We construct oblique multi-wavelets bases which encompass the orthogonal multi-wavelets and the biorthogonal uni-wavelets of Cohen, Deaubechies and Feauveau. These oblique multi- wavelets preserve the advantages of orthogonal and biorthogonal wavelets and enhance the flexibility of wavelet theory to accommodate a wider variety of wavelet shapes and properties. Moreover, oblique multi-wavelets can be implemented with fast vector-filter-bank algorithms. We use the theory to derive a new construction of biorthogonal uni-wavelets.

The pyramid algorithm for computing single wavelet transform coefficients is well-known. The pyramid algorithm can be implemented by using tree-structured multirate filter banks. In this paper, we propose a general algorithm to compute multiwavelet transform coefficients, by adding proper pre multirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be though of as a discrete vector-valued wavelet transform for certain discrete-time vector-valued signals. The proposed algorithm can be also though of as a discrete multiwavelet transform for discrete-time signals. We then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms.

A new approach on the construction of wavelet-frames (affine frames) is presented. It uses a new concept of affine pseudo frames, and a generalized multiresolution structure (GMS). The freedom gained from using pseudo frames allows us to design filters of rapid decay. A fast numerical algorithm for the affine frame decomposition and reconstruction is also a natural consequence of the GMS structure. Examples on applications to data compression and noise suppression are presented. Results are encouraging.

The set of all multiresolution analyses in L²(R) is parameterized by a set of unitary operators which satisfy certain commutation properties.

In this paper we present the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets which does not use the Fourier transform. In contrast with earlier papers we introduce lifting purely from a wavelet transform point of view and only consider the wavelet basis functions in a later stage. We show how lifting leads to a faster, fully in- plane implementation of the wavelet transform. Moreover, it can be used in the construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one function. A typical example of the latter are wavelets on the sphere.

We investigate a general subset of 2D, orthogonal, compactly supported wavelets. This subset, which has a simple parameterization, includes all wavelets with a corresponding wavelet (polyphase) matrix, that can be factored as a product of factors of degree--1, in one variable. In this paper we consider a particular wavelets with vanishing moments. The number of vanishing moments that can be achieved increases with the increase of the McMillan degrees of the wavelet matrix. We design wavelets with the maximal number of vanishing moments for given McMillan degrees, by solving a set of nonlinear constraints, and discuss their relation to regular, smooth wavelets. Design examples are given for two fundamental sampling schemes, the quincunx and the four-band separable sampling.

We construct a multiresolution analysis and wavelets on the manifolds S² and SO(3). For C¹-functions on S² we derive the differentiability conditions for the singular points of their coordinate representation depending on polar coordinates. This leads in a natural way to tensor products of E-spline wavelets and wavelets on the interval. By using quaternion parameterization the case of SO(3) is reduced to the case of S³, which can be dealt with in the same way as S² using a special parameterization.

A method for deriving Divergence-free wavelets is presented. The approach, in 2D, produces the same wavelets as mentioned by Battle. However, Battle claims that to produce the 3D equivalent construct is still an open problem. Our method solves this problem in a simple and intuitive manner. We discuss potential applications for 2D and 3D divergence-free vector wavelets in image processing problems: particularly using volumetric CT or MRI data.

The Gabor scheme is generalized to incorporate several window functions as well as kernels other than the exponential. The properties of the sequence of representation functions are characterized by an approach based on the concept of frames. the frame operator associated with the multi-window Gabor-type frame, is examined for a rational oversampling rate by representing the frame operator as a finite order matrix-valued function in the Zak Transform domain. Completeness and frame properties of the sequence of representation functions are examined in relation to the properties of the matrix-valued function. Calculation of the frame bounds and the dual frame, as well as the issue of tight frames are considered. It is shown that the properties of the sequence of representation functions are essentially not changed by replacing the widely-used exponential kernel with other kernels. The issue of a different sampling rate for each window is also considered. The so-called Balian-Low theorem is generalized to consideration of a scheme of multi-windows, which makes it possible to overcome the constraint imposed by the original theorem in the case of a single window.

Recently connections between the wavelet transform and filter banks have been established. We show that similar relations exist between the Gabor expansion and DFT filter banks. We introduce the `z-Zak transform' by suitably extending the discrete-time Zak transform and show its equivalence to the polyphase representation. A systematic discussion of parallels between DFT filter banks and Weyl-Heisenberg frames (Gabor expansion theory) is then given. Among other results, it is shown that tight Weyl-Heisenberg frames correspond to paraunitary DFT filter banks.

The short-time Fourier transform (STFT) leads to a highly redundant linear time-frequency signal representation. In order to remove this redundancy it is usual to sample the STFT on a rectangular grid. For such regular sampling the basic features of the reconstruction problem are well understood. In this paper, we consider the problem of reconstructing a signal from irregular samples of its STFT. It may happen that certain samples of the STFT from a regular grid are lost or that the STFT has been purposely sampled in an irregular way. We investigate that problem using Weyl-Heisenberg frames, which are generated from a single atom by time- frequency-shifts (along the sampling set). We compare various iterative methods and present typical numerical experiments. Whereas standard frame iterations are doing not very well it turns out that for many reasons the conjugate gradient algorithm behaves best, most often even better than one might expect from the observations made for general frame operators.

Reassignment is a technique which consists in moving the computed value of a time-frequency or time-scale energy distribution to a different location in the plane, so as to increase its readability. In the case of scalograms (squared modulus of wavelet transforms), a general form is given for the reassignment operators and their properties are discussed with respect to the chosen wavelet. Characterization of local singularities after reassignment is investigated by simulation and some examples (from mathematics and physics) are presented in order to support the usefulness of the approach. Since reassigning a scalogram amounts to compute two extra wavelet transforms, it is finally shown how this can be achieved in a fast and efficient way within a multiresolution framework.

