Adapted wave form analysis, refers to a collection of FFT like adapted transform algorithms. Given an image these methods provide special matched collections of templates (orthonormal bases) enabling an efficient coding of the image. Perhaps the closest well known examples of such coding method is provided by musical notation, where each segment of music is represented by a musical score made up of notes (templates) characterized by their duration, pitch, location and amplitude, our method corresponds to transcribing the music in as few notes as possible. The extension of images and video is straightforward. We describe the image by collections of oscillatory patterns of various sizes, locations and amplitudes using a variety of orthogonal bases. These selected basis functions are chosen inside predefined libraries of oscillatory localized functions (trigonometric and wavelet-packets waveforms) so as to optimize the number of parameters needed to describe our object. These algorithms are of complexity N log N opening the door for a large range of applications in signal and image processing, such as compression, feature extraction denoising and enhancement. In particular we describe a class of special purpose compressions for fingerprint images, as well as denoising tools for texture and noise extraction.

Given a band-limited signal, we consider the sampling of the signal and its first K derivatives in a periodic manner. The mathematical concept of frames is utilized in the analysis of the sequence of sampling functions. It is shown that the frame operator of this sequence can be expressed as matrix-valued function multiplying a vector-valued function. An important property of this matrix is that the maximum and minimum eigenvalues are equal (in some sense) to the upper and lower frame bounds. We present a way for finding the dual frame and, thereby, a way for reconstructing the signal from its samples. Using the matrix approach we also show that if no sampling of the signal itself is involved, the sampling scheme can not be stabilized by oversampling. The formalism can be extended to two dimensions to permit representation of images.

We use approximation theory to generalize the classical sampling procedure of Shannon. We give a link between this theory and the theory of wavelet transforms. As an application, we give a general method for constructing scaling and wavelet functions with specifiable properties.

The Papoulis-Gerchberg (PG) algorithm is well known for band-limited signal extrapolation. We consider the generalization of the PG algorithm to signals in the wavelet subspaces in this research. The uniqueness of the extrapolation for continuous-time signals is examined, and sufficient conditions on signals and wavelet bases for the generalized PG (GPG) algorithm to converge are given. We also propose a discrete GPG algorithm for discrete-time signal extrapolation, and investigate its convergence. Numerical examples are given to illustrate the performance of the discrete GPG algorithm.

Most orthogonal signal decompositions, including block transforms, wavelet transforms, wavelet packets, and perfect reconstruction filterbanks in general, can be represented by a paraunitary system matrix. This paper considers the general problem of finding the optimal P X P paraunitary transform that minimizes the approximation error when a signal is reconstructed from a reduced number of components Q < P. This constitutes a direct extension of the Karhunen-Loeve transform which provides the optimal solution for block transforms (unitary system matrix). General solutions are presented for the optimal representation of arbitrary wide sense stationary processes. This work also investigates a variety of suboptimal schemes using FIR filterbanks. In particular, it is shown that low-order Daubechies wavelets and wavelet packets (D2 and D3) are near optimal for the representation of Markov-1 processes.

We use a natural pixel type representation of an object to construct almost orthogonal basis functions. The coefficients of expansion of an object into these basis functions can be computed from the projection (i.e. the strip integral) data by using the Wavelet Transform. This enables us to formulate a multiscale tomographic reconstruction technique wherein the object is reconstructed at multiple scales or resolutions. A complete reconstruction can be obtained by combining these. The almost orthonormal behavior of the basis functions results in a system matrix, relating the input (the object coefficients) and the output (the projection data), which is extremely sparse. The system matrix, in addition to being sparse, has a symmetric block-Toeplitz structure for which fast inversion algorithms exist. The multiscale reconstruction technique can find applications in object feature recognition directly from projection data, tackling ill-posed imaging problems where the projection data are incomplete and/or noisy, and construction of multiscale stochastic models for which fast estimation algorithms exist.

In this paper we introduce an algorithm for fast updating of projection reconstruction MRI images. This algorithm makes use of range characterizations of the Radon transform, and sampling techniques of computerized tomography. Our algorithm differs from other attempts to image time-varying phenomena in that it does not average over time, but rather localizes in time for 'snap-shot' imaging. This uses the fact that image features from different scales can be updated at different rates. We also introduce a technique for spatial localization of MRI data. This technique shows promise for fast updating of locally changing phenomena in MR. Clinical applications of this technique include dynamic imaging of time dependent physiological processes like oxygen usage and blood flow in the brain.

