A crucial problem in image analysis is to construct efficient low-level representations of an image, providing precise characterization of features which compose it, such as edges and texture components. An image usually contains very different types of features, which have been successfully modeled by the very redundant family of 2D Gabor oriented wavelets, describing the local properties of the image: localization, scale, preferred orientation, amplitude and phase of the discontinuity. However, this model generates representations of very large size. Instead of decomposing a given image over this whole set of Gabor functions, we use an adaptive algorithm (called matching pursuit) to select the Gabor elements which approximate at best the image, corresponding to the main features of the image. This produces compact representation in terms of few features that reveal the local image properties. Results prove that the elements are precisely localized on the edges of the images, and give a local decomposition as linear combinations of `textons' in the textured regions. We introduce a fast algorithm to compute the matching pursuit decomposition for images with a complexity of (Omicron) (N log² N) per iteration for an image of N² pixels.

Filtering a sequence of images for target detection is discussed using multiresolution wavelet techniques and nonlinear size limited band filters based on partial differential operators associated with curvature of image level sets. In the latter procedure, evolution of the image under the differential operator is analogous to morphological erosion using structure sets, while backwards time evolution corresponds to dilation. The composition of these operations is subtracted from the original image to give the size limited `tophat' operator. Application of this procedure is demonstrated using synthesized targets embedded in actual infrared camera data.

Multiresolution surface reconstruction from volume data is very useful in medical imaging, data compression and multiresolution modeling. This paper presents a hierarchical structure for extracting multiresolution surfaces from volume data by using a 3-D wavelet transform. The hierarchical scheme is used to visualize different levels of detail of the surface and allows a user to explore different features of the surface at different scales. We use 3-D surface curvature as a smoothness condition to control the hierarchical level and the distance error between the reconstructed surface and the original data as the stopping criteria. A 3-D wavelet transform provides an appropriate hierarchical structure to build the volume pyramid. It can be constructed by the tensor products of 1-D wavelet transforms in three subspaces. We choose the symmetric and smoothing filters such as Haar, linear, pseudoCoiflet, cubic B-spline and their corresponding orthogonal wavelets to build the volume pyramid. The surface is reconstructed at each level of volume data by using the cell interpolation method. Some experimental results are shown through the comparison of the different filters based on the distance errors of the surfaces.

We present a generalization of the scale substraction filter (SSF) of Gregoris et al. and apply it to the problem of filtering objects, within an image, by size.

We briefly survey how to use libraries of (orthonormal) bases of well-behaved waveforms, including wavelets and lapped orthogonal transforms, so as to obtain fast numerical algorithms for the expansion of functions and operators in these bases. The most important applications are fast approximate matrix multiplication, and application of matrices to vectors.

Signal decomposition techniques are an important tool for analyzing nonstationary signals. The proper selection of time-frequency basis functions for the decomposition is essential to a variety of signal processing applications. The discrete wavelet transform (DWT) is increasingly being used for signal analysis, but not until recently has attention been paid to the time-frequency resolution property of wavelets. This paper describes additional results on our procedure to design wavelets with better time-frequency resolution. In particular, our optimal duration-bandwidth product wavelets (ODBW) have better duration-bandwidth product, as a function of wavelet-defining filter length N, than Daubechies' minimum phase and least- asymmetric wavelets, and Dorize and Villemoes' optimum wavelets over the range N equals 8 to 64. Some examples and comparisons with these traditional wavelets are presented.

The optical wavelet circular harmonic function minimum average correlation energy (WCHF- MACE) filter is proposed. The proposed WCHF-MACE filter uses the wavelet transformed image by Mexican-hat wavelet function for circular harmonic function and the multiple harmonic components of circular harmonic function are used for MACE filter synthesis. The proposed filter has good discrimination compared with the conventional circular harmonic function filter and conventional circular harmonic MACE filter about the limited rotated images. And the filter is made of the type of optical wavelet matched filter (WMF), so that the proposed filter can use the conventional 4f correlator.

Visual table of contents provides users with easier access to imagery databases. Reduced resolution images are collected in a portfolio, presented to the user to browse through and select an image based on this thumbnail image. This avoids having to select an image from a list of file names. This paper shows the results of using Daubechies wavelets to produce the reduced resolution images and improvements to the selection process. An image is passed through a Daubechies wavelet several times to produce a four (4) to eight (8) time reduction. The low-pass vertical/low-pass horizontal sub band produces a thumbnail sketch. The remaining three sub bands are compressed with a standard lossless or lossy compression algorithm. Instead of selecting an entire image to retrieve at full resolution, the user can select the level of expansion required. This can be up to the entire image at full resolution. Providing the user with these options reduces the amount of imagery to decompress and to pass along the network.

Using wavelet decompositions to obtain compressed input data for optical character classifiers is a natural idea. However, extensive character preprocessing, maybe by other, different, techniques is usually necessary to achieve adequate recognition rates. We show how, prior to that compression, wavelet techniques can also be used in tasks such as character localization, filtering and scaling that are essential to obtain good recognition performances. We propose a fast two pass bottom-up procedure to perform these tasks over the Haar multiresolution decomposition of single black and white characters. Starting at the lowest resolution, filtering is performed first, yielding as a by-product a first set of character localization and scaling parameters. Carrying those steps upwards in the resolution decomposition allows finer grain filtering and more precise localization and scaling values. Once the desired decomposition level has been reached, actual scaling is performed transforming the resulting wavelet coefficients in terms of a mapping of the new scale basis functions upon those of the original one. Our algorithms have essentially the same order complexity as multiresolution decompositions and can be carried out in integer arithmetic.

This paper is an overview of the Advanced Technology Program (ATP) and describes the essential features, competitions, and industry input of the ATP; the existing focused programs; and how you can make your voice heard in the ATP process. The ATP provides partial funding for high-risk technologies for for-profit companies. Focused programs concentrate development of these high-risk technologies in specific areas. Since industry is a source of abundant ideas for programs to be funded, the process for submitting program ideas to the ATP is straightforward.

In this paper, we present (1) a design concept of a fingerprint scanning system that can reject severely blurred inputs for retakes and then de-smear those less blurred prints. The de-smear algorithm is new and is based on the digital filter theory of the lossless QMF (quadrature mirror filter) subband coding. Then, we present (2) a new fingerprint minutia feature extraction methodology which uses a 2D STAR mother wavelet that can efficiently locate the fork feature anywhere on the fingerprints in parallel and is independent of its scale, shift, and rotation. Such a combined system can achieve high data compression to send through a binary facsimile machine that when combined with a tabletop computer can achieve the automatic finger identification systems (AFIS) using today's technology in the office environment. An interim recommendation for the National Crime Information Center is given about how to reduce the crime rate by an upgrade of today's police office technology in the light of the military expertise in ATR.

Symmetric quadrature mirror filters (QMFs) offer several advantages for wavelet-based image coding. Symmetry and odd-length contribute to efficient boundary handling and preservation of edge detail. Symmetric QMFs can be obtained by mildly relaxing the filter bank orthogonality conditions. We describe a computational algorithm for these filter banks which is also symmetric in the sense that the analysis and synthesis operations have identical implementations, up to a delay. The essence of a wavelet transform is its multiresolution decomposition, obtained by iterating the lowpass filter. This allows one to introduce a new design criterion, smoothness (good behavior) of the lowpass filter under iteration. This design constraint can be expressed solely in terms of the lowpass filter tap values (via the eigenvalue decomposition of a certain finite-dimensional matrix). Our innovation is to design near- orthogonal QMFs with linear-phase symmetry which are optimized for smoothness under iteration, not for stopband rejection. The new class of optimally smooth QMF filter banks yields high performance in a practical image compression system.

We study the problem of choosing an image based optimal wavelet basis with compact support for image data compression and provide a general algorithm for computing the optimal wavelet basis. We parameterize the mother wavelet and the scaling function of wavelet systems through a set of real coefficients of the relevant quadrature mirror filter (QMF) banks. We further introduce the concept of decomposition entropy as an information measure to describe the distance between a given digital image and its projection into the subspace spanned by the wavelet basis. The optimal basis for the given image is obtained through minimizing this information measure. The resulting subspace is used for image analysis and synthesis. A gradient based optimization algorithm is developed for computing the image based optimal wavelet basis. Experiments show improved compression ratios due to the application of the optimal wavelet basis and demonstrate the potential applications of our methodology in image compression. This method is also useful for constructing efficient wavelet based image coding systems.

