A scaling function is the solution to a dilation equation Φ(t) = Σc_kΦ(2t-K), in which the coefficients come from a low-pass filter. The coefficients in the wavelet W(t) = Σd_kΦ(2t-k) come from a high-pass filter. When these coefficients are matrices, Φ and W are vectors: there are two or more scaling functions and an equal number of wavelets. Those 'multiwavelets' open new possibilities. They can be shorter, with more vanishing moments, than single wavelets. We determine the conditions to impose on the matrix coefficients c_k in the design of multiwavelets, and we construct a new pair of piecewise linear orthogonal wavelets with two vanishing moments.

It is commonly acknowledged that there is a tradeoff between schedule quality and schedule robustness. In general terms, schedules that are of high quality tend to be of low robustness, and schedules that are robust tend to be of low quality. To better manage the robustness/quality tradeoff, we have developed an algorithm that implements what we call 'Just-in-Case' scheduling; this algorithm explicitly considers the way in which scheduled actions might fail and how such failures can impact the executability of a schedule. 'Just-in-Case' scheduling is able to build schedules that are robust and of high quality. The 'Just- In-Case' algorithm is motivated in this paper by a specific telescope scheduling problem, and the paper presents the results of an experiment, carried out using real telescope scheduling data, that illustrates the performance improvement one can expect from using it.

A wavelet variation of the frequency decomposition multigrid method of Hackbusch is presented. The perfect reconstruction property of the wavelet system enable us to perform the convergence analysis of the frequency decomposition method. Some applications of this method are also presented.

A theory of wavelet packets is developed for nonlinear operators consisting of a composition, generalizing a sigmoidal operation, followed by convolutions with filter pairs H₀ and H₁. The pyramidal wavelet packet structure is defined by bit reversal trees. The reconstruction theorem, from which the original signal is obtained from frequency localized data at other nodes of the three, requires fixed point theory as well as conditions on H₀ and H₁ resembling those defining quadrature mirror filter pairs. Applications will be to biological systems and neural networks where such nonlinearities occur.

The application of multiscale and stochastic techniques to the solution of linear inverse problems is presented. This approach allows for the explicit and easy handling of a variety of difficulties commonly associated with problems of this type. Regularization is accomplished via the incorporation of prior information in the form of a multiscale stochastic model. We introduce the relative error covariance matrix (RECM) as a tool for quantitatively evaluating the manner in which data contributes to the structure of a reconstruction. In particular, the use of a scale space formulation is ideally suited to the fusion of data from several sensors with differing resolutions and spatial coverage (e.g. sparse or limited availability). Moreover, the RECM both provides us with an ideal tool for understanding and analyzing the process of multisensor fusion and allows us to define the space-varying optimal scale for reconstruction as a function of the nature (resolution, quality, and coverage) of the available data. Examples of our multiscale maximum a posteriori inversion algorithm are demonstrated using a two channel deconvolution problem.

In this paper we present 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 will outline methods which allow one to send optical pulses with arbitrary shapes through fiber optics.

In communication applications, particularly ones using spread spectrum (SS) techniques, transform domain processing can be utilized to suppress undesired interference and, consequently, improve system performance. Since traditional applications requiring transform domain processing perform excision only in the Fourier domain, one of the main objectives of this paper is to extend transform domain processing to include wavelets as the basis functions. Specifically, the use of wavelets in the excision of jamming signals from SS communications will be investigated. Simulations have been performed using several basis functions for an SS receiver with narrowband jamming. Results of this simulation are presented, including BER figures, and compared with conventional Fourier domain processing. Implementation of the exciser using multirate digital filtering filters is also discussed. An intercept receiver which employs wavelet transform domain excision is also described. The receiver detects DS-BPSK spread spectrum signals in the presence of narrow-band interference by employing adaptive interference rejection techniques. The improvement in the system performance over that of conventional radiometric detection only is shown by presenting numerical simulation results of probability of detection versus false alarm as the receiver-operating- characteristic (ROC) for an enhanced total power detector.

This paper examines the use of subband decompositions for channel equalization. Performance results are reported for maximally decimated filter banks. This approach is compared to a conventional fractionally spaced equalizer. Performance comparisons are done using a worst case voiceband telephone network channel with severe amplitude and delay distortion.

Recent work has suggested the possibility and potential gains of using homogeneous signals, which can be represented in terms of orthonormal wavelet basis sets, as the modulating waveforms in communication systems. We present performance analysis, in terms of both bandwidth efficiency and probability of error, for one communication system where a fractal modulated signal is transmitted over a sinusoidal carrier. In addition, we present some preliminary results from a fractal modulation simulator indicating that a practical implementation is quite feasible while maintaining near-theoretical performance for the test case of transmission over the additive white Gaussian noise (AWGN) channel. Preliminary results indicate that fractal modulation achieves a reasonable improvement over some existing modulation schemes.

