or infinite) number of levels such that the information content decreases towards higher levels; (ii) each step towards a higher level is constituted by an (information-reducing) analysis operator, whereas each step towards a lower level is modeled by an (information-preserving) synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. In this report, several examples are described of linear as well as nonlinear (e.g., morphological) pyramid decomposition schemes. Some of these examples are known from the literature (Laplacian pyramid, morphological granulometries, skeleton decomposition) and some of them are new (morphological Haar pyramid, median pyramid). Furthermore, the report makes a distinction between single-scale and multiscale decomposition schemes (i.e. without or with sample reduction).

Abstract: A novel algorithm for ST-segment analysis is developed using the multi-resolution wavelet approach. The system detects the QRS complexes and analyses each beat using the wavelet transform to identify the characteristic points (fiducial points). These fiducial points are, iso-electric level, the J point, and onsets and offsets of the QRS complex and T wave. The algorithm determines the T onset by looking for a point of inflection between the J point and the T peak. Furthermore, detection of characteristic points by the wavelet technique reduces the effect of noise. The results show that the proposed approach gives very accurate ST levels, as compared to the conventional (empirical) technique, at higher heart rates and with different morphologies. The algorithm detects the ST-segment length in 92.3% beats with an error of 4 ms, and in 97.3% beats the error is within 8 ms. The algorithm has been implemented on a TMS320C25 based add-on DSP card connected to a PC to provide the on-line analysis and display of ST-segment data.

Abstract: In this work, adaptive filtering is applied to the different resolutions of the Laplacian pyramid of a noisy image. For natural images, each layer is characterized by an SNR that increases as resolution decreases. Therefore, each filter may be adjusted to achieve adaptivity also across scales. Theoretical frameworks are developed for signal-dependent noise models. A rational Laplacian pyramid is employed to handle multiplicative noise. Experiments carried out on gray-level images corrupted by true and simulated noise, show the potentiality of the approach compared to other established algorithms, in terms both of SNR improvements and of enhancement in visual quality.