We describe a novel adaptive wavelet-based compression scheme for images which takes advantage of image redundancy across scales. As with standard wavelet transform coders, our compressed image representation consists of a set of quantized wavelet coefficients and quantized wavelet subtrees. Instead of having a fixed subtree codebook, however, we construct a codebook from the image being compressed. Subtrees are quantized to contracted isometries of coarser scale subtrees. This codebook drawn from the image is effective for quantizing locally smooth regions and locally straight edges. We prove that this self- quantization enables us to recover the fine scale wavelet coefficients of an image given its coarse scale coefficients. We show that this self-quantization algorithm is equivalent to a fractal image compression scheme when the wavelet basis is the Haar basis. The wavelet framework greatly simplifies the analysis of fractal compression schemes and places fractal compression in the context of existing wavelet subtree coding schemes. We obtain a simple convergence proof which strengthens existing fractal compression results, we describe a new reconstruction algorithm which requires O(N) operations for an N pixel image, and we derive an improved means of estimating the error incurred in decoding fractal compressed images. We show the effect of self-quantization for a test image.

Top-down tree search algorithms with non-additive information cost comparisons as decision criteria have recently been proposed by Taswell for the selection of near-best bases in wavelet packet transforms. Advantages of top-down non-additive near-best bases include faster computation speed, smaller memory requirement, and extensibility to biorthogonal wavelets in addition to orthogonal wavelets. A new compression scheme called parameterized-model coding was also proposed and demonstrated for one-dimensional signals. These methods are extended here to two-dimensional signals and applied to the compression of images. Significant improvement in compression while maintaining comparable distortion is demonstrated for parameterized-model coding relative to quantized-scalar coding. In general, the lossy compression scheme is applicable for low bit rate coding of the M largest packets of wavelet packet decompositions with wavelet packet basis libraries and the M atoms of matching pursuit decompositions with time-frequency atom dictionaries.

This wavelet based data compression algorithm addresses a growing need to handle large quantities of image data quickly and efficiently. This wavelet technique has been coded based on the assumption that small coefficients computed by the two-dimensional orthogonal wavelet transform are principally associated with image noise, and only the largest values are required to capture the information content of the respective source image. The approach has been successfully applied to one-dimensional signals in the design of signal classifiers. This particular algorithm for wavelet-based image compression has been designed and compared to the Joint Photographic Expert Group (JPEG) still picture image compression standard. Examples are shown of side scan sonar images, x rays, and laser line scan images.

In this paper, we describe an improved version of our previous approach for low bit rate near- perceptually transparent image compression. The method exploits both frequency and spatial domain visual masking effects and uses a combination of Fourier and wavelet representations to encode different bands. The frequency domain masking model is based on the psychophysical masking experimental data of sinusoidal patterns while the spatial domain masking is computed with a modified version of Girod's model. A discrete cosine transform is used in conjunction with frequency domain masking to encode the low frequency subimages. The medium and high frequency subimages are encoded in the wavelet domain with spatial domain masking. The main improvement over our previous technique is that a better model is used to calculate the tolerable error level for the subimages in the wavelet domain, and a boundary control is used to prevent or reduce the ringing noise in the decoded image. This greatly improves the decoded image quality for the same coding bit rates. Experiments show the approach can achieve very high quality to nearly transparent compression at bit rates of 0.2 to 0.4 bits/pixel for the image Lena.

Wavelet image decompositions generate a tree-structured set of coefficients, providing an hierarchical data-structure for representing images. Recent efforts in image compression have focused on new ways for exploiting dependencies within this hierarchy of wavelet coefficients, and these efforts are achieving significant improvements in coding performance over traditional subband coding approaches. This paper presents a statistical framework for understanding the efficiency of these new zerotree wavelet coding algorithms, and describes our own wavelet coding algorithm which optimally couples scalar and zerotree quantization modes. Building from this framework, we describe a radically new approach to image compression (dubbed morphological representation of wavelet data or MRWD) based on the statistical characterization of morphological sets of wavelet coefficients (a morphological set is a set of coefficients related to other coefficients via morphological operators). Preliminary simulation of the new algorithm shows very promising results.

An image representation based on a generalization of quad trees, related to the definition of Haar wavelets and the resulting multi-resolution decomposition of L²(Rⁿ) is presented. The basic Haar theory is presented, and then generalized to the case of multiple dilation matrices, allowing for the definition of (almost) arbitrary tilings. The image representation is translation invariant with a computational complexity of O(Nlog_aN) where a is determined by the mixing of the individual dilation matrices. The technique has been applied to the unsupervised clustering of magnetic resonance imagery and is being studied for the lossless coding of imagery.

Wavelet transform (WT) is concatenated with the modified Hadamard-structured discrete cosine transform (MHDCT) to encode image signals. As the outer code, the WT decomposes the image signals into uncorrelated low resolution and detailed signal components. As inner codes, the MHDCT adaptively encodes the low resolution signal, while the detailed signals are encoded using pulse code modulation (PCM). The concatenated coding approach offers the possibility of regulated coding for progressive transmission of the image signals over communication networks. Results of image coding using the concatenated WT/MHDCT coder are presented.

A new Riesz product is formulated in terms of waveletpacket pyramidal tree structures. A path within such a structure is determined by a lacunarity criterion and a sequence (H_(epsilon (j))) of filters, where each H_(epsilon (j)) is one or the other element of a prescribed subband coding pair (H₀, H₁) of FIR filters. The Riesz product associated with a given path is a continuous radon measure. Path based criteria are given to determine singular and absolutely continuous pairs of such measures. The measures have full support, and their approximants exhibit fractal behavior. These properties can be used to design a secure transmission scheme in communications theory.

In this paper we present basic background material concerning the propagation of pulses through fiber optics, and current transmission schemes in fiber optics. We discuss the value of using different signal processing techniques, and time-frequency bases in fiber optic transmission. Finally we outline issues which we believe to be of current interest in fiber optic transmission.

Frequency domain signal processing has been used for some time to suppress narrowband interference in spread spectrum signals. Essentially, narrowband interference energy appears in only a few of the transform bins whereas the wideband spread spectrum signal energy appears in most, if not all, of the bins. A simple exciser, which sets bins that are primarily interference to zero, can then remove most of the interference energy while removing only a small portion of the spread spectrum signal energy. Recently, this concept has been extended to include wavelet transform domain excision where the time-frequency localization properties of the wavelet transform are utilized to improve the effectiveness of the excision process. This paper generalizes this work by considering wavelet-based linear transforms and multirate filterbanks and replacing the simple exciser with continuously variable adaptive tap weights on each transform bin. These tap weights can be derived using a transform domain adaptive filter structure wherein the weights are adapted to optimize the detection performance in the mean- square sense, effectively producing a transform domain decision-directed equalizer. The paper presents an analytical result for bit-error-rate (BER) performance that is applicable for arbitrary linear transforms and generic multirate filterbank analysis/synthesis structures. Both adaptive weighting and simple exciser schemes are considered. Through both analytical and Monte-Carlo simulation results, the adaptive filter is shown to provide significantly better performance than the simple exciser for tone interference.

In communications systems, the message signal is sometimes spread over a large bandwidth in order to realize performance gains in the presence of narrowband interference, multipath propagation, and multiuser interference. The extent to which performance is improved is highly dependent upon the spreading code implemented. Traditionally, the spreading codes have consisted of pseudo-noise (PN) sequences whose chip values are limited to bipolar values. Recently, however, alternatives to the PN sequences have been studied including wavelet based and PR-QMF based spreading codes. The spreading codes implemented are the basis functions of a particular wavelet transform or PR-QMF bank. Since the choice of available basis functions is much larger than that of PN sequences, it is hoped that better performance can be achieved by choosing a basis tailored to the system requirements mentioned above. In this paper, a design method is presented to construct a PR-QMF bank which will generate spreading codes optimized for operating in a multiuser interference environment. Objective functions are developed for the design criteria and a multivariable constrained optimization problem is employed to generate the coefficients used in the filter bank. Once the filter bank is complete, the spreading codes are extracted and implemented in the spread spectrum system. System bit error rate (BER) curves are generated from computer simulation for analysis. Curves are generated for both the single user and the CDMA environment and performance is compared to that attained using gold codes.

We examine the question of how to choose a time-varying filter bank representation for a signal which is optimal with respect to an additive cost function. The tree structure gives segmentations of the signal in frequency, while the time-varying nature of the tree gives segmentations in time. We present an efficient algorithm which finds the optimal basis, given the constraint that the time and frequency segmentations are binary. Extension to multiple dimensions is simple. We verify that the algorithm indeed produces a lower cost representation than any of the wavelet packet representations for compression of images using a simple rate-distortion cost.

In this paper we introduce a new class of communications systems called wavelet transform domain (WTD) systems. WTD systems are transmultiplexer (TMUX) structures in which information to be communicated over a channel is encoded, via an inverse discrete wavelet transform (IDWT), as the wavelet coefficients of the transmitted signal, and extracted at the receiver by a discrete wavelet transform (DWT). WTD constructs can be used for covert, or low probability of intercept/detection (LPI/D) communications, baseband bandwidth efficient communications, or code-division multiple access (CDMA). This paper concentrates on the spread spectrum applications.