We use the theory of the continuous wavelet transform to derive inversion formulas for the Radon transform. These inversion formulas are local in even dimensions in the following sense. In order to recover a function f from its Radon transform in a ball of radius R > 0 about a point x to within error (epsilon) > 0, we can find (alpha) ((epsilon) ) > 0 such that this can be accomplished by knowing the projections of f only on lines passing through a ball of radius R + (alpha) ((epsilon) ) about x. We give explicit a priori estimates on the error in the L2 and L(infinity ) norms.

The performance of a wavelet-based edge detector is characterized by a set of digital filters that implement the wavelet transform. In this paper we clarify the issue whether an implementation filter can be made symmetric or antisymmetric with respect to the origin while still maintaining the desired spatial localization. It is shown that, by analyzing the asymptotical convergence of the filter coefficients, these two conditions are not compatible if the filter possesses an antisymmetry. However, when the filter is symmetric with respect to the origin, the optimal spatial localization can be obtained.

Wavelet theory provides very general techniques that can be utilized to perform many tasks in signal and image processing applications. The zero-crossings of a wavelet transform of a signal, using a particular class of wavelets, provide the locations of the sharp variation points of the signal at the different resolutions. These points provide meaningful features for characterizing the signals. In this paper, we present a new approach to recognize a 2D object of general shape based on its wavelet transform zero-crossing representation. This is performed in two stages. The first stage consists of building a 1D signal representation of the 2D boundary of the object followed by obtaining the zero-crossing of the wavelet transform of the resulting representation. The second stage is the matching procedure for object recognition. Our algorithm uses only a few intermediate resolution levels for matching thus making it computationally efficient while being less sensitive to noise and quantization errors. A normalization process is implemented for matching objects of different scales to their models both in noisy and noise-free situations. Our algorithms have been tested using simulated object boundaries and have been successful in recognizing the objects with results being invariant under translation, rotation and scaling.

Interaction of edges in a detection process often cause displacements of detected edges at large scales. We consider a wavelet-based multiscale edge detector for reducing the degree of edge interactions. A symmetric scaling function corresponding to (c₀, c₃) equals (0.05, 0.05) is selected from solutions to the four-coefficient dilation equations, which leads to a four- coefficient filter yielding a minimum interaction between edges. Experimental results are presented to show its detection and localization performance.

In this paper, we propose a multiple-channel Gabor pyramid approach for extracting image flow of non-homogeneous motion from two image frames. We integrate motion information from multiple channels and combine them to obtain the image flow. To illustrate the effectiveness of the Gabor-wavelet pyramid for extracting image flow, two image sequence containing non-homogeneous motion are used. The result shows the performance of the Gabor-wavelet pyramid is satisfied.

This paper discusses the selection of projection functions used in an iterative implementation of the Gabor expansion. We show that the optimal support-limited projection function corresponds to a truncated version of Bastiaans' biorthonormal projection function for the case of a harmonic lattice. For various support widths, the lower bound of the optimal convergence factor is calculated. It is shown that Gabor's original projection function, which corresponds to the central lobe of Bastiaans' biorthonormal projection function, is truncated too severely, producing a significant overlap with elementary functions from high frequency channels. As a result, the lower bound for the optimal convergence factor and the rate of convergence will approach zero as the signal bandwidth (and the highest frequency Gabor channel) is increased. This work also determines the lower bound of the optimal convergence factor for projection functions implemented using log-polar lattices. For both the harmonic and log-polar lattices, we investigate the trade-off between spread of convergence and the size of the projection function.

In this paper, the theory and computations for the Gabor transform are discussed. The Gabor coefficients can be computed with a biorthogonal function or the Zak transform. Relations between a window function and its biorthogonal function are discussed. The formulas derived for the continuous variable Gabor transform with Zak transforms can be applied to the discrete Gabor transform by replacing the Zak transforms with the discrete Fourier transforms. The generalized Gabor transform are also discussed. Relations between a window function and its biorthogonal functions are presented. In the case of the generalized Gabor transform, the biorthogonal functions are not unique. The optimal biorthogonal functions are discussed. A relation between a window function and its optimal biorthogonal function is presented based on the Zak transform when T/T' is rational. The finite discrete generalized Gabor transform is also derived. The relations between a window function and its optimal biorthogonal function derived for the continuous variable generalized Gabor transform can be extended to the finite discrete case.