This paper discusses several current attempts to use acoustic and electromagnetic wave propagation for modeling physical phenomena and the role that wavelet analysis is playing in these efforts. The first problem involves geophysical modeling of the ocean floor using acoustic waves, and wavelets have recently been shown to play an important role. The second problem involves modeling of SAR radar images in the context of automatic target recognition efforts. The third problem is global illumination in computer graphics, i.e., simulation of reflected and absorbed light for everyday environments. The role of wavelets is more embryonic in these latter two areas, but there are some common principles in all of these modeling efforts, and the methodology of wavelets seems well suited to certain aspects of these problems.

Principal orthogonal decomposition can be used to solve two related problems: distinguishing elements from a collection by making d measurements, and inverting a complicated map from a p- parameter configuration space to a d-dimensional measurement space. In the case where d is more than 1000 or so, the classical O(d³) singular value decomposition algorithm becomes very costly, but it can be replaced with an approximate best-basis method that has complexity O(d² log d). This can be used to compute an approximate Jacobian for a complicated map from R^p to R^d in the case where p is much less than d.

A form of nonlinear multiresolution analysis is described in which, by appropriate scaling and choice of analyzing wavelet, information concerning the structure of a signal or image is derived from local maxima and minima in the data transformed to position-scale, or wavelet, space. There results a data coding as a discrete sum of 'signal wavelets' that are generated adaptively to give a good local fit to the data, rather than being specified a priori as in standard applications of the wavelet transform. Although the wavelets generated in this manner are not in general mutually orthogonal, an orthogonal set of basis functions is derived subsequently in the form of linear combinations of appropriately weighted wavelets. A further novel aspect of the method is the introduction of a non-Gaussian statistical model which is fitted to the data and used to categorize individual data samples and to focus attention upon unusual events. Statistical distributions at different scales are related by means of fractal exponents. Applications are described to remote- sensing imagery, where the 'unusual events' are typically man- made artifacts in natural environmental backgrounds, and to a medically oriented signal in the form of heart rate, where the method can be used to draw attention to unusual patterns of heart beats.

Many papers have been published recently about the characterization of time-dependent processes through techniques using wavelet approach. Our work takes into account a particular class of time-dependent processes in nonlinear realm. We want to characterize chaotic dynamics from the standpoint of its unstable periodicities. For this aim we introduce a new technique able to stabilize such unstable orbits. We illustrate this technique both from the theoretical and the experimental standpoint. As a further step, we want to deal with the problem of detecting and removing noise from chaotic dynamics. In this paper, firstly, we show how our technique is able to distinguish with very high sensitivity between a purely chaotic dynamics and a chaotic dynamics with noise even though the noise percentage is very low (of the order of 1 percent only Secondly, we apply our technique to remove noise from this dynamics. Finally, we compare both from the theoretical and experimental standpoint our technique with the well known wavelet technique. This work is a part of 'Skynnet' international project supported by the Italian National Institute for Nuclear Physics (INFN) and partially devoted to the application of new chaotic techniques instantiated in neural architectures for compressing, storing and transmitting information to earth from satellites.

Non-Fourier electrodynamics of plasmalike and waveguidelike media are advanced by means of new exact analytical solutions of Maxwell equations without traditional Fourier transformations. These solutions describe the non-sine fields in strongly dispersive media and are distinguished from usual traveling sine waves by unequal periods of oscillations of electric and magnetic components and enhanced spreading of magnetic components as compared with electric ones. Using orthogonal Hermitian pulses for description of non-sine waveforms incidenting on such a dispersive interface we consider the shape-dependent reflection of these waveforms and controlled shaping of ultrashort pulses.

A new discrete Gabor function provides subpixel resolution of phase while overcoming many of the computational burdens of current approaches to Gabor function implementation. Applications include hyperacuity measurement of binocular disparity and optic flow for stereo vision. Convolution is avoided by exploiting band-pass to subsample the image plane. A general purpose front end processor for robot vision, based on a wavelet interpretation of this discrete Gabor function, can be constructed by tessellating and pyramiding the elementary filter. Computational efficiency opens the door to real-time implementation which mimics many properties of the simple and complex cells in the visual cortex.

This paper presents a binocular disparity computation method by using Gabor wavelet pyramid to recover depth structure of the scene. In our previous work, differential Gabor filters were used to extract image flow from two image frames. With Newton-Raphson method, the differential Gabor approach can be extended to compute binocular disparity and, hence, to recover the depth structure of the scene. To illustrate the effectiveness of the Gabor wavelet pyramid for recovering depth structure, we synthesized a pair of stereo random-dot images with three centered pyramidal planes. Experimental result shows that the performance of the Gabor wavelet pyramid approach for depth recovery is good and the convergence rate is very fast. An architecture of the differential Gabor wavelet pyramid for binocular disparity computation is proposed. The computational structure of the pyramid is very simple and it is easy to implement with parallel processor.