A class of waveforms developed for covert, or low probability of intercept/detection (LPI/D) communications has been tested for its applicability to code-division multiple access (CDMA) communications. This paper reports some results of the investigation. The waveforms, constructed by techniques based on wavelet transforms, show the characteristics of bandpass Gaussian noise under a number of statistical tests. We refer to these as single-scale waveforms because of an aspect of their construction. By a combination of analysis and simulation, the single-scale waveforms have been compared to conventional CDMA techniques, and it is concluded that on the additive, white Gaussian noise channel, they outperform the conventional by 75 - 200% in capacity (number of supportable users per unit bandwidth at fixed SNR and bit error rate).

The joint University of Rochester/Rochester Institute of Technology `Center for Electronic Imaging Systems' (CEIS) is designed to focus on research problems of interest to industrial sponsors, especially the Rochester Imaging Consortium. Compression of tactical images for transmission over an rf link is an example of this type of research project which is being worked on in collaboration with one of the CEIS sponsors, Harris Corporation/RF Communications. The Harris digital video imagery transmission system (DVITS) is designed to fulfill the need to transmit secure imagery between unwired locations at real-time rates. DVITS specializes in transmission systems for users who rely on hf equipment operating at the low end of the frequency spectrum. However, the inherently low bandwidth of hf combined with transmission characteristics such as fading and dropout severely restrict the effective throughput. The problem at designing a system such as DVITS is particularly challenging because of bandwidth and signal/noise limitations, and because of the dynamic nature of the operational environment. In this paper, a novel application of wavelets in tactical image coding is proposed to replace the current DCT compression algorithm in the DVITS system. THe effects of channel noise on the received image are determined and various design strategies combining image segmentation, compression, and error correction are described.

This work presents a generalized signaling strategy for orthogonally multiplexed communication. Wavelet packet modulation (WPM) employs the basis functions from an arbitrary pruning of a full dyadic tree structured filter bank as orthogonal pulse shapes for conventional QAM symbols. The multi-scale modulation (MSM) and M-band wavelet modulation (MWM) schemes which have been recently introduced are handled as special cases, with the added benefit of an entire library of potentially superior sets of basis functions. The figures of merit are derived and it is shown that the power spectral density is equivalent to that for QAM (in fact, QAM is another special case) and hence directly applicable in existing systems employing this standard modulation. Two key advantages of this method are increased flexibility in time-frequency partitioning and an efficient all-digital filter bank implementation, making the WPM scheme more robust to a larger set of interferences (both temporal and sinusoidal) and computationally attractive as well.

The existing multiplexing techniques, such as TDM, FDM, and CDM, use different time slots, different frequency bands, and orthogonal codes, respectively, to distinguish each user. In this paper, a multiplexing model based on wavelet packets is presented. This model employs another kind of orthogonality -- the waveform orthogonality. In this approach, the channels are assigned to different users based on the mutual orthogonality of wavelet packets. After a brief review of the properties of wavelet packets, we can see that multiplexing based on the wavelet packets is a generalized form of TDM and FDM and it can be easily realized by the multiresolution algorithm developed for wavelet packets. In the multiplexing techniques, synchronization is an important problem. In this paper, the interference and cross- talk due to non-perfect synchronization between the receiver and transmitter are analyzed. It is demonstrated that with a suitable wavelet function, digital signals can be well recovered under noisy environments even when the synchronization is not accurate. The analysis on the synchronization also leads to the design of optimum wavelets for the application. Simulation results are presented and compared for the use of different wavelets.

We discuss the problem of waveform selection for wideband radar imaging. We show how this problem can be solved in a finite-dimensional setting. We discuss an algorithm for imaging an unknown target from a collection of targets belonging to M classes where M > 2. We present two possible algorithms for simultaneous imaging and classification of unknown targets. We describe how the radar simulations can be done in these cases.

In this paper we investigate the problem of fast and accurate classification of naval targets from radar returns. The algorithms can be applied to both 1-D and 2-D data (i.e., high range resolution and imaging radar returns). We describe the structure of these algorithms and report experimental results on their performance with synthetic returns from ships. We have successfully addressed the problem of reducing the target model representations with respect to viewpoint variations and other sensor parametric variations. Our method can be viewed as a quantization of the space of sensing operations. The resulting multiresolution aspect graph is a (relational) graph representation of this quantization. Aspect graphs of target radar returns are generated algorithmically. Since our off-line model/parameter tuning methods are based on general vector quantization, our methods extend naturally and efficiently to multi-sensor data: LADAR, TV, mmWave, SAR, etc. We describe new results on the `continuity' of the aspect graph, new properties and improvements of our algorithmic constructions. Our basic classification algorithm utilizes a cascade of a wavelet preprocessor followed by a tree- structured clustering algorithm; a learning mode can also be added. We develop a high performance parallel progressive classification algorithm and report on its performance and complexity. We show experiments illustrating that the parallel algorithm outperforms a compound version (which is the more intuitive choice) and that it has performance close to Bayes optimal classification (via comparison to learning vector quantization). We outline an analytical framework for establishing these results theoretically. We also discuss similar experiments from face recognition and medical image classification problems.

This paper applies wavelet transform methods to the detection of continuous wave (cw) radar signals in Gaussian white noise. The method applies to radar signal processing in typical, semi-active missile systems. The usual detection procedure consists of Fourier transforming the sampled data using the FFT, performing a CFAR operation, and thresholding. However, a detection loss occurs when the signal Doppler exists near the FFT bin center. This loss results from the spectral spreading of the signal, due to mismatch between the signal and FFT basis functions. The spectral spreading causes the signal to appear as a larger scale fluctuation relative to the small scale fluctuations of the white noise. It is shown that application of the single-level wavelet transform captures the signal through exploitation of these scale differences. Furthermore, it is shown that improved detection performance may be obtained by first applying the single-level wavelet transform to the amplitude spectrum. Then, the CFAR process is performed in the wavelet domain, followed by thresholding. A family of wavelet- based detectors is presented, which offer a trade-off between peak detection performance, and average detection performance over Doppler frequencies. That is, a slight detection performance loss near the FFT basis frequencies is traded for a significant detection performance gain near the FFT bin center. ROC curves, generated by Monte-Carlo simulation, are presented which sweep out detector performance. The computational complexity of the proposed detectors is discussed. The design of wavelets matched to this application, using Vaidyanathan's lattice decomposition, is also presented.

We consider several new applications of wavelet transforms: general time-frequency analysis, detection and range-Doppler processing. These differ from the standard data compression and encoding applications generally considered. We note problems with standard and dyadic wavelet transforms concerning shift-invariance, frequency resolution, detection in noise at high frequencies where the bandwidth of the wavelet subband is large, and in range-Doppler processing. We consider Gabor wavelets and advance a new shift-invariant continuous wavelet filter with only O(N) processing required; we also advance new non-dyadic wavelet filters and an optical implementation for the general use (especially for range-Doppler processing).

Inverse synthetic aperture radar (ISAR) uses target's motion to generate images on the range- Doppler plane. The conventional ISAR uses Fourier transform to compute Doppler spectrum for each range cell. Due to the target irregular translational and rotational motion, the Doppler frequency in fact is time-varying. By using Fourier transform, the reconstructed image becomes blurred. To represent time-varying Doppler spectrum, time-frequency transform should be utilized. Adaptive time-frequency wavelet transform is a very useful tool in analysis of signals with time-varying spectrum. We applied adaptive time-frequency wavelet transform to ISAR image reconstruction and developed a simulation procedure to describe the characteristics of the algorithm. By replacing the conventional Fourier processor with the adaptive wavelet processor, a 2-D range-Doppler Fourier ISAR frame becomes a 3-D time- range-Doppler wavelet ISAR cube. By sampling in time, a time sequence of 2-D range- Doppler images can be viewed. Each individual wavelet ISAR image provides not only superior resolution but also the temporal information within each frame time. Both simulated and real ISAR data have been tested. The result from simulated ISAR data is illustrated in this paper.

We investigate a new signal model based on the wavelet packet representation to achieve high resolution beam-forming. With a properly selected signal model, we perform a robust iterative algorithm to extrapolate the data outside of the observation interval and, therefore, increase the resolution of the beam patterns for a 1-D linear array. The wavelet packet sign model is particularly attractive in representing nonstationary signals such as the chirp signal. The superior performance of the resulting beamformer is demonstrated by numerical experiments.