A way of increasing the spatial resolution of SPOT multispectral images (XS) using the corresponding panchromatic image (P) is presented here. Existing methods for merging P and XS are analyzed, before presenting a new method which aims at simulating 10 m resolution multispectral images that contain the same spectral information at the XS images. This method, called ARSIS after its French name 'Amelioration de la Resolution Spatiale par Injection de Structures', is based upon multiresolution analysis and wavelet transform. Different versions have been implemented, which differ on the model that describes the similarity of the spatial variability on P and XS. ARSIS can also be applied to other sensors, featuring different spectral bands and spatial resolutions.

We present an iterative multiresolution algorithm for the translational and rotational alignment of digital images. An image is represented by an interpolating spline. Coarser versions of this continuous image model are obtained by using spline approximations at various scales (polynomial spline pyramid). We use a coarse-to-fine updating strategy to compute the alignment parameters iteratively, using a variation of the Levenberg-Marquardt non-linear least-squares optimization method. This approach yields very precise image registration with subpixel accuracy. It is also much faster and more robust than a comparable single-scale implementation, because the resolution of the underlying image mode is adapted to the step size of the algorithm.

This paper analyzes the concept of zerotree coding in the context of successive approximation by considering a simple model for the interdependence of the wavelet coefficients corresponding to the same location at different scales and demonstrating why predicting insignificance across scales is easier than predicting significance.

A hybrid algorithm for high rate image compression based on spline wavelet packets and vector quantization is presented in this paper. The algorithm consists of (1) a fast projection of an image signal onto the spline-wavelet space, (2) an efficient spline-wavelet packet decomposition algorithm, (3) a method to select the optimal number of component subimages, (4) criteria for thresholdings, and (5) vector quantization of subimages. We have compressed a test image to 0.08 bpp from 8 bpp with PSNR of 27 db.

This paper presents the Local Cosine Transform as a new method for the reduction and smoothing of the blocking effect that appears at low bit rates in image coding algorithms based on the Discrete Cosine Transform. In particular, the blocking effect appears in the JPEG baseline sequential algorithm.

Some of the most remarkable image compression results published recently are the Barlaud et al, where an entropy estimate is used as the measure of coded bit rate. The rates they quoted are not achievable in a real image compression system because the number of codewords in the lattice vector quantizer codebook used by Barlaud et al. can be orders of magnitude greater than the number of quantizer source vectors output from the wavelet transform of a single image. The root of this problem is that lattice vector quantizers are best suited for use with uniform source probability distributions. In this paper we propose a novel quantizer that compands the codebook lattice in a piecewise fashion, is based on the independent Laplacian source model of wavelet transform coefficients, and achieves excellent rate-distortion results for quantization of wavelet transform coefficients. Simulations compare the use of this quantizer on wavelet transform coefficients with the quantizer originally proposed by Barlaud et al.

Wavelet transform coding has the potential to offer a lot of benefits with respect to image compression and image filtering as well. The later aspect is of importance in machine processing of high entropy images such as Synthetic Aperture Radar images obtained by the European Remote sensing Satellite (ERS-1). A mathematical model is developed to study the relation between a specified Mean Square Error, the value of the wavelet coefficient threshold and the maximum number of resolutions used. The wavelet compression method is compared with other compression methods like JPEG standard and Vector Quantization for a number of different (in a statistical sense) images amongst others LENA.

In this paper we evaluate the performance of subband/wavelet coders for image compression. The objective is to determine any possible connection between the suitability of the used FIR filters and their regularity index. First, we discuss the tool of (orthogonal and biorthogonal) multiresolution analysis. Then we investigate its relation to subband coding techniques. We show that dyadic multiresolution analysis is basically the same as octave band subband coding, except for the additional regularity requirement on the used filters. We cite techniques for calculating the regularity of arbitrary FIR subband filters. In the experimental part, we compare the performance of image coders employing several `classical' subband filters and `true' wavelet filters. We find that the regularity order of a filter and its performance are not strongly correlated and that the regularity of a filter can not be used as a selection criterion for the choice of filters in image coding applications.

In this paper we show how wavelets can be used for data segmentation. The basic idea is to split the data into smooth segments that can be compressed separately. A fast algorithm that uses wavelets on closed sets and wavelet probing is presented.

A tree-structured wavelet transform has been developed for texture classification in our previous work. The new transform, which offers a non-redundant representation and can be interpreted as the decomposition of a 2D function with the wavelet packet basis, is able to zoom into dominant frequency channels containing significant information of textures. In this research, we extend our work to the texture segmentation problem. A new multiscale texture segmentation algorithm based on the tree-structured wavelet transform and a fuzzy clustering technique is proposed. Numerical experiments are given to demonstrate the performance of the proposed algorithm.