In shape from shading, iterative algorithms are often used to compute the surface derivatives. These algorithms are, however, very time-consuming when the iterations are performed on the original image. In this paper we propose a multiresolution method which makes use of the orthonormal wavelet transform to construct a multiresolution pyramid and let most iterations be performed on the low resolution images to give good predictions of the initial values of the surface derivatives of higher resolution images and thus save many computations. On the other hand, the nonlinearity of imaging makes the direct reduction of the image resolution not an optimal way of utilizing the multiresolution method. Instead, we construct the pyramid of the norm T= √(p²+q²) of the surface direction (p, q, 1) . We prove that this strategy gives out excellent results when the surface is smooth and the support length of the wavelet is small. Factors that may affect the selection of the wavelet in the multiresolution shape from shading are also studied. Experiments show the superiority of this strategy to other methods.

The FBI has developed a specification for the compression of gray-scale fingerprint images to support paperless identification services within the criminal justice community. The algorithm is based on a scalar quantization of a discrete wavelet transform decomposition of the images, followed by zero run encoding and Huffman encoding.

We discuss bases formed by smooth local trigonometric functions and their applications to image compression. It is known that these bases can reduce the blocking effect that occurs in JPEG. We present and compare two generalizations of the original construction of Coifman and Meyer: biorthogonal and equal parity folding. They have the advantage that constant and linear components can be represented efficiently. We show how they reduce the blocking effect and improve the mean square error.

We study the problem of choosing the optical wavelet basis with compact support for signal representation and provide a general algorithm for computing the optimal wavelet basis. We first briefly review the multiresolution property of wavelet decomposition and the conditions for generating a basis of compactly supported discrete wavelets in terms of properties of quadrature mirror filter (QMF) banks. We then parametrize the mother wavelet and scaling function through a set of real coefficients. We further introduce the concept of information measure as a distance measure between the signal and its projection onto the subspace spanned by the wavelet basis in which the signal is to be reconstructed. The optimal basis for a given signal is obtained through minimizing this information measure. We have obtained explicitly the sensitivity of dilations and shifts of the mother wavelet with respect to the coefficient set. A systematic approach is developed here to derive the information gradient with respect to the parameter set for a given square integrable signal and the optimal wavelet basis. A gradient-based optimization algorithm is developed in this paper for computing the optimal wavelet basis.

This paper describes real-time implementation of a novel wavelet- based audio compression method. This method is based on the discrete wavelet (DWT) representation of signals. A bit allocation procedure is used to allocate bits to the transform coefficients in an adaptive fashion. The bit allocation procedure has been designed to take advantage of the masking effect in human hearing. The procedure minimizes the number of bits required to represent each frame of audio signals at a fixed distortion level. The real-time implementation provides almost transparent compression of monophonic CD quality audio signals (samples at 44.1 KHz and quantized using 16 bits/sample) at bit rates of 64-78 Kbits/sec. Our implementation uses two ASPI Elf boards, each of which is built around a TI TMS230C31 DSP chip. The time required for encoding of a mono CD signal is about 92 percent of real time and that for decoding about 61 percent.

We describe an algorithm to estimate a discrete signal from its noisy observation, using a library of orthonormal bases (consisting of various wavelets, wavelet packets, and local trigonometric bases) and the information-theoretic criterion called minimum description length (MDL). The key to effective random noise suppression is that the signal component in the data may be represented efficiently by one or more of the bases in the library, whereas the noise component cannot be represented efficiently by any basis in the library. The MDL criterion gives the best compromise between the fidelity of the estimation result to the data (noise suppression) and the efficiency of the representation of the estimated signal (signal compression): it selects the 'best' basis and the 'best' number of terms to be retained out of various bases in the library in an objective manner. Because of the use of the MDL criterion, our algorithm is free from any parameter setting or subjective judgments. This method has been applied usefully to various geophysical datasets containing many transient features.

The error in the reconstructed data of a wavelet decomposition by using a finite number of taps in quadrature mirror filter (QMF) and the computational costs are analyzed in time (or space) domain in this paper. In order to avoid the reconstruction error based on the error analysis and the number of taps in QMF being set to three, two QMFs for wavelet decomposition and reconstruction are obtained. The derived mother wavelet is based on a modified Haar function. A pair of fast and parallel 2D digital wavelet multiresolution decomposition and reconstruction algorithms are presented in this paper. The computational costs and some characteristics of the algorithms are also studied.

The wavelet decomposition algorithm can be used to break down a signal into many components before processing. Single processor- based wavelet algorithms have been used in signal and image analysis with great success. However, serial algorithms are inadequate to meet the demand for speed of processing in many real-time applications. An alternative to this problem is to parallelize computing steps in the wavelet computation to meet the real time computing requirements. We have constructed parallel algorithms for wavelet decomposition and reconstruction, and have implemented them in a MasPar parallel computer. Preliminary results indicate a two-order increase in processing speed is achieved.