Time-domain electrodynamics of dispersive and lossy media is advanced by means of new exact explicit solutions of Maxwell equations, obtained beyond the scope of traditional Fourier expansions. These non-stationary solutions describe the fast dispersive spreading and attenuation of unharmonic fields in these media without using traditional concepts of frequency, wavelength, decrement, attenuation length, etc. Using orthogonal Laguerre transients for description of arbitrary waveforms, incidenting on the interfaces of dispersive and absorbing media, we find analytically the simple expressions for coefficients of waveforms reflection from wide classes of dielectric and conducting materials.

Ground-based S-band radars with center frequencies in the 3 GHz range, and bandwidths of about 2 MHz, are currently being considered for the classification/identification of targets in the 50 - 250 km range (low signal-to-noise (SNR) environment). Its main drawback is that it is time and computation intensive, and, for targets at long range, it usually requires a number of looks before an ID decision is made. This leaves open the possibility of alternative approaches to S-band radar target classification. In this paper, we propose a method of classification based on correlation in a nonlinear feature space; our features are generated by a combination of wavelet transform (WT) processing and compression, and Fourier transform (FT)-based methods. The classification itself can be done explicitly by metric criteria, e.g., nearest neighbor or matched-filtering schemes, or implicitly using artificial neural networks; our simulation tests show successful classification with simple matched-filtering in our feature space. While our proposed method could be used to build an entirely different approach to target classification, it makes more sense to incorporate it into the existing classification structure. Since the frequency-domain processing required here is already performed by the existing algorithms, our method would add a minimum of extra computational burden, while opening the possibility of an earlier decision.

Several different new Gabor wavelet filters are described: the Gabor transform (GT) filter consists of real, imaginary, and clutter filters; the Gabor basis function (GBF) filter uses a Gabor basis function for each training image; the morphological wavelet transform (MWT) filter includes a Gabor transform clutter map filter that locates clutter regions of a scene. These filters ar all shift-invariant and distortion-invariant. They are employed for detection: location of the positions of all object regions of interest (ROIs) in an input scene. Fusion of multiple filter outputs is used to reduce false alarms. This paper emphasizes the role for Gabor wavelet filters in detection and for producing a clutter map. Major emphasis is given to the final version of the Gabor wavelet clutter map portion of our MWT algorithm (this is our best detection algorithm). New detection and fusion results with a consistent database and thresholds are provided.

The wavelet transform (WT) can be used for pattern recognition. One scheme is to extract the wavelet features in the 4-D space-scale joint representation of the 2-D pattern for statistical pattern recognition. Another scheme is the wavelet matched filters (WMF), that uses the WT to enhance the edge features and make correlation between the WT of the input image and the WT of the reference image. This approach uses the optical shift invariant continuous WT and implements the WT and the matching of two WTs in a single step of the correlation. Several adaptive WTs and the matching pursuits have been proposed that use the best basis functions to the signal decomposition. The basis is selected from a library of dictionary waveforms to minimize an energy or an entropy in such a way that the signal expansion with those bases is the best for signal representation or classification. Pattern recognition emphasizes the classification. Fast numerical algorithms are given for the signal expansion with the best adaptive discrete orthogonal bases. Most approaches use a fixed shape basic wavelet with varying shift and dilation parameters. Szu et al., proposed adaptive wavelets that are linear combination of wavelets, called the `super-wavelet.' The super-wavelets can be continuous and redundant. The shape of the super-wavelets can be adaptively changed for the particular applications. They show the adaptive WT of the 1-D speech signals. In this paper we show the adaptive WT with continuous 2-D wavelets, whose shape is adaptively changed to achieve the pattern recognition invariant to continuous shift and scale changes. We show why such an adaptive WT is needed and how to construct the composite wavelet matched filter (CWMF) with the adaptive super-wavelet for the continuous scale invariant pattern recognition. The real-time complex valued optical filters implementation is reported in this paper.

An adaptive time-frequency classifier algorithm is detailed and tested on a data set of acoustic backscatter from a metallic manmade object and natural clutter with synthetic reverberation noise. The algorithm is improved over previous versions in that it operates directly on time signals rather than their wavelet transforms, and in that the features measure time-frequency energy and are insensitive to phase differences (due to signal variations). The time-frequency features are initialized by selecting wavelet transform coefficients with the highest Fisher ratios. Similarities and differences of this classification algorithm with matching pursuit, a representation algorithm, are discussed. This is done to show the need to treat classification differently than representation for realistic cases where class boundaries significantly overlap. The adaptive classifier is shown to have significantly higher detection rates than Fourier transform and wavelet transform methods, as compared by receiver operating characteristics.

A low bit rate speech coder based on Gaussian adaptive wavelets is described. This speech coder ameliorates the problem of using two different mother wavelets to model voiced and unvoiced sounds. In addition, it has been demonstrated in this paper that Gaussian adaptive wavelets are better suited to model both voiced and unvoiced sounds as compared to Morlet and Daubechies' wavelets. The bit rate and speech quality of this speech coder is compared with the speech coder based on Morlet wavelet. Experimental results using TIMIT speech database are discussed with examples.

Image enhancement techniques based on a composition of uncorrelated subband coefficients called `wavelet' features are demonstrated on printed circuit board images for fault detection objectives as applied to an automated inspection system. The wavelets are used to preprocess the imagery for enhancing and filling the holes on the images of the conducting connections. Several image enhancement techniques are demonstrated and described to make visual performance comparisons.

Automatic classification of different modulation schemes is an important topic in the receiver design for a non-cooperative communication system. The modulation classifier often serves as a pre-processing unit for a monitoring or interception system. The main difficulty of the problem lies in the uncertainty of the waveform and communication parameters of received signals. In this research, we first make a thorough comparison among four MPSK (M-ary phase shift keying) classifiers, i.e., the M-th power classifier (MLC), phase histogram classifier (PHC), phase difference histogram classifier (PDHC) and phase change classifier (PCC), in terms of their error classification probability and assumptions on the availability of various communication parameters. Then, we propose a novel MPSK classification algorithm based on the estimated phase change rate detected by using the continuous wavelet transform (CWT). Numerical experiments are used to demonstrate the performance of the proposed algorithm.

We present a methodology for classifying and/or identifying unknown radio transmitters by analyzing turn-on transient signals. Since an expedited signal classification and identification is desirable, we developed an automated, fast signal classification and identification method using wavelet-based feature extraction combined with an artificial neural network (ANN). The environment we considered is that there are n radio frequency (rf) transmitters given m finite duration signals (m > n, several signals may be emitted from the same transmitter). We preprocess unknown transient signals using wavelet decomposition and extract multiresolution features (statistical and energy content) to provide efficient signal characterization. An ANN, trained on known signals and selected wavelets, is then used for classifying and identifying the extracted feature characteristics of the unknown signals. Our wavelet preprocessing combined with the ANN provide a robust and adaptive classifier and identifier. We also provide an example of transmitter classification and identification using transient signals collected from three different transmitters.

In automatic speech or speaker recognition, the features used for classification are usually derived from a power spectrum representation. Only a few models present a spectral representation derived from a phase analysis of the signal. This paper presents a new transformation of the time-scale plane, obtained by a quasi-continuous wavelet transform, into a time-frequency plane: the synchrosqueezed plane representation. This analysis is `phase- oriented' and it formalizes naturally the IFD, SBS, and EIH models which are algorithmic representations for a speech signal derived from auditory nerve models. The representation is applied for closed-set speaker identification in the context of the modulation model. This leads to the introduction of a `phase-oriented' cepstral parameter: the wastrum. Experimental results are presented on KING database. The analysis is inverted to extract the primary components of a speech signal in the framework of the modulation model. Extensions to other time- frequency analyses are also discussed.

In this paper we consider wavelet analyses of acoustic signatures of ground vehicles. We select two test cases, a squeak emanating from a tank track and the clatter of a tank running on pavement. We examine various cost functions that can be utilized in determining which set of wavelet basis functions give the most accurate representation of a given signal. We compare different orthogonal and biorthogonal wavelet transforms with each other and with the local cosine transform. We found that the local cosine transform performed better for the squeak than the particular wavelet packet transforms that we used, while they both gave approximately equivalent results for the clatter signal.

In this paper, we propose a first-order fused HMM-ANN (hidden Markov model and artificial neural net) classifier using feature vectors extracted from ground vehicle acoustic signals. The feature vectors applied in this paper are Fourier power spectrum and scale-invariant wavelet coefficients. Our fused classifier network robustly provides a better performance for a variety of ground vehicle acoustic signals when compared to a classifier with either HMM or ANN alone. We emphasize the use of scale-invariant wavelet transforms to extract scale-invariant wavelet coefficient features because they play a vital role in classifying and identifying unknown ground vehicle acoustic signals that are time-varying in scale structure.