This paper uses a wavelet transform approach to actively image an object continuously distributed in range and velocity. It is shown that by transmitting a high resolution, i.e. large time-bandwidth product, signal into the environment and operating on the echo with a wavelet transform, an estimate of the delay-time-scale representation or wideband spreading function of the object can be obtained in the wavelet domain. The wideband spreading function (WBSF) is a characterization of the time-varying propagation and scattering associated with the channel being imaged. It is shown that the linear operator that acts on the wideband spreading function to form the echo is in the form of an inverse wavelet transform and the adjoint operator is in the form of a forward wavelet transform. Thus, the wavelet transform is a natural transform for the investigation of wideband spreading functions. By combining information extracted from wavelet estimates of the WBSF associated with independently located sensors, it is possible to estimate vectors which describes the physical characteristics of the object in the channel. Specifically, the support curve of the WBSF in the wavelet domain can be directly related to the projections of these vectors along the line of sight of each of the sensors. Therefore, it is necessary to obtain a wavelet estimate of the WBSF which highly resolves the support curve. The wavelet transform estimate is shown to be limited by the resolution capabilities of the auto-wavelet transform of the transmitted signal therefore establishing the need for high resolution transmit signals. Results of wavelet transform estimates of the WBSF of a rough, rotating sphere using wideband and narrowband signals are compared and discussed.

Short, high frequency bursts are found in flow noise data from a transient regime in a pipe flow experiment. These bursts are directly detectable in accelerometer measurements, but processing is necessary to detect them in synchronous hydroacoustic measurements. These signals can be detected and recovered using wavelet thresholding. New variations on methods of choosing the threshold and exploiting translations of the wavelet basis are described, then applied to the estimation of these bursts.

Electroencephalographic (EEG) signal texture content analysis has been proposed for early warning of an epileptic seizure. This approach was evaluated by investigating the interrelationship between texture features and basic signal informational characteristics, such as Kolmogorov complexity and fractal dimension. The comparison of several traditional techniques, including higher-order FIR digital filtering, chaos, autoregressive and FFT time- frequency analysis was also carried out on the same epileptic EEG recording. The purpose of this study is to investigate whether wavelet transform can be used to further enhance the developed methods for prediction of epileptic seizures. The combined consideration of texture and entropy characteristics extracted from subsignals decomposed by wavelet transform are explored for that purpose. Yet, the novel neuro-fuzzy clustering algorithm is performed on wavelet coefficients to segment given EEG recording into different stages prior to an actual seizure onset.

It is shown that analyses based on Symmetric Daubechies Wavelets (SDW) lead to a multiresolution form of the Laplacian operator. This property, which is related to the complex values of the SDWs, gives a way to new methods of image enhancement applications. After a brief recall of the construction and main properties of the SDW, we propose a representation of the sharpening operator at different scales and we discuss the `importance of the phase' of the complex wavelet coefficients.

As new remote sensing systems are deployed, we will see an increase in the amount of image data available at different wavelengths. Also, images from a single sensor over the same area often exhibit clouds, forcing analysts to switch among several images or to mosaic the images by manually defining cutlines to eliminate clouds. The ability to fuse multiple images over the same area, and to have the fused product exhibit, in a single image, the important details visible in individual bands has become crucial in dealing with the large volume of data available. We describe an approach to image fusion using the wavelet transform. When images are merged in wavelet space, we can process different frequency ranges differently. For example, high frequency information from one image can be combined with lower frequency information from another, for performing edge enhancement. We have built a prototype system that allows experimentation with various wavelet array combination and manipulation methods for image fusion, using a set of basic operations on wavelet frequency blocks. Problems caused by image misregistration and processing artifacts are described. Examples of wavelet fusion results are shown, along with test images that clarify behavior of the wavelet fusion methods used.

Sampling `theorems' in `wavelet spaces' are very useful for the integration of `wavelet multiresolution' techniques with various time-domain methods for data processing. In this paper, the application potential of `wavelet sampling theorem' will be illustrated through a few examples of dynamical data analysis and filtering. The main results among our recent applications of the wavelet sampling principles for data processing include the `compact- harmonic wavelets' and a new technique for time-frequency analysis. This new analysis technique provides localized wavelet filters with arbitrarily adjustable frequency-resolution, and the exact reconstruction capability. These filter qualities are both useful and essential for the accurate representation of local power-spectra, and segmentation of signals. These results and underlying ideas are also applicable to the fields of imaging and data compression.

We present a method to automatically register images presenting both global and local deformations. The image registration process is performed by exploiting a multi-level/multi- image approach whereby after having wavelet transformed the images, the subband images at different levels are used in a non-feature-based way to determine the motion vectors between the reference and the target images. The crude motion field determined by block matching at the coarsest level of the pyramid is successively refined by taking advantage of both the orientation sensitivity of the different subbands and the contribution of the adjacent levels.

This paper presents a multiresolution approach to removing unwanted backscatter from high frequency underwater acoustic signals and compares it to high pass filtering of the same signals. Since the unwanted backscatter typically concentrates in the low frequencies, high pass filters are often applied but with limited effectiveness. It turns out that some of the backscattering actually appears across several frequencies, and so a more flexible filtering approach is needed. The filtering approach presented applies wavelet transforms for signal recovery and denoising of high frequency acoustic signals. Wavelet transforms are applied since they perform a multiresolution decomposition in time and frequency and therefore are well suited for removing specific unwanted signal components that may vary spectrally. It is shown that by computing a wavelet transform of the returned signals, applying a denoising technique, and then reconstructing the signals, additional unwanted backscatter can be removed.

We propose a nonlinear, wavelet-based method to efficiently improve the performance of various coding schemes for lossy image data compression. Coarse quantization of the transform coefficients often results in some undesirable artifacts, such as ringing effect, contouring effect and blocking effect, especially at very low bit rate. The decoding can be viewed as a typical statistical estimation problem of reconstructing the original image signal from the decomposed image, a noisy observation, using the classical signal processing model of `signal plus additive noise'. We perform the wavelet-domain thresholding on the decompressed image to attenuate the quantization noise effect while maintaining the relatively sharp features (e.g. edges) of the original image. Experimental results show that de-noising using the undecimated discrete wavelet transform achieves better performance than using the orthonormal discrete wavelet transform, with an acceptable computational complexity (O(MNlog₂(MN)) for an image of size M X N). Both the objective quality and the subjective quality of the reconstructed image are significantly improved with the reduction of coding artifacts. In addition, dithering technique can be embedded in the encoding scheme to achieve further improvement of the visual quality.