We introduce a new method for fingerprint image recognition that combines Fourier transform and wavelet transform. We use Fourier transform to produce fingerprint image spectrum, and use wavelet transform as a processor. After Fourier transform and wavelet transform, the postprocessing and fingerprint image recognition become simple and easy. We also propose a 3f multiple optical correlator with a bank of wavelet filters to implement wavelet transforms in parallel. The wavelet filters are obtained by computer generated hologram. Experimental results show that the introduced method is efficient in complicated fingerprint image recognitions.

An algorithm for real-time implementation of affine frames generated by 'oversampling' is introduced. This notion of oversampled frames differs from the more well-known concept of over-sampling in that no extra data information is required in the implementation of our oversampled frame algorithm. The advantages of this fast-frame algorithm over the standard wavelet decomposition algorithm in signal processing include its ability to reduce the destructive effects due to aliasing and the presence of uncorrelated noise, as well as to compensate for the lack of shift-invariance in discrete wavelet decomposition.

In this paper, we briefly discuss two approaches for enhancing the resolution enhancement of range-Doppler images (including SAR and ISAR images). The first approach can be used in conjunction with current radar systems that use a fixed narrow band waveform to acquire target data. It measures Doppler shifts more accurately than the traditional FFT based techniques. It performs well in low signal-to-clutter regimes regardless of the statistical structure of the clutter. The second technique assumes that the radar can transmit different waveforms that are matched to the imaging task under consideration. It is based on the fact that the most accurate reconstruction of a range-Doppler target density function from N waveforms and their echoes is obtained by transmitting the singular functions corresponding to the N largest singular values of two kernels derived from the target density. We discuss two strategies for selecting the radar waveforms. The first strategy uses fixed waveforms that act as approximate singular functions for the kernels corresponding to wide classes of target densities. The second strategy adaptively selects the transmitted waveforms by solving a simultaneous target classification and image reconstruction problem.

In this paper we investigate the problem of fast and accurate classification of high range resolution radar returns. In addition we investigate the problem of efficient organization of large databases of pulsed high resolution radar returns from naval vessels in order to economize memory requirements and minimize search time. We use both synthetic radar returns from ships as well as real ISAR returns as the experimental data. We develop a novel algorithm for hierarchically organizing the database, which utilizes a multiresolution wavelet representation working in synergy with a Tree Structured Vector Quantizer (TSVQ), utilized in its clustering mode. The tree structure is induced by the multiresolution decomposition of the pulses. The TSVQ design algorithm is of the 'greedy' type. We demonstrate that our algorithm automatically computes the aspect graph (i.e. the simultaneous representation of compressed pulses as functions of aspect and elevation) for single target or for a group of targets. We also develop a novel optimization framework for the simultaneous design of the wavelet basis, the Tree-Structured Vector Quantizer and the Classification rule. We show that an efficient implementation consists of an adaptive Wavelet Transform - Tree Structured Vector Quantization with Learning. We show experimental results on the performance of the algorithm as measured by: (a) memory requirements; (b) search time; (c) scaling with respect to size; (d) accuracy in recovery. We also show experimental results with respect to variations in the mother wavelet and the design of the tree, as well as their impact on the performance of the algorithm. The results indicate that the combined algorithm results in orders of magnitude faster data search time, with negligible performance degradation (as measured by rate-distortion curves).

Discrete nonorthogonal wavelet transforms play an important role in signal processing by offering finer resolution in time and scale than their orthogonal counterparts. The standard inversion procedure for such transforms is a finite expansion in terms of the analyzing wavelet. While this approximation works quite well for many signals, it fails to achieve good accuracy or requires an excessive number of scales for others. This paper proposes several algorithms which provide more adequate inversion and compares them in the case of Morlet wavelets. In the process, both practical and theoretical issues for the inversion of nonorthogonal wavelet transforms are discussed.

In a series of papers, Donoho and Johnstone develop a powerful theory based on wavelets for extracting non-smooth signals from noisy data. Several nonlinear smoothing algorithms are presented which provide high performance for removing Gaussian noise from a wide range of spatially inhomogeneous signals. However, like other methods based on the linear wavelet transform, these algorithms are very sensitive to certain types of non-Gaussian noise, such as outliers. In this paper, we develop outlier resistant wavelet transforms. In these transforms, outliers and outlier patches are localized to just a few scales. By using the outlier resistant wavelet transform, we improve upon the Donoho and Johnstone nonlinear signal extraction methods. The outlier resistant wavelet algorithms are included with the 'S+WAVELETS' object-oriented toolkit for wavelet analysis.

This paper reviews the relation between the radar ambiguity function and the matched filter, then introduces a geometric concept of the matched filter as a slice of the ambiguity function along a frequency-shift line. With the help of the geometric concept, the conventional matched filter can be generalized to a time-varying filter, which is equivalent to the optimum time-frequency correlator and can be implemented with wavelet transforms. The application of the optimum wavelet correlator to inverse synthetic aperture radar (ISAR) will be illustrated.