A new method of adaptive waveform analysis, called shape adaptive wavelet analysis, is presented where the basis vectors (wavelet packets) are initially considered as functions of continuous variables, in particular position/time, frequency, and scale. A discrete subset of these wavelet packets is chosen from local maxima of the modulus of their expansion coefficients, in general resulting in a non-orthogonal basis. Redundancy removal, when required, is implemented using an algorithm loosely based on Gram-Schmidt orthogonalization, where basis vectors are chosen to maximize the modulus of the projection of the residual perpendicular to the space spanned by the basis vectors chosen previously, thus maximizing `information content.' In cases where this representation is not sufficiently accurate an additional process of feature substraction is available enabling representations of arbitrary accuracy. It is shown, however, that there are sometimes advantages in retaining redundancy in the wavelet packet characterization of the signal, in particular where noise is present. The method presented in this paper is compared with the wavelet packet transform of Coifman et al. and with the matching pursuit algorithm of Mallat et al., and is shown to provide an improved characterization under circumstances which are of relevance in surveillance, detection, and classification in noisy or cluttered backgrounds. The method is shown to be well suited to characterizations in higher-dimensional wavelet packet spaces, where additional shape parameters, for example chirp angle, characterize the wavelet packets in addition to position, frequency, and scale.

The electromagnetic and acoustic wavelets recently introduced by the author are here redefined in a much simplified and generalized form. An application to remote sensing is proposed, resulting in a formulation of radar and sonar as problems in the calculus of variations which reduce to the usual wideband ambiguity function formalisms when the reflectors are assumed to move with a uniform velocity. The new wavelets also lend themselves naturally to multiresolution analysis of electromagnetic and acoustic signals, which could have applications for communication.

We present a highly powerful, modular, and interactive software tool for the analysis of time- frequency coherent signals via wavelet transformations. A major design goal of the Wavelet Signal Processing Workstation (WSPW) is to maximize ease of use while minimizing programming complexity. As such, the WSPW makes ample use of graphical mouse driven user interfaces and, in turn, allows powerful signal processing, classification, and identification techniques to be rapidly implemented and tested. Because it has been developed using MATLAB, the WSPW is easily extensible and inherently portable between varying system architectures. Although the emphasis of this paper is on the wavelet representation of signals, the WSPW has proven itself a valuable tool in applications including radar source identification and signal classification.

The pitch of voiced speech sounds provides very important information in speech analysis. Pitch estimation is a difficult task when unprevented noise exists. However experimental results have shown that even robust pitch detection techniques fail in noisy environment with periodic patterns such as noise generated by machines. Wavelet transform, with its special properties in time frequency relation, can be used to detect pitch with remarkable advantage in noise resistance. In wavelet signal analysis, the modulus of the transform have been used extensively, however, we found that the phase information is equally important especially for pitch detection. Since the phase spectrum is always intensive to noise, a more promising pitch period can be obtained from the phase diagram. Properties of the phase pattern in wavelet transform are investigated and the result is applied to construct a robust pitch detector. In our first test, the detector is employed to detect the pitches of a set of speech signals with white noise. We found that our approach clearly outperforms other non-wavelet methods with low signal-to-noise ratio. Sinusoidal noise with different frequency levels is used in the second test. Simulation results have shown that our system works quite stable in such an environment.

Information regarding the pitch period of a speech signal is an important parameter in designing a speech recognition system for tone based languages such as Chinese. Chinese speech has four fundamental tone types: the tone for a given word is mainly characterized by its vowel section. The four tone types are distinct and can be recognized by examining the tone contour. Researchers have developed several different techniques for speech pitch detection. One recent technique is an event-based pitch detection scheme utilizing the dyadic wavelet transform (DyWT). This pitch detector compares maxima across wavelet scales to locate the beginning of each pitch period. We extend this pitch detection technique to a set of Chinese tone utterances using both the cubic spline wavelet and the Daubechies 4 wavelet. We investigate the impact of applying two preprocessing techniques, center clipping and half wave rectification, as well as several pitch decision logic methods. The DyWT performance is compared to an autocorrelation based pitch detector. The results show that the DyWT pitch detector compares favorably with the autocorrelation pitch detector, and has better performance for some cases. A variety of pitch decision logic techniques improve the DyWT pitch detector performance over the original method.

Due to the fact that linearly dependent wavelet transforms are inefficient in computation and storage, the most popular discrete wavelet transforms (DWT) use orthonormal basis decompositions (ODWT). However, there are several problems related to ODWT: as a time- frequency analyzer, the time-frequency resolution and localization of ODWT are poor; from a spectral analysis point of view, the ODWT produces a distorted spectrum with energy leaking, aliasing, and magnitude distortion; to produce a reasonable time-varying spectrum, the signal is required to be octave-distributed with a spectral shape matching the spectra of the wavelets. To overcome these problems, a new approach for designing the optimal wavelet is proposed. Using the multiresolution parametric spectral estimator, the new method continuously tracks the time-varying signal to adapt the optimal wavelets, and yields high resolution, localization, and fidelity for the resulting time-frequency decomposition. When the optimal wavelets act as matched filters they can greatly reduce broadband background noise in the decomposition process.

A novel approach for noise reduction is presented. Similar to Donoho, we employ thresholding in some wavelet transform domain but use a nondecimated and consequently redundant wavelet transform instead of the usual orthogonal one. Another difference is the shift invariance as opposed to the traditional orthogonal wavelet transform. We show that this new approach can be interpreted as a repeated application of Donoho's original method. The main feature is, however, a dramatically improved noise reduction compared to Donoho's approach, both in terms of the l₂ error and visually, for a large class of signals. This is shown by theoretical and experimental results, including synthetic aperture radar (SAR) images.

A novel idea of scale-translation filtering based on the orthonormal wavelet transform is developed and demonstrated.

The large variance of the Wigner-Ville distribution makes smoothing essential for producing readable estimates of the time-varying power spectrum of noise-corrupted signals. Since linear smoothing trades reduced variance for increased bias of the signal components, we explore two nonlinear estimation techniques based on soft thresholding in an orthonormal basis representation. Soft thresholding provides considerable variance reduction without greatly impairing the time-frequency resolution of the estimated spectrum.

A modified technique is presented for projecting a large class of nonlinear partial differential equations with respect to (x, t) onto a finite number of ordinary differential equations with respect to t. Improved description compared to standard finite-difference or Fourier spectral methods involves using an orthonormal basis of wavelet functions (psi) _(nu ,n)(x). Whereas Fourier projection represents the interaction between spatial scales throughout the x- domain, wavelet representation does the same locally. This technique is applied to solving the BDO-Burgers equation, extending previous results for the Burgers equation.

The electromagnetic measurement of range to a topographic surface from a satellite platform, such as that performed by the Earth Resources Satellite-1 (ERS-1) synthetic aperture radar (SAR), results in a geometrically distorted image which displays characteristics not readily identifiable with natural terrain features. This paper hypothesizes that SAR signals may be examined as a set of singularities which arise from a scale independent topography where the radar returns originate with variations in local terrain slope. The hypothesis is explored through the use of fractal geometry, SAR image simulations, ERS-1 SAR data, and analysis using the continuous wavelet transform. By using the language of fractal geometry, the topographic descriptors of slope and elevation are transformed into singularity strengths through numerical simulations modeling range foreshortening. Input data are from (1) a digital elevation model (DEM) and (2) fractally interpolated DEMs. Continuous wavelet transforms (CWT) are employed in a scale-space analysis of the SAR simulations to analyze and explain the relationships between singularity strength and the various models of topography. By comparing the results to an identical analysis of actual SAR data (ERS-1 SAR SGC), the CWT modulus maxima show that the overall singularity behavior of SAR image simulations derived from fractally interpolated DEMs more closely resembles actual SAR data than do simulations from the original DEM. At detail scales, measurements from the modulus maxima of SAR image simulations better approximate actual SAR data as the complexity of the DEM increases. At coarser scales, the generalizations depicted by the modulus maxima from the simulations correspond closely to the actual data.

Wavelet transform (WT) is a new and powerful tool for the analysis of nonhomogeneous or non stationary signals. It permits an evolutionary spectral decomposition of the signal, which was almost impossible to get with the traditional Fourier signal decomposition. In this study a WT is applied to analyze wind speed signals measured at a frequency of 21 Hz above the Amazon forest and pasture. Surface boundary layer measurements are analyzed. The results show the essentially nonhomogeneous behavior of the turbulent fields above those vegetation covers, which appear to be organized in extremely localized structures.

We describe a new method for the computation of a disparity map between two stereoscopic satellite images. The disparities are computed along the x and y axis respectively at each point of the image, without the use of the epipolarity condition. In order to compute the disparity field, first a set of ground control points is detected in both images. Next, a mapping of the disparities over the entire image is done using the Kriging method. Finally, the two stereoscopic images are geometrically registered using the disparity maps.