Donoho and Johnstone's WaveShrink procedure has proven valuable for signal de-noising and non-parametric regression. WaveShrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise. WaveShrink has very broad asymptotic near- optimality properties. In this paper, we introduce a new shrinkage scheme, semisoft, which generalizes hard and soft shrinkage. We study the properties of the shrinkage functions, and demonstrate that semisoft shrinkage offers advantages over both hard shrinkage (uniformly smaller risk and less sensitivity to small perturbations in the data) and soft shrinkage (smaller bias and overall L₂ risk). We also construct approximate pointwise confidence intervals for WaveShrink and address the problem of threshold selection.

Complex signals need to be adaptively expanded over families of waveforms selected to match their different structures. A matching pursuit is a greedy algorithm that expands signals over vectors that are selected among redundant dictionaries of waveforms. The properties of this algorithm are reviewed. Applications to sound and image processing with dictionaries of time- frequency atoms are described.

Multifractal signals are characterized by a local Holder exponent that may change completely from point to point. We show that wavelet methods are an extremely efficient tool for determining the exact Holder exponent of a function, or at least, for getting some information about this Holder exponent, such as the Spectrum of Singularities. We construct functions that have a given Holder exponent in a deterministic setting and also in a probabilistic setting (we then obtain the Multifractional Brownian Motion); we also study the Multifractal Formalism for Functions and give some results about its validity.

Fractal image compression was one of the earliest compression schemes to take advantage of image redundancy in scale. The theory of iterated function systems motivates a broad class of fractal schemes but does not give much guidance for implementation. Fractal compression schemes do not fit into the standard transform coder paradigm and have proven difficult to analyze. We introduce a wavelet-based framework for analyzing fractal block coders which simplifies these schemes considerably. Using this framework we find that fractal block coders are Haar wavelet subtree quantization schemes, and we thereby place fractal schemes in the context of conventional transform coders. We show that the central mechanism of fractal schemes is an extrapolation of fine-scale Haar wavelet coefficients from coarse-scale coefficients. We use this insight to derive a wavelet-based analog of fractal compression, the self-quantization of subtrees (SQS) scheme. We obtain a simple SQS decoder convergence proof and a fast SQS decoding algorithm which simplify and generalize existing fractal compression results. We describe an adaptive SQS compression scheme which outperforms the best fractal schemes in the literature by roughly 1 dB in PSNR across a broad range of compression ratios and which has performance comparable to some of the best conventional wavelet subtree quantization schemes.

We explore the relationship between random processes and wavelets in multiple dimensions and their application to statistical signal processing. To this end, we introduce a multiresolution Wiener filter (MWF) that is applied to the wavelet coefficients of a random process. The MWF is based upon the multiresolution Wiener-Hopf (MWH) equation, which is derived using orthogonal projection theorem on a Hilbert space. The MWH is applied to the solution of the signal estimation problem for both stationary and fractional Brownian motion (fBm) processes. A theoretical mean square error is calculated for the MWF and its values compared to experimental data.

We present an algorithm for speech compression which uses the wavelet packet transform, vector quantization, entropy coding and postfiltering of the decoded speech. We address the following issue: obtaining the best speech quality for a given bit rate with minimal algorithmic delay (applying it on the possible shortest segment). The wavelet packet transform provides good compression since it is based on a very close relation between the transform and the actual physical processes in the human ear. The experimental results demonstrate that we can compress speech by factor of 6 - 10 and still have reasonable intelligibility and perceivability of the output speech using an algorithmic delay of 8 msec (64 speech samples). In addition, the proposed algorithm fits well DSP architecture and can be easily ported into any current 40MIPS DSP. By comparing the proposed algorithm in this paper with new CELP-oriented algorithm one can conclude that the former has less delay with higher compression ratio. The postfiltering was found to improve the quality of the decoded speech. We see that by using fixed size segments with 64 samples with wrap-around in the segments border does not degrade the performance in comparison to FIR-implementation without wrap-around. In addition, it is useful to implement different filter in each level of the decomposition.

In this research, we improve Shapiro's EZW algorithm by performing the vector quantization (VQ) of the wavelet transform coefficients. The proposed VQ scheme uses different vector dimensions for different wavelet subbands and also different codebook sizes so that more bits are assigned to those subbands that have more energy. Another feature is that the vector codebooks used are tree-structured to maintain the embedding property. Finally, the energy of these vectors is used as a prediction parameter between different scales to improve the performance. We investigate the performance of the proposed method together with the 7 - 9 tap bi-orthogonal wavelet basis, and look into ways to incorporate loseless compression techniques.

Wavelet transform is becoming increasingly important in image compression applications because of its flexibility in representing nonstationary signals. In wavelet-based compression, coding performance can be improved by exploiting human visual system characteristics. In this paper, we propose an algorithm, optimal in the weighted mean square sense, to quantize the wavelet coefficients. The proposed algorithm provides a superior coding performance.

The sampling theorem of Bar-David provides an implicit representation of bandlimited signals using their crossings with a cosine function. This cosine function is chosen in a way that guarantees a unique representation of the signal. Previously, we extended Bar-David's theorem to periodic functions on an interval, leading to a multiplicative representation involving a Riesz product whose roots form a unique and stable representation of the signal. We also presented numerical algorithms for the analysis and synthesis of 1D signals. In this paper, we extend our previous results by developing algorithms for 2D signals and incorporating the wavelet transform into the cosine crossing representation.

In this contribution we introduce a new family of wavelets named Circular Harmonic Wavelets (CHW), suited for multiscale feature-based representations, that constitute a basis for general steerable wavelets. The family is based on Circular Harmonic Functions (CHF) derived by the Fourier expansion of local Radial Tomographic Projections. A multiscale general feature analysis can be performed by linearly combining the outputs of CHW operators of different order. After a survey on the general properties of the CHFs, we investigate the relationship between CHF and the wavelet expansion, stating the basic admissibility and stability conditions with reference to the Hankel transform of the radial profiles and describing some fundamental mathematical properties. Finally some applications are illustrated through examples.