We introduce an adaptive approach for texture feature extraction based on multi-channel wavelet frames and 2D envelope detection. Representations obtained from both standard wavelets and wavelet packets are evaluated for reliable texture segmentation. Algorithms for envelope detection based on edge detection and the Hilbert transform are presented. Analytic filters are selected for each technique based on performance evaluation. A K-means clustering algorithm was used to test the performance of each representation feature set. Experimental results for both natural textures and synthetic textures are shown.

An approach for characterizing the properties of basis functions which constitute a finite scheme of discrete Gabor representation is presented in the context of oversampling. The approach is based on the concept of frames and utilizes the Piecewise Finite Zak Transform (PFZT). The frame operator associated with the Gabor-type frame is examined by representing the frame operator as a matrix-valued function in the PFZT domain. The frame property of the Gabor representation functions are examined in relation to the properties of the matrix-valued function. The frame bounds are calculated by means of the eigenvalues of the matrix-valued function, and the dual frame, which is used in calculation of the expansion coefficients, is expressed by means of the inverse matrix. DFT-based algorithms for computation of the expansion coefficients, and for the reconstruction of signals from these coefficients are generalized for the case of oversampling of the Gabor space.

In this paper we introduce the concept of a local Hilbert space frame and develop theory for the representation and reconstruction of signals using local frames. The theory of global frames is due to Duffin and Schaeffer. Local frames are defined with respect to a global frame and a particular element from a Hilbert space H. For any signal f* (epsilon) H, H may be decomposed into two signal dependent subspaces: a finite dimensional one which essentially contains the signal f* and one to which the signal is essentially orthogonal. The frame elements associated with the former subspace constitute the local frame around f*.

We present two methods for generating frames of a Hilbert space H. The first method uses bounded operators on H to transform a frame into another frame of H₁ C H. The other method uses bounded linear operators on l₂ to generate frames of H. We characterize all the mappings that transform frames into other frames. We also show how to construct all frames of a given Hilbert space H starting from any given frame. We show how to apply the theory to obtain shift-invariant tight frames, and shift-invariant tight multiresolution. We also show how to obtain scaling function and wavelets with prescribed properties. Finally, we discuss the noise reduction properties of frames.

The discrete wavelet transform (DWT) is adapted to functions on the discrete circle to create a discrete periodic wavelet transform (DPWT) for bounded periodic sequences. This extension also offers a solution to the problem of non-invertibility that arises in the application of the DWT to finite length sequences and provides the proper theoretical setting for the completion of some previous incomplete solutions to the invertibility problem. It is proven that the same filter coefficients used with the DWT to create orthonormal wavelets on compact support in l^(infinity ) (Z) may be incorporated through the DPWT to create an orthonormal basis of discrete periodic wavelets. By exploiting transform symmetry and periodicity we arrive at easily implementable and fast synthesis and analysis algorithms.

In this paper we present orthogonal transforms as a signal analysis and processing tool with the capability of variable multiresolution time-spectral decomposition of discrete signals. Our prime interest is in the representation of square summable sequences in terms of wavelet packet matrices used as discrete orthogonal systems, and we concentrate on the fast transform algorithms for such systems. We analyze polyphase and lattice-tree structures which are common for multiple block-size orthogonal transforms, multiresolution multirate filter banks, and wavelet packet transforms. The purpose of this present paper is to compare and contrast the wavelet packet based approach to the traditional techniques for fast orthogonal transform algorithms. We consider results from this technique that influence the design of filter banks and we indicate some results from lattice-structured filter banks which can be useful for the design of fast wavelet packet transform algorithms. A time-varying structure that is based on fast algorithms of orthogonal transforms and their orthogonal sub-transforms is presented. In this case, orthogonal bases consist of a finite collection of wavelet packets which provide a fairly rich family of orthogonal decompositions of the time-scale plane. In particular, the time- sequence plane representation, one of many possible time-spectral and time-scale plane representations is discussed.

Wavelet processing has shown great promise for a variety of image and signal processing applications. Wavelets are also among the most computationally expensive techniques in signal processing. It is demonstrated that a wavelet engine constructed with residue number system arithmetic elements offers significant advantages over commercially available wavelet accelerators based upon conventional arithmetic elements. Analysis is presented predicting the dynamic range requirements of the reported residue number system based wavelet accelerator.