A discrete biorthogonal wavelet transform is used to extract water regions from radar imagery. A family of discrete scaling vectors and wavelets are formulated by direct construction and used to create a multiresolution image pyramid from an input radar image. The scaling vectors and wavelets each have eight nonzero elements. A second set of vectors is used to perform the inverse transform and obtain an image. Various sections of the pyramid are used to reconstruct images which resemble the original image except at reduced resolution. In this way a small image `cube' is built up. Feature vectors are formed from the image cube such that the dimension of each feature vector is equal to the number of images in the cube. These feature vectors serve as input data to two standard clustering routines. The clustering routines are the minimax and the K-means. These clustering routines are used to place each pixel of the original image into one of two classes: water or land. Finally a simple routine is performed to try and correct any obviously misclassified pixels. This technique was applied to synthetic aperture radar images. It was found that when the image cube was calculated from sections of the wavelet transform high up in the pyramid the shadow regions on land blended in with the rest of the terrain but large water bodies could still be extracted intact. The results of applying this procedure to two synthetic aperture radar images are shown.

In this paper we develop fast numerical algorithms for representations of periodic functions in biorthogonal multiresolution analyses in two dimensions. We show that in the Fourier domain the decomposition and reconstruction algorithms have a matrix representation in terms of permutations and block diagonal matrices. We present some illustrative examples using biorthogonal spline wavelets applied to meteorological satellite images (METEOSAT) compression.

This paper describes two approaches for accomplishing interactive feature analysis by overcomplete multiresolution representations. We show quantitatively that transform coefficients, modified by an adaptive non-linear operator, can make more obvious unseen or barely seen features of mammography without requiring additional radiation. Our results are compared with traditional image enhancement techniques by measuring the local contrast of known mammographic features. We design a filter bank representing a steerable dyadic wavelet transform that can be used for multiresolution analysis along arbitrary orientations. Digital mammograms are enhanced by orientation analysis performed by a steerable dyadic wavelet transform. Arbitrary regions of interest (ROI) are enhanced by Deslauriers-Dubuc interpolation representations on an interval. We demonstrate that our methods can provide radiologists with an interactive capability to support localized processing of selected (suspicion) areas (lesions). Features extracted from multiscale representations can provide an adaptive mechanism for accomplishing local contrast enhancement. By improving the visualization of breast pathology can improve changes of early detection while requiring less time to evaluate mammograms for most patients.

More than 50 million women over the age of 40 are currently at risk for breast cancer in the United States. Computer-aided diagnosis, used as a `second opinion' to radiologists, will aid in decreasing the number of false readings of mammograms. A novel feature extraction method is presented that provides increased classification power. Wavelets, previously only exploited for their segmentation benefits, are explored as features for classification. Daubechies₄, Daubechies₂₀, and biorthogonal wavelets are each investigated. Applied to 94 difficult-to- diagnose digitized microcalcification cases, performance is 74 percent correct classifications. Feature selection techniques are presented which further improve wavelet classification performance to 88 percent correct classification.

A wavelet-based technique WISP is used to discriminate normal brain activity from brain activity during epileptic seizures. The WISP technique is used to exploit the noted difference in frequency content during the normal brain state and the seizure brain state so that detection and localization decisions can be made. An AR-Pole statistic technique is used as a comparative measure to base-line the WISP performance.

We discuss the advantages and disadvantages of using a Karhunen-Loeve (K-L) expansion of a training set of images to reduce the number of encodes required for a magnetic resonance (MR) image of a new object. One form of this technique has been proposed and another implemented. We evaluate the error likely to be achieved as a function of the number of encodes and two technical problems: reduced SNR in the images and smoothing of the K-L functions in practice. As an alternative, we propose the use of joint best bases derived from the local trigonometric library as an approximation to the K-L basis. These bases approach the rate-distortion characteristic achieved by the K-L basis, but they are easier to use in MRI and can be applied with existing methods for fast acquisition.

In this paper we present a method for reconstructing a function f:R² yields R from limited angle tomographic data. This reconstruction problem occurs in may physical systems, where physical limitations prohibit the gathering of tomographic data at certain angles. We begin by reviewing some of the classical work on singular value decompositions and POCS in the context of this problem. We then review some of the classical work by G. Szego and others on finite Toeplitz operators. We consider the implications of this work toward a classical inversion of the problem. We introduce a new inversion technique which utilizes multiresolution analysis, induced correlations caused by non-linear constraints, and non-linear filtering to mollify the reconstruction process. We show that the uncertainty principles generated in recent works of Donoho, Stark et al. guarantee the invertibility of this alternative inversion technique. We also utilize the noise reduction techniques of Donoho and Johnstone to reduce the effects of noise on the process.

We propose a new experimental data acquisition method which acquires multiple exposures through a windowed x-ray radiation profile. The tapered profile is localized at the target region with a reduced radiation dosage but must be repeated at a few discrete shifts outside the target region to cover the whole slice. Such an approximated spatial window-convolution projection, a weighted-shifted radon transform, is ideally suited for the wavelet transform (WT) which is a bank of constant-Q matched filters creating a local space-scale joint representation. It is more efficient than the Fourier transform (FT) for any local signal analysis. It requires fewer basis terms and provides better signal-to-noise ratio enhancement than FT. Thus, the WT version of the central slice theorem (CST) yields an image reconstruction method. Because of the compact support and the design flexibility of the wavelet bases, this new approach has the potential of requiring fewer of the weighted x-ray projections, in which a lower radiation dosage is used, than standard tomography. Despite the sparser data set, a reconstructed image of similar quality to that created by standard FT tomography can be achieved. Moreover, the WT using the tapered radiation cuts can provide a multi-resolution feature extraction for use in artificial neural network-based automatic pattern classification.

Interictal `spikes' are abnormal transient features appearing in the electroencephalogram (EEG) and electrocortiocogram (ECoG) of epileptic patients. Although tentative characterizations of these transients have been offered in terms of their temporal sharpness (second derivative with respect to time), duration, and amplitude, the implementations of detectors for these transients have only achieved limited success because of the amounts of false positive detection's that occur. It has been suggested that the measurement of temporal sharpness at varied temporal resolution scales can provide a comprehensive measurement of the typical temporal pattern observed by these phenomena. In this study we apply a wavelet transformation to the digitized ECoG of epileptic patients towards that goal. A mother wavelet provides the basic measure of sharpness, so that its replicas at D progressively increasing dilations will capture the components of sharpness at progressively decreasing temporal resolutions. Then, patterns are sought in the D-dimensional space that indicate a simultaneous increase in many levels of sharpness. When such patterns occur the detection of target feature is suspected. Typically, true interictal spikes display temporal sharpness in a wide range of temporal resolution scales, while other transients (which may be erroneously detected by other detection approaches) display sharpness at the high temporal resolution level only.

The presence of a high level of noise is a characteristic in some tomographic imaging techniques such as positron emission tomography (PET). Wavelet methods can smooth out noise while preserving significant features of images. Mallat et al. proposed a wavelet based denoising scheme exploiting wavelet modulus maxima, but the scheme is sensitive to noise. In this study, we explore the properties of wavelet phase, with a focus on reconstruction of emission tomography images. Specifically, we show that the wavelet phase of regular Poisson noise under a Haar-type wavelet transform converges in distribution to a random variable uniformly distributed on (0, 2(pi) ). We then propose three wavelet-phase-based denoising schemes which exploit this property: edge tracking, local phase variance thresholding, and scale phase variation thresholding. Some numerical results are also presented. The numerical experiments indicate that wavelet phase techniques show promise for wavelet based denoising methods.

Wavelet analysis is a recently developed mathematical theory and computational method for decomposing a nonstationary signal into components that have good localization properties both in time and frequency domains and hierarchical structures. Wavelet transform provides local information and multiresolution decomposition on a signal that cannot be obtained using traditional methods such as Fourier transforms and distribution-based statistical methods. Hence the change in complex biological signals can be detected. We use wavelet analysis as an innovative method for identifying and characterizing multiscale electroencephalogram signals in this paper. We develop a wavelet-based stationary phase transition method to extract instantaneous frequencies of the signal that vary in time. The results under different clinical situations show that the brian triggers small bursts of either low or high frequency immediately prior to changing on the global scale to that behavior. This information could be used as a diagnostic for detecting the onset of an epileptic seizure.