The theory of orthogonal polynomials is used to construct a family of orthogonal wavelet bases of L²(R) which are compactly supported, continuous, and piecewise polynomial and have arbitrary approximation order.

We demonstrate some wavelet-based image processing applications of a class of simplicial grids arising in finite element computations and computer graphics. The cells of a triangular grid form the set of leaves of a binary tree and the nodes of a directed graph consisting of a single cycle. The leaf cycle of a uniform grid forms a pattern for pixel image scanning and for coherent computation of coefficients of splines and wavelets. A simple form of image encoding is accomplished with a 1D quadrature mirror filter whose coefficients represent an expansion of the image in terms of 2D Haar wavelets with triangular support. A combination the leaf cycle and an inherent quadtree structure allow efficient neighbor finding, grid refinement, tree pruning and storage. Pruning of the simplex tree yields a partially compressed image which requires no decoding, but rather may be rendered as a shaded triangulation. This structure and its generalization to n-dimensions form a convenient setting for wavelet analysis and computations based on simplicial grids.

Multisplines provide a method to get piecewise polynomial representations that zoom in on details. Since they use multiple spline-based multiresolution simultaneously, they offer control on the polynomial order (of the piecewise polynomials) as well as the number of continuous derivatives. We explore some of the properties of multisplines and their relationship to piecewise polynomials. We show that instead of using wavelets to transcend resolutions, we can use the even translates of the scaling function.

Periodic spline wavelets provide an efficient tool to analyze periodic signals. Wavelet analysis gives its best performance when it is applied to detect transients or local events in the signal. However, it is not well suited to characterize stationary phenomena. To overcome this problem we propose a new family of periodic spline functions, capable of playing the role of trigonometric wave-forms. They lead us to an orthogonal decomposition of the signal into quasi-monochromatic spline waves. Further, a full collection of periodic spline wavelet packets is also proposed. These elemental functions can be organized in a large library of orthonormal bases. Thus, one can analyze any periodic signal in accordance with a well adapted strategy.

We present a unifying framework for the design of discrete algorithms that implement continuous signal processing operators. The underlying continuous-time signals are represented as linear combinations of the integer-shifts of a generating function p_i with (i equals 1,2) (continuous/discrete representation). The corresponding input and output functions spaces are V(p_i) and V(p₂), respectively. The principle of the method is as follows: we start by interpolating the discrete input signal with a function s₁ (epsilon) V(p₁). We then apply a linear operator T to this function and compute the minimum error approximation of the result in the output space V(p₂). The corresponding algorithm can be expressed in terms of digital filters and a matrix multiplication. In this context, we emphasize the advantages of B-splines; and show how a judicious use of these basis functions can result in fast implementations of various types of operators. We present design examples of differential operators involving very short FIR filters. We also describe an efficient procedure for the geometric affine transformation of signals. The present formulation is general enough to include most earlier continuous/discrete signal processing techniques (e.g., standard bandlimited approach, spline or wavelet-based) as special cases.

We use group representation theory to establish common theoretical foundation for wavelet, Fourier-Wigner, Gabor and short-time Fourier transforms as well as for the narrow and wideband ambiguity functions. These transforms are coefficients of unitary representations of either affine or Heisenberg groups. From this fact, many important properties of these transforms and ambiguity functions, including volume conservation and admissibility conditions, follow. In this paper we use a generalization of the Frobenius-Shur-Godement theorem (generalized resolution of identity) to derive the reproducing kernels associated with these transforms and ambiguity functions. This result has several new applications to the well- established reproducing kernel Hilbert space theory. First of all, it establishes the conditions for the general resolution of identity and identifies spaces on which transforms are invertible. These results can be used to solve inverse problems that arise in remote sensing and characterization of stochastic propagation and scattering channels. Since reproducing kernels are positive definite functions, they can be used as approximating functions, analogously to the radial bases functions, for neural network expansions, interpolation and optimization. Because auto-wavelet and auto-Fourier-Wigner transforms are reproducing kernels on a well defined space of functions, we have a powerful method for generating a rich set of 2n dimensional positive definite functions for multi-dimensional interpolation, approximation, and sampling.

This paper provides classes of unitary operations of L²(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L²(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We also develop an equivalence relation between multiresolution analyses of L²(R). This relation called unitary equivalence is created by the action of a group of unitary operators contained in all the classes mentioned previously, in a way that the multiresolution structure and the Decomposition and Reconstruction algorithms remain invariant. A characterization of this relation in terms of the scaling functions is given. Distinct equivalence classes of multiresolution analyses are derived. Finally, we prove that B-splines give rise to non-equivalent examples.

Often, the Discrete Wavelet Transform is performed and implemented with the Daubechies wavelets, the Battle-Lemarie wavelets or the splines wavelets whereas in continuous time wavelet decomposition a much larger variety of mother wavelets are used. Maintaining the dyadic time-frequency sampling and the recursive pyramidal computational structure, we present various methods to obtain any chosen analyzing wavelet (psi) _w, with some desired shape and properties and which is associated with a semi-orthogonal multiresolution analysis or to a pair of bi-orthogonal multiresolutions. We explain in details how to design one's own wavelet, starting from any given Multiresolution Analysis or any pair of bi-orthogonal multiresolutions. We also explicitly derive, in a very general oblique (or bi-orthogonal) framework, the formulae of the filter bank structure that implements the designed wavelet. We illustrate these wavelet design, techniques with examples that we have programmed with Matlab routines, available upon request.

We consider a class of Gabor-type matrices and develop simplified Gabor-type matrix operations. As applications to discrete Gabor transforms, we propose `superfast' algorithms for determining the inverse of Gabor frame operators and the square roots of the Gabor frame operators as well as the dual Gabor and tight Gabor atoms. Besides, we summarize briefly some additional results.