The wavelet transform is applied to signal processing of synthetic aperture radar, and techniques for determining the range, cross-range, and rotation of the target are studied. The wavelet transform was also implemented optically in the laboratory for real-time processing of the radar data.

We have measured the time-resolved photoconductive response of a strained InGaAs/InGaAsP/InP multiple quantum well laser structure as a function of temperature and bias. It is found that the hole escape is dominated by tunneling at reverse biases of greater than -0.5 V at all temperatures. Under forward bias, recombination is dominant at temperatures below approximately equals 90 K while thermal escape processes prevail at higher temperatures. From Arrhenius plots of the hole escape time, the activation energy from the ground level has been determined as a function of bias and is in good agreement with a valence band offset of 75 percent of the total band offset. The intercepts of the plots yielded a scattering parameter of 6 ps. The carrier dynamics within the well were simulated using a simple model of thermionic emission and gave good qualitative agreement. The calculations indicate that the structures have the potential for extremely fast detection.

Mathematical formalism and computational considerations have led to the development of nonstationary filtering concepts that use coherent frames to exploit waveform frequency content that is localized in time. This paper deals with nonstationary filtering concepts that utilize phase space information provided by the Weyl-Heisenberg and wavelet coherent frames. In both cases, the frame formulations possess optimum simultaneous localization in phase space that follows from the use of Gaussian based frame functions. A filtering procedure is presented that first formulates a noise-free waveform signature template in phase space, and then uses this template for nonstationary filtering. The nonstationary filtering operation can be applied either before or after classical matched filtering to obtain improved peak signal-to-root mean square (rms) noise ratio (SNR) performance for enhanced detection or waveform feature extraction. The advantage arises from exploiting the time varying spectrum of the waveform. The manner in which the SNR improvement comes about is examined so that it can be properly interpreted in the context of candidate applications. The procedure is described and the performance is demonstrated using both the Weyl- Heisenberg and wavelet coherent frames applied to examples of linear FM and Barker coded waveforms. The nonstationary filter performance used with the matched filter is compared to classical stationary matched filter performance for the case of additive white Gaussian noise.

To compute the optimal expansion of signals in redundant dictionary of waveforms is an NP complete problem. We introduce a greedy algorithm, called matching pursuit, that performs a suboptimal expansion. The waveforms are chosen iteratively in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions, a matching pursuit defines an adaptive time-frequency transform. We derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit is a chaotic map, whose attractor defines a generic noise with respect to the dictionary. We derive an algorithm that isolates the coherent structures of a signal and an application to pattern extraction from noisy signals is described.

When data records contain energy at periods longer then the record standard Fourier techniques do not yield accurate spectra because of bias errors. The wavelet transform can, with proper choice of analyzing wavelet, accurately examine some of these biased signals. Each wavelet generates its own cone of influence from the endpoints that contain inaccurate wavelet coefficients. Additionally, wavelet analysis performs a scale projection from Fourier wavelength to wavelet scale that depends upon the analyzing wavelet. Taking these two effects into account, a single parameter is found that describes how well a particular wavelet can reveal a stationary red signal.

The Backus-Gilbert (BG) method provides an algorithm for solving the moment problem. Its performance can be greatly improved by incorporating an appropriate signal model, e.g., the bandlimited signals. In this research, we introduce a practical signal model called the scale-time limited signal spaces and generalize the Backus-Gilbert (BG) method for this class of signals. Since the model proposed in this work includes general wavelet bases such as the modulated Gaussians and the Mexican hat which have simple analytic forms, the required computation can be reduced.

Due to the increasing amount and diversity of remotely sensed data, image registration is becoming one of the most important issues in remote sensing. In the near future, remote sensing systems will provide large amounts of data representing multiple- time or simultaneous observations of the same features by different sensors. The combination of data from coarse-resolution satellite sensors designed for large-area survey and from finer- resolution sensors for more detailed studies will allow better analysis of each type of data as well as validation of global low-resolution data analysis by the use of local high-resolution data analysis. This integration of information from multiple sources starts with the registration of the data. The most common approach to image registration is to choose, in both input image and reference image, some well-defined ground control points (GCPs), and then to compute the parameters of a deformation model. The main difficulty lies in the choice of the GCPs. In our work, a parallel implementation of decomposition and reconstruction by wavelet transforms has been developed on a single-instruction multiple-data (SIMD) massively parallel computer, the MasPar MP-1. Utilizing this framework, we show how maxima of wavelet coefficients, which can be used for finding ground control points of similar resolution remotely sensed data, can also form the basis of the registration of very different resolution data, such as data from the NOAA Advanced Very High Resolution Radiometer (AVHRR) and from the Landsat/Thematic Mapper (TM).