The wavelet transform is implemented by volume hologram in the frequency domain. 2D Harr's wavelet was encoded with polarizer. Different types and different dilation factors of the 2D Harr's wavelets are recorded in the volume holograms via an angular multiplexing technique. The corner and edge Harr wavelets are used together to extract different features of an input image. Experimental results and computer simulation are compared and presented in this paper. The results show that we can use the proposed optical system to realize wavelet transforming, and then to extract different local features (such as edge, corner) of the image with multiresolution and realize the task of pattern recognition. The limit and the feasibilities of the system are discussed.

In the paper, we proposed an optimal receiver based on discrete wavelet transform. To characterize transients in time-varying signal, a more generalized representation which can reflect both the time and the frequency behavior of signals is desirable. Wavelet was used as an orthonormal basis to represent signals in the joint time-frequency domain. We derive a sufficient statistic for the wavelet-based signal detection. An architecture of the optimum wavelet receiver was given. We discussed some issues on the detection of known signals, signals with unknown parameters and parameter estimation. We also evaluated the performance of the optimum wavelet receiver. It shows that the performance of the wavelet receiver is optimum in the sense that a specific criterion is satisfied.

The detection of transients in noisy time series is an important part of modern signal analysis because of the importance of its civil and military applications. We present a new denoising procedure, the output of which gives a very reasonable guess for the component of the input signal which was buried in noise.

In this paper, a wavelet based approach to the detection of anomalies in an image is described. In this approach, the anomalies are detected through hypothesis tests on the wavelet coefficients of the input image. In the development of this approach, some results on the correlation structure of the wavelet expansion of wide-sense stationary (WSS) processes are established. Namely, the wavelet coefficients are WSS and weakly within correlated a resolution level, uncorrelated when separated by more than one resolution levels, almost uncorrelated when separated by one resolution level. Experimental results on both synthetic and real-world images (sandpaper defect detection) and comparison with results obtained by neural network demonstrate the efficacy of the wavelet approach.

Wavelet analysis offers a technique for signal detection particularly suited for non-stationary signals. Other methods of signal analysis, namely Fourier techniques, make explicit assumptions about the signals to be analyzed. If these assumptions hold, then the results can be striking. Conversely, if the signal does not ideally match the properties assumed by the analysis function then the results can be indecisive or ambiguous. The standard Fourier transform assumes a basis function of sines and cosines underlies all signals and analyzes the fluctuation frequency of the signal function in equal bandwidth units. Wavelet theory offers a more general technique for solving a wide variety of signal processing problems using an octave bandwidth approach. The Wavelet transform offers an alternative method for analyzing functions that have variable periodicity over the duration of a data set, either through localization or a change in the oscillation period. The Wavelet transform is shown to be a tool for the analysis of general signals. Astrophysical data sets often provide interesting applications of signal processing techniques. In this paper the Morlet Wavelet transform is applied to the preliminary analysis of radio pulsar dynamic spectra.

In this paper, a method using the wavelet transform and the multiresolution analysis to remove the speckle effect in Synthetic Aperture Radar imagery is proposed. This technique allows the filtering of the most speckled structures and the improvement of the performances of a classical Wiener filter. Results on one example are presented. They confirm the efficiency of the multiresolution approach, as described in Cauneau and Ranchin' .The choice of the algorithm of wavelet transform and of the number of scales treated are discussed. The efficiency of the "a trous" algorithm is due to the isotropy of the wavelet used. This non-directional wavelet allows an improvement of the radiometric resolution without degrading significantly the geometrical resolution. Perspectives for an improvement of the quality of the speckle reduction are explored.

This paper develops a signal detection scheme based on the wavelet packet theory, which has been introduced recently as an effective signal analysis tool by M.V. Wikerhauser. As we all know, for the detection of transient signals with unknown arrival times, there is no uniformly tool because the detector performance usually depends on the signal representation. If all the signal components are similar, our method, called wavelet packet detection algorithm (WPDA), provides the possibility of detecting different arrival times correctly, even in the case that their waveforms overlap too much. As a comparison, the Improved Gabor Representation Method is utilized. The experiments show that WPDA can achieve more satisfactory performance.

A new wavelet transform is introduced for analyzing image compression and signal processing. We introduce Wavelet Stieltjes transform (WST) which provides a unified framework to analyze both continuous and discrete signals. Some properties of WST are summarized. Moreover, we obtain multiresolution analysis and encode WST coefficients for WST. It is shown that one can distinguish images or signals by WST methods while they are not discriminated by regular wavelet transform. Theory for regular wavelet transform may be extended to WST.