We first review a method for the characterization of fractal signals introduced by Muzy et al. This approach uses the continuous wavelet transform (CWT) and considers how the wavelet values scale along maxima lines. The method requires a fine scale sampling of the signal and standard dyadic algorithms are not applicable. For this reason, a significant amount of computation is spent evaluating the CWT. To improve the efficiency of the fractal estimation, we introduced a general framework for a faster computation of the CWT. The method allows arbitrary sampling along the scale axis, and achieves O(N) complexity per scale where N is the length of the signal. Our approach makes use of a compactly supported scaling function to approximate the analyzing wavelet. We discuss the theory of the fast wavelet algorithm which uses a duality principle and recursive digital filtering for rapid calculation of the CWT. We also provide error bounds on the wavelet approximation and show how to obtain any desired level of accuracy. Finally, we demonstrate the effectiveness of the algorithm by using it in the estimation of the generalized dimensions of a multi-fractal signal.

The Wavelet Transform Modulus Maxima method is used to analyze the fractal scaling properties of DNA sequences. This method, based on the definition of partition functions which use the values of the wavelet transform at its modulus maxima, allows to determine accurately the singularity spectrum of a given singular signal. By considering analyzing wavelets that make the wavelet transform microscope blind to `patches' of different nucleotide compositions which are ubiquitous to genomic sequences, we demonstrate and quantify the existence of long-range correlations in the noncoding regions. The fluctuations around the patchy landscapes of the DNA walks reconstructed from both the noncoding and coding regions are found to have Gaussian statistics. Whereas the fluctuations from the former behave like fractional brownian motions, those of the latter cannot be distinguished from uncorrelated random brownian walks.

This paper interprets 1D intensities in images of Synthetic Aperture Radar (SAR) as interwoven sets of mathematically defined singularities, and the spectrum of singularity strengths has been calculated using wavelets. Superposition of spectra from a transect of SAR data from the European Earth Resources Satellite (ERS-1) with the spectra from a SAR simulation based only on local terrain elevation variation shows that there exists a range of singularity strengths in ERS-1 data which are closely described by a topographic model. Modeling the SAR spectra using singularity spectra from Cantor sets and multiplicative cascades only partially helps in defining the SAR signal as a multifractal. The paper proposes that identifying and isolating the singularities in a SAR image for the purposes of classifying images can be assisted by using results from a singularity spectrum. The object of this classification is the resolution of images into regions with stable pixels representing purely a response to topographic relief and distinguishable from responses from land use and terrestrial cover types.

A redundant wavelet filtering method is used in conjunction with spectrogram computations to address a component of the problem of predicting epileptic seizure activity. It is shown that spectrograms of seizure episodes exhibit multiple chirps consistent with the relatively simple almost periodic behavior of the observed time series. Scalograms corresponding to a redundant (non-dyadic) wavelet analysis are used to provide finer information about these chirps, including their evolution in preseizure intervals. Detection of the origin of such periodicities are useful in the prediction problem.

We argue that an important aspect of the human speech signal is scaling in the frequency domain. We discuss the two physical mechanisms responsible for the scaling. The first mechanism is that when we have a harmonic signal whose fundamental is frequency modulated then the spectrum is the sum of scaled functions. The second comes about from the consideration that while different speakers have very different size vocal tracts (for example an adult and a child), we none the less produce speech which is similar in some sense. We will argue and present evidence to show that the speaker differences result in scaling in the frequency domain. We further discuss how one can handle scale processing.

We consider linear discriminant analysis in the setting where the objects (signals/images) have many dimensions (samples/pixels) and there are relatively few training samples. We discuss ways that time frequency dictionaries can be used to adaptively select a small set of derived features which lead to improved misclassification rates.

Speech and Handwriting interfaces to computing devices have received increased attention recently as alternate human-computer media. Automatic recognition of unconstrained handwritten text must be provided as a capability in handwriting computer interfaces. Variation in writing styles of a single writer at different times and between multiple writers makes unconstrained on-line handwriting recognition a challenging task. On-line handwriting is recorded as a sequence of coordinates as the writer's pen moves along the recording device. Isolated character and word recognition have been addressed by several researchers. More recently, attention has been focussed on the recognition of unconstrained text streams. The dynamic changes in handwriting styles observed in everyday use requires development of methods that are adaptive to local variations. We present a novel application of wavelet-based analysis of pen position, velocity and acceleration time-sequences for segmentation and recognition of text components.

The Time-Frequency and Time-Scale communities have recently developed a large number of overcomplete waveform dictionaries. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed--including the Method of Frames, Matching Pursuit, and, for special dictionaries, the Best Orthogonal Basis. Basis Pursuit is a principle for decomposing a signal into an `optimal' superposition of dictionary elements-- where optimal means having the smallest l¹ norm of coefficients among all such decompositions. We give examples exhibiting several advantages over the Method of Frames, Matching Pursuit and Best Ortho Basis, including better sparsity, and super-resolution. Basis Pursuit in highly overcomplete dictionaries leads to large-scale optimization problems. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate gradient solver.

In this paper we will begin by presenting basic background material concerning the propagation of pulses through optical fibers, and current transmission schemes for optical fiber communication networks. We will then propose a time-frequency network design for high bandwidth fiber optic communications. We will discuss the crucial links in the system, which are: (1) an optical temporal Fourier transform which enables the optical computation of signal correlations, (2) the design of the input waveforms in order to avoid the harmful effects of Brillouin scattering and phase noise, and (3) the use of time frequency bases to lower the requirements on electronic modulators and detectors in high bandwidth communications.

In this paper, we investigate the performance of baseband modems based on biorthogonal and semiorthogonal wavelets and compare their performance in terms of bandwidth efficiency and bit-error-rate with orthogonal wavelet based systems and conventional polar signalling schemes. The proposed modems are shown to conserve bandwidth at the cost of increased per- bit signal to noise ratio.

A highly efficient texture compression scheme is proposed in this research. With this scheme, energy compaction of texture images is first achieved by the wavelet packet transform, and an embedding approach is then adopted for the coding of the wavelet packet transform coefficients. By comparing the proposed algorithm with the JPEG standard, FBI wavelet/scalar quantization standard and the EZW scheme with extensive experimental results, we observe a significant improvement in the rate-distortion performance and visual quality.