Managing massive databases of scientific images requires new techniques that address indexing visual content, providing adequate browse capabilities, and facilitating querying by image content. Subband decomposition of image data using wavelet filters is offered as an aid to solving each of these problems. It is fundamental to a vidual indexing scheme that constructs a pruned tree of significant subbands as a first level of index. Significance is determined by feature vectors including Markov random field statistics, in addition to more common measures of energy and entropy. Features are retained at the nodes of the pruned subband tree as a second level of index. Query images, indexed in the same manner as database images, are compared as closely as desired to database indexes. Browse images for matching images are transmitted to the user in the form of subband coefficients, which constitute the third level of index. These coefficients, chosen for their unique significance to the indexed image, are likely to contain valuable information for the subject area specialist. This paper presents the indexing scheme in detail, and reports some preliminary results of selecting subbands for reconstruction as browse images based on their significance for indexing purposes.

Global temperature has risen significantly during the past century, based on measurements at meteorological stations. The two important factors in climate forcings are anthropogenic activities and changes of incident solar irradiance. The 14 years of Nimbus-7 Earth radiation budget measurements, which started in November 1978 and continued through January 1993, provide an important long-term record of solar irradiance, absorbed solar energy, and outgoing long wave and net radiation. Wavelet transforms are powerful techniques to decompose time series in time and frequency domains, and to isolate relevant characteristics. Transforms based on Morlet wavelets and Mexican Hat wavelets are used to examine the periodicities in the 14-year Nimbus-7 measurements and the 9-year Solar Maximum Mission (SMM) measurements. Short-term variations with periods ranging from a few days to 30 or 40 days are identified. The importance of selecting wavelet kernels is illustrated, pointing toward the need of wavelet transform and adaptive wavelet transform. The superiority of wavelet analyses over short-time Fourier transform and Gabor transform is also demonstrated.

Powerful new data analysis techniques based on wavelets are proving extremely useful in the reduction and interpretation of time series data. The goals of these methods include denoising, characterizing, modeling, and compressing of time series data. The multi-scale nature of wavelet analysis makes it especially useful for detection and characterization of self-similar or 'scaling' behavior, such as is common for chaotic processes. This paper describes how wavelet techniques led to a transient-chaos model for a rapidly fluctuating celestial X-ray source. The methods described here are freely available in a new software package called TeachWave, developed by David Donoho and Iain Johnstone of Stanford University (anonymous ftp to playfair.stanford.edu; the software is in directory /pub/software/wavelets, and a number of related technical papers are in /pub/reports).

This paper presents a methodology for the development of software for classifying power system disturbances by type from the transient waveform signature. The implementation of classification capability in future transient recorders will enable such features as selective storage of transient data (to better utilize limited storage media) and automated reporting of disturbances to central control facilities. The wavelet transform provides an effective and efficient means of decomposing voltage and current signals of power system transients to detectable and discriminant features. Similarities of power system transients to wide-band signals in other domains, the simultaneous presence of a resonant frequency, its harmonics, and impulse (high-frequency, time-localized) components, make this technique extendible to other classification systems. The classification algorithm uses statistical pattern recognition on features derived from the extreme representation of the transient waveform after processing the transient waveform by a non-orthogonal, quadratic spline wavelet. Training and classification testing use simulated waveforms of a 200 mile, three-phase transmission line produced by the Electromagnetic Transients Program (EMTP). A simple Bayesian classifier identifies an unknown transient waveform as a capacitor switching or fault transient, and locates the point of disturbance from one of two possible locations on the transmission line. Due to the effectiveness of the wavelet transform preprocessing, the classification system currently performs at 100 percent accuracy on four transient classes.

Thickness evaluation of materials via ultrasound can have special requirements that make the application of wavelets desirable. The thickness information is extracted from ultrasound echoes, and it is the distance between these echoes which must be estimated. Due to typical rates involved in real-time applications, only a minimum amount of processing can be done. In this paper, the multiscale analysis induced by the wavelet transform is employed and compared against conventional techniques (matched filtering by direct and a FFT based approach) in terms of computational complexity. To meet the data rate involved, the fast wavelet transform is used in a manner to obtain the necessary position information. Steps are taken to compensate for the translation variance of the wavelet transform.

The design of large-scale systems requires methods of analysis which have the flexibility to provide a fast interactive simulation capability, while retaining the ability to provide high-order solution accuracy when required. This suggests that a hierarchical solution procedure is required that allows us to trade off accuracy for solution speed in a rational manner. In this paper, we examine the properties of the biorthogonal wavelets recently constructed by Dahlke and Weinreich and show how they can be used to implement a highly efficient multiscale solution procedure for solving a certain class of one-dimensional problems.