In this paper the qualitative properties of an inviscid, incompressible two-dimensional fluid are examined by numerical methods based on the compactly supported wavelets (the wavelet- Galerkin method). In particular, we examine the behavior of the spatial gradients of the vorticity. The growth of these gradients is related to the transfer of enstrophy (integral of squared vorticity) to the small-scales of the fluid motion. Implicit time differencing and wavelet-Galerkin space discretization allow a direct investigation of the long time behavior of the inviscid fluid. The effects of hyperviscosity on the long time limit are examined. To solve boundary problems we developed a new numerical method for the solution of partial differential equations in nonseparable domains. The method uses a wavelet-Galerkin solver with a nontrivial adaptation of the standard capacitance matrix method. The numerical solutions exhibit spectral convergence with regard to the order of the compactly supported, Daubechies wavelet basis. Furthermore, the rate of convergence is found to be independent of the geometry. We solve the Helmholtz equation since, for the indefinite case, the solutions have qualitative properties that well illustrate the applications of our method.

In this work, we describe how wavelets may be used for the temporal discretization of ODEs and PDEs. A major problem associated with the use of wavelets in time is that initial conditions are difficult to impose. A second problem is that a wavelet-based time integration scheme should be stable. We address both of these problems. Firstly, we describe a general method of imposing initial conditions, which follows on from some of our recent work on initial and boundary value problems. Secondly, we use wavelets of the Daubechies family as a starting point for the development of stable time integration schemes. By combining these two ideas we are able to develop schemes with a high order of accuracy. More specifically, the global error is O(h^p-1), where p is the number of vanishing moments of the original wavelet. Furthermore, these time integration schemes are characterized by large regions of absolute stability, comparable to increasingly high order BDF methods. In particular, Daubechies D4 and D6 wavelets give rise to A-stable time-stepping schemes. In the present work we deal with single scale formulations. We note, however, that the standard multiresolution analysis for orthogonal wavelets on L²(R) applies here. This opens up interesting possibilities for treating BVPs and IVPs at multiple scales.

In this research, we introduce vector-valued multiresolution analysis and vector-valued wavelets for vector-valued signal spaces. We construct vector-valued wavelets by using paraunitary vector filter bank theory. In particular, we construct vector-valued Meyer wavelets that are band-limited. We classify and construct vector-valued wavelets with sampling property. As an application of vector-valued wavelets, multiwavelets can be constructed from vector-valued wavelets. We show that certain linear combinations of known scalar-valued wavelets may yield multiwavelets. We then present discrete vector wavelet transforms for discrete-time vector-valued (or blocked) signals, which can be thought of as a family of unitary vector transforms. In applications of vector wavelet transforms in two dimensional transform theory, the nonseparability can be easily handled.

We describe the general finite field wavelet transform, which operates on vectors of length p^k for p a prime, k >= 0. This transform allows one to choose not only the analyzing wavelet, but the wavelet `machine' by which a tight frame is produced from the analyzing wavelet. We discuss an implementation of the transform, which uses an algorithm that reveals the multiresolution character of the transform. Finally, its applications to denoising and cryptography are discussed.

For many applications, general frame wavelets are more attractive than orthonormal wavelets owing to the extent to which the former may be tailored to suit the application. The mother wavelet may be chosen to cause, during wavelet transform, congealing of selected features into localized regions in the wavelet domain or separation of chosen features from one another during transform. Thus choice of mother wavelet impacts feature dimensionality as well as separability among features. Owing to the special importance of frames, there is a need to identify and characterize new frame wavelets in order to make available a rich variety of mothers which run the gamut of the diverse applications. This paper discusses three classes of frame wavelet mothers, to include new mothers as well as a few familiar ones, approached from the viewpoints of residues, sigmoid, and Hermitivity.

We present a fast and parametric algorithm for discrete Gabor expansions (DGE). The fast algorithm allows us to obtain any different pseudo-dual window within a fraction of a second based on the (usual) dual window. We discuss important roles of pseudo-duals in general, and in particular for DGEs. An example of using pseudo-duals to achieve the dimension invariance for DGEs is presented. A new multi-Gabor expansion scheme is also defined and characterized. Various other issues on DGEs are also discussed.

In this paper we introduce an adaptive collocation scheme for the solution of time independent partial differential equations. Such a technique is based on the interpolating multiresolution analysis. The method defines a sequence of grids, each one computed by looking at the interpolating wavelet transform of the solution of the problem on the previous grid. The computational cost of the solution of each problem is kept low by either applying a Schur complement technique or by coupling the refining procedure with an iterative solver. The method is tested on some two dimensional test problems, both of linear and non linear type.

Because multiresolution analyses and wavelet bases are generated by translating and dilating scaling and wavelet functions, these functions must satisfy some special equations involving Toeplitz and Toeplitz-like operations. These relations can be exploited when such functions are to be constructed or their properties are studied. In the case of multiwavelets, where more than one scaling functions generate the multiresolution analysis, the corresponding operators are block Toeplitz-like. We discuss here some basic properties of these operators and compare them to the classical case of one scaling function. Using these observations we study the convergence of the cascade algorithm and derive sufficient conditions for existence of solution to a two-scale dilation equation.

In this paper, continuous-time wavelet transform (CTFT) is applied to analyze non- bandlimited continuous-time signals. Orthogonal CTWT based on Daubechies finite-support wavelets is used for signal analysis and reconstruction. A procedure to compute CTWT coefficients for a given scale and translation is described. Both noise-free and noisy signals are used in the specific examples that we consider here.

Let g be a Gabor window of length N, (a,b) be a pair of lattice constants. The time and frequency translates {M_mbT_nag} of g forms so-called Gabor family (or Weyl-Heisenberg wavelet system). In this note, we present the structural properties of the discrete Gabor transforms, and determine the best approximation of a signal (chi) (epsilon) $CBAR^N in a very general case by linear combinations from a given Gabor family (we do not assume here whether this family forms a frame or not). For this task, we are determining the (generalized) dual Gabor atom. We propose the conjugate-gradient (CG)- method with O(N) complexity for fixed lattice constants (a,b) to solve the problem.

This paper is concerned with the development of an equivalence relation between two multiresolution analyses of L²(R). The relation called unitary equivalence is created by the action of a unitary operator in a way that the multiresolution structure and the decomposition and reconstruction algorithms remain invariant. A characterization in terms of the scaling functions of the multiresolution analyses is given. Distinct equivalence classes of multiresolution analyses are derived. Finally, we prove that B-splines give rise to non- equivalent examples.

Based on the fact that fuzzy systems can be represented as a linear combination of fuzzy basis functions (FBF), it is proposed that spline wavelets scaling functions be used as FBF. A fuzzy system built from dilations and translations of a wavelet scaling function can universally approximate any function in a Hilbert space. Such fuzzy systems would also enjoy the excellent numerical and computational characteristics of the fast wavelet transform, while retaining the approximation properties of spline functions.

The authors have constructed a wavelet processing board that implements a 14-level wavelet transform. The board uses a high-speed analog-to-digital (A/D) converter, a hardware queue, and five fixed-point digital signal processing (DSP) chips in a parallel pipeline architecture. All five processors are independently programmable. The board is designed as a general purpose engine for instrumentation applications requiring near real-time wavelet processing or multiscale filtering. The present application is the processing engine of a magnetic field monitor that covers 305 Hz through 5 MHz. The monitor is used for the detection of peak values of magnetic fields in nuclear power plants. This paper describes the design, development, simulation, and testing of the system. Specific issues include the conditioning of real-world signals for wavelet processing, practical trade-offs between queue length and filter length, selection of filter coefficients, simulation of a 14-octave filter bank, and limitations imposed by a fixed-point processor. Test results from the completed wavelet board are included.

We are presenting a fast Fourier transform (FFT) based architecture for computing discrete wavelet transform (DWT) of a one dimensional sequence. The scheme is non recursive unlike a dyadic decomposition and DWT coefficients at all resolutions can be generated simultaneously without waiting for generation of coefficients at a lower octave level. This architecture is faster than time domain convolution based architectures proposed for DWT decomposition and can be fully pipelined. Complexity of control circuits for this architecture is much lower as compared to time domain convolution based architectures. Also problems associated with maintaining streams of intermediate results is eliminated or very much reduced. The speed advantage comes by using FFT based frequency domain convolutions and is more pronounced for long filters. But this comes at the price of increased hardware requirements.

This paper focuses on the experimental combination of two topical technologies: (1) the joint transform correlator (JTC) and (2) the wavelet transform (WT). The result is an optical wavelet processor utilizing liquid crystal televisions (LCTV) as phase-only spatial light modulators (SLM) in a JTC architecture. The mathematical formalism of the optical wavelet processor is presented followed by simulations involving amplitude- and phase-only SLM operation.