Wavelet based compression schemes belong to the general class of transform coding schemes. We show how the genetic programming approach can be used to optimize such a compression scheme in the sense of rate-distortion. The results of optimized wavelet based compression scheme are compared with the JPEG compression standard. A prototype implementation of the method is realized as a distributed, parallel implementation on a heterogeneous Unix network.

Multiresolution pyramid decomposition of images for data compression and transmission have been successfully employed using the common frame of linear subband filtering techniques involving wavelet transform, and Laplacian of Gaussian while multiresolution morphological pyramid decomposition represent a different class of nonlinear filters that maybe used as an optimal predictor of an image. To achieve a desired compression ratio for a specific class of images, a compression algorithm needs to be optimized at all stages from initial mapping to final encoding. We demonstrate the superiority of an optimized wavelet transform based compression algorithm over the standard JPEG from a number of distortion measure criteria for radiographic images. We also describe here a hybrid technique for noisy images where a combination of multiresolution morphological and wavelet filters dramatically reduce the inherent noise and hence increase the peak signal to noise ratio at a particular compression level. Noisy synthetic aperture radar images are chosen as illustrations.

A new transform which combines the key features of the Radon transform with the localization abilities of the wavelet transform is presented. The transform was developed in response to the problem of wake detection in open water synthetic aperture radar (SAR) images. Consider the fact that wakes are generally small in size relative to the original SAR image. Further, ship wakes have linear features. So even though the orientation and size of the wake is unknown a priori, a transform which has parameters for rotation and spatial localization may aid in the detection of these features.

3D data acquisition is increasingly used in positron emission tomography (PET) to collect a larger fraction of the emitted radiation. A major practical difficulty with data storage and transmission in 3D-PET is the large size of the data sets. A typical dynamic study contains about 200 Mbyte of data. PET images inherently have a high level of photon noise and therefore usually are evaluated after being processed by a smoothing filter. In this work we examined lossy compression schemes under the postulate not induce image modifications exceeding those resulting from low pass filtering. The standard we will refer to is the Hanning filter. Resolution and inhomogeneity serve as figures of merit for quantification of image quality. The images to be compressed are transformed to a wavelet representation using Daubechies12 wavelets and compressed after filtering by thresholding. We do not include further compression by quantization and coding here. Achievable compression factors at this level of processing are thirty to fifty.

The information content in many signal processing applications can be reduced to a set of linear features in a 2D signal transform. Examples include the narrowband lines in a spectrogram, ship wakes in a synthetic aperture radar image, and blood vessels in a medical computer-aided tomography scan. The line integrals that generate the values of the projections of the Radon transform can be characterized as a bank of matched filters for linear features. This localization of energy in the Radon transform for linear features can be exploited to enhance these features and to reduce noise by filtering the Radon transform with a filter explicitly designed to pass only linear features, and then reconstructing a new 2D signal by inverting the new filtered Radon transform (i.e., via filtered backprojection). Previously used methods for filtering the Radon transform include Fourier based filtering (a 2D elliptical Gaussian linear filter) and a nonlinear filter ((Radon xfrm)**y with y >= 2.0). Both of these techniques suffer from the mismatch of the filter response to the true functional form of the Radon transform of a line. The Radon transform of a line is not a point but is a function of the Radon variables (rho, theta) and the total line energy. This mismatch leads to artifacts in the reconstructed image and a reduction in achievable processing gain. The Radon transform for a line is computed as a function of angle and offset (rho, theta) and the line length. The 2D wavelet coefficients are then compared for the Haar wavelets and the Daubechies wavelets. These filter responses are used as frequency filters for the Radon transform. The filtering is performed on the wavelet pyramid decomposition of the Radon transform by detecting the most likely positions of lines in the transform and then by convolving the local area with the appropriate response and zeroing the pyramid coefficients outside of the response area. The response area is defined to contain 95% of the total wavelet coefficient energy. The detection algorithm provides an estimate of the line offset, orientation, and length that is then used to index the appropriate filter shape. Additional wavelet pyramid decomposition is performed in areas of high energy to refine the line position estimate. After filtering, the new Radon transform is generated by inverting the wavelet pyramid. The Radon transform is then inverted by filtered backprojection to produce the final 2D signal estimate with the enhanced linear features. The wavelet-based method is compared to both the Fourier and the nonlinear filtering with examples of sparse and dense shapes in imaging, acoustics and medical tomography with test images of noisy concentric lines, a real spectrogram of a blow fish (a very nonstationary spectrum), and the Shepp Logan Computer Tomography phantom image. Both qualitative and derived quantitative measures demonstrate the improvement of wavelet-based filtering. Additional research is suggested based on these results. Open questions include what level(s) to use for detection and filtering because multiple-level representations exist. The lower levels are smoother at reduced spatial resolution, while the higher levels provide better response to edges. Several examples are discussed based on analytical and phenomenological arguments.

Many problems in applied mathematics involve recovering a function f from measurements of Lf, where L is a known operator. If L is a linear operator with a bounded inverse, then f can recovered from noisy data Lf + n via standard techniques with little difficulty. The recovery of f from noisy data Lf + n is much more difficult if L is an operator whose inverse is unbounded. One such problem is the recovery of a function f : R2 —R, from limited knowledge of its Fourier transform f. Thus we are interested in reconstructing f from Lf where L is an operator which reflects the limited knowledge of the Fourier transform of f. This problem is motivated by the limited angle tomography problem. This recovery problem is ill-posed and unsolvable without a priori assumptions about f. With proper a priori knowledge, however, this problem is solvable. For example, if f is compactly supported, then f will be an analytic function, and therefore f will be uniquely determined by its values on any region containing a limit point [3]. Classical analytic continuation techniques are not numerically feasible [22] since the operator L will generally not have a bounded inverse. A technique for doing analytic continuation was introduced by Papoulis in [20]. This algorithm depends upon the eigenfunctions of the continuous time and band-limiting operators, which were extensively studied in the classical works of Slepian, Pollack, and Landau [28]. The convergence of the algorithm is dependent upon the eigenvalues of the joint time- and band-limiting operators, and except in extreme cases, this algorithm is not feasible numerically [9]. We will model these data recovery problems as the inversion of a compact operator L ( for limited data ); where Lf = Xs(Xcj), S is the common support of the functions under consideration, C is the set where we can measure the Fourier transform of f, and Xs is a characteristic, or indicator function on the set S. One tool for analyzing the spectra of these discrete approximations to L will be the theory of finite Toeplitz forms, originally introduced by Siegö [4]. We will show that the study of these finite Toeplitz forms will give us some clues concerning the construction of an accurate, stable inversion for L, even when the continuous spectra of L suggests that it is not invertible. Possibly more importantly, we will examine the addition of nonlinear constraints into this problem in order for further mollification

The motivation of this paper is to give a direct way for determining the crucial Gabor coefficients. Based on the characterized Gabor-Gram matrix structures, we are able to propose fast and direct computational algorithms. We derive also a new way for dual Gabor atoms.