In this paper, we discuss a novel multiscale algorithm for the detection of moving objects in a sequence of images. We model the apparent image motion (optical flow) with a class of multiscale stochastic models and use these to develop efficient estimation algorithms to compute the optical flow estimates, and to detect the presence of small moving objects within this flow. The algorithm has five clear advantages over existing approaches. First, it produces estimates of optical flow and associated covariances at multiple resolutions using explicit, multiscale statistical models. Second, it achieves substantial computational savings compared with existing optical flow techniques due to the fact that it has per pixel computational complexity--independent of image size. Third, it can track pixel-level motion over time using simple temporal dynamic models, improving the results over static edge and target detection algorithms. Fourth, it employs the multiscale error covariance information to identify the optimal resolution for flow estimation across the field of view, thus pinpointing regions in which motion can be localized to finer resolutions than other areas. Finally, it can generate spatial measurement residuals that highlight and enhance localized areas of optical flow discontinuity due to target motion. In this paper, the multiscale optical flow algorithm is defined and illustrated using a digital, grayscale image sequence of a helicopter. The results of this research clearly show the feasibility of enhanced target detection through the coherent processing of image sequences.

We developed an approach combining morphological processing and wavelet transforms to detect multiple objects in an input scene. The input scene contains different types of background clutter regions and multiple objects in different classes, with different object aspect views, different object representations, hot/cold/bimodal/partial object variations, and high/low object contrast variations. Our approach provides high detection rates and low false alarm rates. The most computationally demanding operations required are realizable on an optical correlator.

Daubechies wavelets are applied to the fixed pattern noise measured by a 336-by-165 Honeywell uncooled microbolometer infrared sensor and by an Amber AE 4128 128-by-128 indium antimonide (InSb) staring array. The main hypothesis presented in this paper is that Daubechies wavelets are the proper filter for the noise, because they decorrelate the temporal pixel responses on both arrays. These results are an experimental verification of the basic wavelet research of Flandrin with significant additions by Tewfik and Kim.

This paper describes a wavelet-based system for computing localized velocity fields associated with time-sequential imagery. The approach combines the mathematical rigor of the multiresolution wavelet analysis with well known spatiotemporal frequency flow computation principles. The foundation of the approach consists of a unique, nonhomogeneous multiresolution wavelet filter bank designed to extract moving objects in a 3D image sequence based on their location, size and speed. The filterbank is generated by an unconventional 3D subband coding scheme that generates twenty orientation tuned filters at each spatial and temporal resolution. The frequency responses of the wavelet filter bank are combined using a least-squares method to assign a velocity vector to each spatial location in an image sequence. Several examples are provided to demonstrate the flow computation abilities of the wavelet vector motion sensor.

Results are presented of a numerical survey of optical flow algorithms for tracking problems associated with infrared imaging in high-speed missiles. The algorithms tested include those associated with normal flow subject to global smoothness constraints, edge detection via zero crossings of the image after convolution with spatiotemporal filters, and windowed matching techniques. The tracking problems considered in this survey fall into two classes: acquisition, and tracking after acquisition. These classes can be further divided into near and far range problems, characterized by extended and point target images. Other parameters of interest model allowable target and sensors motion and the amount of background clutter. Each of the above methods for determining optical flow can be used in conjunction with a variety of image preprocessing techniques such as kernel smoothing (especially by Gaussian kernels). and evolution under affine invariant partial differential operators. These preprocessing methods can also by used in combination with other approaches such as temporal layering in which successive image are combined to produce images with streaks whose edges are predominantly parallel to the optical flow.

This paper describes a new wavelet-transform-based nonlinear spatial filter called the scale subtraction filter (SSF). The SSF enhances dim target signals relative to clutter in forward- looking infrared (FLIR) imagery using the scale characteristics of the target. The SSF was applied to a database of FLIR image sequences representing different target, sensor, and background clutter scenarios. Detection performance was quantified in terms of the receiver operating characteristics (ROC) for all sequences. The detection performance of the new algorithm was compared to the Holmes double-grated filter method and was found to be equivalent or significantly better depending on the image sequence.

The composite wavelet-matched filter is a nonlinear combination of the wavelet-matched filters that yields correlation between the wavelet transforms of the input image and a combination of edge enhanced training images. The filter is designed with an iteration algorithm to ensure the desired output peak value for each training image. The filter is optically implemented with a phase mostly modulation liquid crystal television and a photographic amplitude modulation mask. This filter shows the best performance compared with the phase-only composite filter and the conventional composite filter. Experimental results are shown.

We propose a hybrid optical/electronic image processing system for the real-time implementation of a wavelet transform. Specifically, the difference-of-Gaussians (DOG) wavelet is synthesized in the hybrid system and used to process an input object. In the implementation, two laser beams with different temporal frequencies are first generated by an acousto-optic modulator. The two optical beams are then combined spatially and used to scan and process the object. The scattered light from the object is then picked up by a photodetector. The photodetector's electrical output signal, after appropriate electronic filtering, represents a scanned and processed version of the original object and can be sent to a monitor for real-time display or to a digital storage device for possible further processing. Because the hybrid system is based on optical scanning, it is an incoherent system and hence has a much better signal-to-noise ratio (SNR) as compared to its coherent counterpart. Experimental results are presented and shown to be in excellent agreement with computer simulations.