An important consideration in the design of VLSI architectures is that of word size and the consequent data precision it confers. The word size affects the dimension of the final circuit as well as the machine's performance, power consumption, I/O requirement, production costs, and so on; while issues concerning the precision relate to how the data words are interpreted -- as integers, fixed-point numbers, standard or custom floating point numbers. In existing architectural designs for the discrete wavelet transform (DWT), little consideration has been given to word size and precision. In this paper we address this problem, showing how the word size expectations can be calculated for a specific problem (based on the range of the input data and the wavelet used). As the depth of the DWT filtering increases the data-word requirements increase, so it is important to investigate how this can affect the potential of the resulting hardware. Although the theory of wavelets is based on perfect reconstruction filters, in a digital implementation there are rounding errors associated with the word size selected and so this must be analyzed to be sure the signal processing benefits of the DWT are not compromised. We present example cases for this analysis, showing the one-dimensional analysis of speech data and two-dimensional analysis of images. Our studies show the tradeoffs involved with respect to precision and perfect reconstruction, and we show simulation results using fixed-point arithmetic with varying word size and precision.

Traditionally, multiresolution analysis is used for the construction of scaling functions and wavelets on the entire line, or, in the higher dimensional case, the entire Euclidean space. In this paper we present a construction of multiresolution analysis on nonsmooth sets in several dimensions and briefly discuss some applications.

Mathematical morphology, as originally described by Matheron and Serra, consists of the application of set theoretical operations between the image set X and a structure element set B. Image skeletons are very efficient representations of shape, and can be directly derived using the morphological operations of erode and open. If done at full image resolution, derivation of a skeleton can be very time consuming without dedicated signal processing hardware. This paper presents an alternative to the standard approach that relies on morphological operations within the wavelet coefficient space. In particular, the skeleton transformation can be done very efficiently at the reduced resolution of the coarse wavelet coefficient levels. We investigate the relationship between wavelet image compression and morphological transforms for the derivation of skeletons. We also report the results of some experimental studies on binary and gray scale images.

This paper looks at a novel technique for designing wavelets. Based only on orthonormality, the technique exploits the degrees of freedom in discrete wavelets when the Strang accuracy conditions are abandoned. For an N order matrix, orthonormality guarantees perfect reconstruction when compression is not used, and N/2 equations remain for specifying the new wavelets. Classically the moment conditions provide polynomial accuracy to degree N/2 - 1. By not using the Strang accuracy conditions, reconstruction degrades more quickly for smooth images, but more slowly for irregular images. One dimensional signals are considered, as well as black and white images using a gray scale range of 256 on 128 X 128 pixels. For this discussion, 2 dimensional images are treated by 1 dimensional slicing. The transform process is that of Mallat forward and backward transformations with compression affected in the usual way by coefficient chopping. The novel feature is the departure from Strang's accuracy conditions.

We present a local-feature-oriented image indexing and retrieval method based on Kwong and Tang's W-transform. Multiresolution histogram comparison is an effective method for content- based image indexing and retrieval. However, most recent approaches perform multiresolution analysis for whole images but do not exploit the local features present in the images. Since W- transform is featured by its ability to handle images of arbitrary size, with no periodicity assumptions, it provides a natural tool for analyzing local image features and building indexing systems based on such features. In our approach, the histograms of the local features of images are used in the indexing system. The system not only can retrieve images that are similar or identical to the query images but also can retrieve images that contain features specified in the query images, even if the retrieved images as a whole might be very different from the query images. The local-feature-oriented method also provides a speed advantage over the global multiresolution histogram comparison method. The feature-oriented approach is expected to be applicable in managing large-scale image systems such as video databases and medical image databases.

A system is presented for compression of hyperspectral imagery. Specifically, DPCM is used for spectral decorrelation, while a 2-D discrete wavelet coding scheme is used for spatial decorrelation. Trellis coded quantization is used to encode the wavelet coefficients. Side information and rate allocation strategies are discussed. Entropy-constrained codebooks are designed using a modified version of the generalized Lloyd algorithm. This entropy constrained system achieves a compression ratio of greater than 70:1 with an average PSNR of the coded hyperspectral sequence approaching 41 dB.

In this research, we propose a video compression scheme that uses the boundary-control vectors to represent the motion field and the embedded zerotree wavelet (EZW) to compress the displacement frame difference. When compared to the DCT-based MPEG, the proposed new scheme achieves a better compression performance in terms of the MSE (mean square error) value and visual perception for the same given bit rate.

Image coding is one of the most visible applications of wavelets. There has been an increasing number of reports each year since the late 1980s on the design of new wavelet coders and variations to existing ones. In this paper, we report some results from our comparative study of wavelet image coders using a perception-based, quantitative picture quality scale as the distortion measure. Coders are evaluated in rate-distortion sense; the influences of different wavelets, quantizers, and encoders are assessed individually. Our results provide an insight into the design issues of optimizing wavelet coders, as well as a good reference for application developers to choose from an increasingly large family of wavelet coders for their applications.

We propose a novel method for simultaneous speckle reduction and data compression based on shrinking, quantizing and coding the wavelet packet coefficients of the logarithmically transformed image. A fast algorithm is used to find the best wavelet packet basis in the rate- distortion sense from the entire library of admissible wavelet packet bases. Soft-thresholding in wavelet domain can significantly suppress the speckles of the synthetic aperture radar (SAR) images while maintaining bright reflections for subsequent detection and recognition. Optimal bit allocation, quantization and entropy coding achieve the goal of compression while maintaining the fidelity of the SAR image.

We present in this paper very simple compression experiments using wavelets of computer address traces which are heavily used in computer architecture simulation. Trace driven simulation is facing a major obstacle with the sizes of traces used during simulation. These sizes are requiring an increasing amount of storage to be stored and an increasing amount of simulation time to be consumed. We report in this paper some preliminary experiments of using the wavelet theory as a tool to reduce this size. Because of the well established locality nature of computation, a program profile exhibits patterns that we expected to meet the specifications of wavelet theory to be applied with interest. We conducted several experiments on a standard suite of benchmarks used for trace-driven simulations using several filters and thresholding policies. Apart from excessive computation time requirements, this approach suffers from unstable behavior with performance varying from poor to excellent.

Wavelet transforms applied to multiresolution analyses of images produce outputs similar in theory to those of matched filters. In order to maximize the output at the location and scale of a signal of interest, it is necessary for the wavelet used in the multiresolution analysis to `match' the signal of interest. Current techniques match a signal to one of several predefined wavelets in a library, which requires wavelets to be designed in advance. Here, we present an alternative by developing a technique for deriving the wavelet directly from the desired signal spectrum in such a way that the mean squared error between their spectra is a minimum. Furthermore, the matched wavelet is designed such that its associated scaling function generates an orthonormal multiresolution analysis. The technique includes an algorithm for finding the scaling function from an orthonormal wavelet, and algorithms for finding the optimal wavelet magnitude and phase from a given input signal. Several examples are shown to demonstrate the performance of the technique for both known orthonormal wavelets and arbitrary signals.

One of the generic image/signal processing problems is how to restore accurately the first order point singularity feature having the discontinuity of the first derivative that is furthermore masked with nosy clutter environment. This is known as the class of ill-condition inverse problem in a real world environment. Specifically, given the noisy sensor data, how to reconstruct the unknown singularity as the feature of the unknown source/object. We consider the rooftop singularity for the noisy sensor inverse problem, and demonstrate that the multiresolution paradigm-wavelet transform is useful to restore the source/object singularity.

A wavelet preprocessed amplitude-modulated phase-only filter has been implemented. It is evaluated in terms of various pattern recognition performance statistics. The effect of thresholding nonlinearity has also been studied.

This paper deals with the compression of electrocardiogram (ECG) signals using a large library of orthonormal bases functions that are translated and dilated versions of Daubechies wavelets. The wavelet transform has been implemented using quadrature mirror filters (QMF) employed in a sub-band coding scheme. Interesting transients and notable frequencies of the ECG are captured by appropriately scaled waveforms chosen in a parallel fashion from this collection of wavelets. Since there is a choice of orthonormal bases functions for the efficient transcription of the ECG, it is then possible to choose the best one by various criterion. We have imposed very stringent threshold conditions on the wavelet expansion coefficients, such as in maintaining a very large percentage of the energy of the current signal segment, and this has resulted in reconstructed waveforms with negligible distortion relative to the source signal. Even without the use of any specialized quantizers and encoders, the compression ratio numbers look encouraging, with preliminary results indicating compression ratios ranging from 40:1 to 15:1 at percentage rms distortions ranging from about 22% to 2.3%, respectively. Irrespective of the ECG lead chosen, or the signal deviations that may occur due to either noise or arrhythmias, only one wavelet family that correlates best with that particular portion of the signal, is chosen. The main reason for the compression is because the chosen mother wavelet and its variations match the shape of the ECG and are able to efficiently transcribe the source with few wavelet coefficients. The adaptive template matching architecture that carries out a parallel search of the transform domain is described, and preliminary simulation results are discussed. The adaptivity of the architecture comes from the fine tuning of the wavelet selection process that is based on localized constraints, such as shape of the signal and its energy.