We present the W-transform for a multiresolution signal decomposition. One of the differences between the wavelet transform and W-transform is that the W-transform leads to a nonorthogonal signal decomposition. Another difference between the two is the manner in which the W-transform handles the endpoints (boundaries) of the signal. This approach does not restrict the length of the signal to be a power of two. Furthermore, it does not call for the extension of the signal thus, the W-transform is a convenient tool for image compression. We present the basic theory behind the W-transform and include experimental simulations to demonstrate its capabilities.

Multisensory data often have different resolutions and measurement noises. To integrate the multiresolution information together while removing their respective noises, a fusion process is performed to improve the sensing quality. Wavelet-based multiresolution analysis provides a new approach to study the data fusion problem. In such an approach, the approximation of a signal at each scale may be interpreted as the state variables and the state transition takes place from a coarse scale to a fine scale. Through the use of Kalman theory, data fusion can be obtained by optimally estimating the finest scale representation from a set of multiscale noisy measurements, in a scale recursive form. In this paper, a generalized fusion algorithm has been developed using the dyadic-tree wavelet transform with either orthogonal or biorthogonal wavelets. We then have more choices of analyzing wavelets to be used for fusing data. Examples are given to illustrate the improved fusion performance.

Learning can often be viewed as the problem of mapping from an input space to an output space. Examples of these mappings are used to construct a continuous function that approximates given data and generalizes for intermediate instances. Generalized Radial Basis Function (GRBF) networks are used to formulate this approximating function. A novel method is introduced that uses the Integrated Wavelet Transform to construct an optimal GRBF network for a given mapping and error bound. Simple 1D examples are used to demonstrate how the optimal network is superior to one constructed using standard ad hoc optimization techniques. The paper concludes with an application of optimal GRBF networks to a multidimensional problem (15 - 20 dimensions), real-time object recognition and pose estimation. The results of this application are favorable and the optimal GRBF network outperforms a GRBF network constructed using a traditional method.

The Airborne Visible InfraRed Imaging Spectrometer (AVIRIS), presently being flown by the Jet Propulsion Laboratory, acquires images of the earth in the visible and reflected infrared. The wavelengths of the measured radiation range from about 400 nm to 2400 nm and are divided into 224 contiguous channels having a nominal spectral bandwidth of 10 nm. This means a high resolution radiance spectrum is acquired for each 20 m X 20 m ground cell in the AVIRIS scene. Geologic mapping from such data is possible by classifying each pixel based on the distinctive spectral signatures recorded in the channels. Artificial neural networks (ANN) have used these spectra successfully to classify an AVIRIS subscene of the Lunar Craters Volcanic Field (LCVF) in Nye County, Nevada. The size and number of spectra in an AVIRIS scene makes classifying these images a computationally intensive task. By classifying the data in a compressed format, savings in computer time may be realized. The wavelet bases have the desirable property of rendering signals similar to the AVIRIS spectra sparse in the wavelet domain. In this investigation, the discrete wavelet transform was applied to the spectra. This produced a set of wavelet coefficients for the spectra that could be made sparse with seemingly little loss of accuracy. Small subsets of the wavelet coefficients were used to classify the LCVF scene by ANN. The degree to which information was lost in the wavelet transform and the elimination of wavelet coefficients from the classification was assessed by making comparisons between the different ANN classifications. The ANN was chosen over more conventional classifiers because of its proven sensitivity in distinguishing subtle but geologically relevant features in these spectra.

Wavelet analysis is a newly developed technique for separating and sorting structures on different time scales at different times, and different spatial scales at different locations. Mathematically, it is related to Fourier spectral analysis. It has been used extensively for image and signal processing, including the processing of seismic and turbulence data in geophysics. Characterizing spatial heterogeneity in the subsurface is an important issue for construction of facilities for waste containment and management, prediction of transport of pollutants through geologic media and remediation of contaminated sites. Localization processes in the subsurface and multiscale effect in geologic sampling data have to be addressed in the site characterization. Current methods can not solve these problems, however. The potential exists for using wavelet analysis as the multiscale spatial heterogeneity analysis method for characterizing detailed geologic structures. In particular, wavelet analysis overcomes significant limitations in standard geostatistics. We first present these limitations, then demonstrate how wavelet analysis overcomes them. We conclude with two demonstrations of geologic data analyses (the spatial variability of permeability in the alluvial fan deposits in DOE Nevada Test Site Pit 3. and cone penetrometer data interpretation) using wavelet analysis, compared to current geostatistical approaches.

The present contribution proposes a new remarkably efficient image compression algorithm for graylevel images based on dyadic wavelet transformations. In order to achieve perfect reconstruction, orthogonal decomposition is applied. Scalar quantization of wavelet coefficients is combined with run-length coding. Code word assignment is performed by semi- adaptive Huffman coding (determined by validity tables). To improve the reconstruction quality of images a new technique of pdf-adaptive reconstruction of wavelet coefficients is used.

We develop a second-generation subband image coding algorithm which exploits the pyramidal structure of transform-domain representations to achieve variable-resolution compression. In particular, regions of interest within an image can be selectively retained at higher fidelity than other regions. This type of coding allows for rapid, very high compression, which is especially suitable for the timely communication of image data over bandlimited channels.

An image segmentation scheme based on multiresolutional, successive approximations of the image histogram is proposed. The algorithm begins with a coarse, initial segmentation of the image obtained by selecting thresholds from a coarse sampling of a low-pass filtered version of the image histograms. This segmentation is refined by selecting thresholds from increasingly better approximations of the histogram. The algorithm is linear in the size of the input image and handles images with multimodal histograms. Preliminary results indicate that the approach shows promise as a simple, computationally efficient algorithm for hierarchical image segmentation. The algorithm may easily be embedded in the `split' phase of any of the well known split-and-merge type segmentation algorithms.

The improvement in the performance of a wavelet packet based compression scheme for single lead electrocardiogram (ECG) data, obtained by prefiltering noise from the ECG signals, is investigated. The removal of powerline interference and the attenuation of high-frequency muscle noise are considered. Selected records from the MIT-BIH Arrhythmia Database are used as test signals. After both types of noise artifact were filtered, an average data rate of 167.6 bits per second (corresponding to a compression ratio of 23.62), with an average root mean-square (rms) error of 15.886 (mu) V, was achieved. These figures represent better than a 9% improvement in data rate and a 13.5% reduction in rms error over compressing the unfiltered signals.

This paper describes a simple method for retrieving images from image databases by a similarity retrieval scheme. With wavelets feature vectors are defined, which `measure the presence' of structures of variable size and orientation; moreover corresponding feature vectors providing a multiresolution analysis of object contours are developed. The significance of these feature vectors for classifying images as `similar' is verified on a set of medical MR- images. Based on these experiments a scheme for retrieving MR-images from databases can be defined and typical retrieval results are reported.

Projections of signals continuous translations into subspaces of multiresolution analysis are considered. Compactly supported orthonormal scaling function N(phi) any translation of which has good approximation by means of integer translations of the same scaling function, is found for N <EQ 10. Two estimation characteristics of translation invariance of the multiresolution analysis are suggested. The translation invariance characteristics of new scaling functions and wavelets are compared with those proposed by Daubechies. For N equals 2, it is proved that the Daubechies compactly supported scaling function 2(phi) with highest number of vanishing moments compatible with its support width has optimal translation invariance.

In this paper we present a family of orthonormal transforms for functions in l² in which basis functions have compact support and verify a self-similarity criterion at different resolution levels. First, we define the discrete-time orthogonal wavelet transform, a transform that verifies a set of orthonormality properties on a self-similar discrete multiresolution analysis. We relax the constrains imposed on this transform generalizing the concept of self- similarity and defining the generalized discrete-time orthogonal wavelet transform. In this case, it is possible to obtain different levels of self-similarity and more degrees of freedom in the design of basis functions. Finally, a discrete-time self-similar multi-function transform and a discrete-time multi-wavelet transform are presented, and the criteria that basis functions must verify to become an orthonormal transform are pointed out.

It is possible to build a multiresolution image encoding technique using the fractal transform. Fractal encoding methods rely on measuring least mean square difference between image blocks at two different spatial scales for most efficient block matching. If we instead, compute the centroid of the blocks to be matched before matching occurs and preserve this centroid information in the encoding process we both speed up the encoding process and have information that will allow us to interpret the shape of the objects in the encoded scenes. This ability may dramatically increase the speed at which pattern identification in large volumes of image data may be performed since less data would have to be processed for searches over large number of images.

This paper gives a refinement of a general method for the construction of multivariate numerical integration formulas that use merely boundary points as evaluation points. Boundary quadrature formulas are constructed by using the optimal dimensionality-reducing expansion and quadrature formulas for the integrals of periodic functions with wavelet weights. Boundary quadrature formulas are also used to solve boundary value problems of partial differential equations.

This paper investigates the application of a multiresolution extended Kalman filter to estimate signal parameters of a passive helicopter signal impinging upon a simple acoustic array of microphones. This approach circumvents some of the problems associated with the initial phase estimate when using an extended Kalman filter to estimate signal parameters of a passive signal. A multiresolution extended Kalman filter uses a wavelet transform to form filter observations at multi-scale resolutions. These wavelet transformed observations are processed from coarser scales to finer scales to jointly estimate all signal parameters including signal phase. By using multi-resolution observations the extended Kalman filter can tolerate larger initial phase errors than a conventional extended Kalman filter. Once the filter has been initialized using the wavelet transformed observations, the filter then uses the highest resolution observations (i.e. not wavelet transformed) from that time on.

Discrete wavelet transformations have become indispensable analytical tools in data compression and data denoising. In this paper we give some empirical accounts of wavelet transformations and propose novel thresholding and wavelet selection methods. This is achieved via connections with measures of inequality, that have been used in economics for a long time. We compare our methods with standard thresholding and wavelet selection procedures.

Many applications, in robotics, require identification of human being. Using complex methods, based on model matching are too computationally expensive and not always justified. We propose a fast and simple method for identification of human being. This method takes profit of the learning capabilities of a neural network. The idea is to train a neural network on some images of persons. In order to reduce the amount of this data (images), we use wavelet multiresolution propriety analysis that allows to bring significant information content of image. This one thus is characterized by its approximation at a given resolution. After the training phase, the generalization capabilities of the network allow it to identify no-learned images. We describe here the proposed method, and we present experimental results obtained on a data base of 437 images.

We present a domain decomposition procedure for solving the Dirichlet problem for the Laplace equation in the union of two intersecting discs in R². Each subdomain problem is solved using the boundary integral technique, at each iteration integrating the product of the prior solution multiplied by the normal derivative of the Green's function. The subdomain problems are solved in parallel, in a Jacobi fashion. Numerically, they correspond to multiplying dense matrices by vectors of boundary values. We use DAUB4 wavelets to replace the dense matrices by their sparse approximations, thus reducing the computational complexity. The procedure iterates in `wavelet space', on the wavelet transform of the solution at `internal' boundary points, i.e. at subdomain boundary points not part of the full domain boundary. When the convergence criterion is met, an inverse wavelet transform is applied, and each subdomain problem is solved in full to yield the complete solution. Numerical results are presented.