The computational efficiency of the adaptive wavelet transform (AWT) is due both to the compact support closely matching with signal characteristics, and to a larger redundancy factor of the superposition-mother (s(x), or in short super-mother, created adaptively by a linear superposition of other admissible mother wavelets. We prove that the super-mother always forms a complete basis, but usually associated with a higher redundancy number than its constituent C.O.N. bases. Then, in terms of Daubechies frame redundancy, we prove that the robustness of super-mother in suffering less noise contamination (since noise is everywhere, and a redundant sampling by band-passings can suppress the noise and enhance the signal). Since the continuous function of super- mother has been created with least-mean-squares (LMS) off-line using neural nets and is formulated in discrete approximation in terms of high-pass and low-pass filter bank coefficients, then such a digital subband coding via QMF saves the in-situ computational time of AWT. Moreover, the power of such an adaptive wavelet transform is due to the potential of massive parallel real-time implementation by means of artificial neural networks, where each node is a daughter wavelet similar to a radial basis function using dyadic affine scaling.

A wavelet network, or Wave-Net is a connectionist network that combines the mathematical rigor and multiresolution character of wavelets with the adaptive learning of artificial neural networks. In this paper, we present some novel techniques for training and adaptation of Wave-Nets, and describe the induction of models that may be physically interpretable, and may provide useful insight into the system being modeled. Learning from empirical data is formulated as a constrained optimization problem. This formulation illustrates the complexity of the learning problem, and highlights the decision variables and the simplifying assumptions necessary for a practical learning methodology. Techniques for Wave-Net training and adaptation are developed for minimizing the L² or L^(infinity) norms. Minimizing the L^(infinity) norm is particularly relevant for solving control problems. The connection between Wave-Net parameters, and the error of approximation is derived using the principles of frame theory. The performance of Wave-Nets for different training methodologies, and basis functions is compared via case studies. Wave-Nets with Haar wavelets as activation functions are well-suited for problems where the output consists of a finite set of discrete values, as in classification problems. The mapping learned by Haar Wave-Nets may be represented as simple if-then rules, which provide an explicit and physically meaningful relationship between inputs and outputs. The relationship of learning by Haar Wave-Nets with other rule induction techniques, such as decision trees is explored.

Supervised learning generally requires the induction of an input- output map from a training set of labeled examples. This induced map is then used in an attempt to correctly classify input features that were not part of the original training set. Current backpropagation neural networks models, however, suffer several shortcomings. These are due to the difficulty in estimating the number of necessary hidden units, the inherent problems of gradient descent computations and the hyperplane classification implicit in most network models. We will present a mathematically sound framework for neural networks simulations in the form of multiresolution analysis. In these multiresolution neural networks (MRNNs), the neuron activation functions are chosen from a set of wavelet basis functions, in such a way that a resulting network represents a wavelet expansion of the underlying input- output map. The networks are constructed by using a modified recursive partitioning algorithm which we call receptive field partitioning (RFP). The RFP algorithm constructs a network by localizing a region of high error and adding nodes whose activation function is taken from a higher resolution space than the current local nodes, and whose support falls within the region of high error. The combination of MRNNs and the RFP algorithm provides a solution to the problems associated with the backpropagation networks.

An adaptive wavelet classifier algorithm is detailed and tested on a data set of acoustic backscatter from a metallic man-made object and from natural and synthetic specular clutter with reverberation noise. The classifier computes the locations, sizes and weights of Gaussian patches in time-scale space that contain the most discriminatory information. This new approach is shown to give higher classification rates than commonly used power spectral features. The new approach also reduces the number of free parameters in the classifier based on all wavelet features, which leads to simpler implementation for applications.

In this paper, we describe a text-independent phoneme-based speaker identification system that uses adaptive wavelets to model the phonemes. This system identifies a speaker by modeling a very short segment of phonemes and then by clustering all the phonemes belonging to the same speaker into one class. The classification is achieved by using a two layer feed forward neural network classifier. The performance of this speaker identification system is demonstrated by considering the phonemes that were extracted from various sentences spoken by three speakers in the TIMIT acoustic-phonetic speech corpus.

We consider the detection of multiclasses of objects in clutter with 3D object distortions and contrast differences present. We use a correlator since shift invariance is necessary to handle an object whose location is not known and to handle multiple objects. The detection filter used is a linear combination of the real part of different Gabor filters which we refer to as a macro Gabor filter (MGF). A new analysis of the parameters for the initial set of Gabor functions in the MGF is given a new neural net algorithm to refine these initial filter parameters and to determine the combination coefficients to produce the final MGF detection filter are detailed. Initial detection results are given. Use of this general neural net technique to design correlation filters seems very attractive for this and other applications.

Neurons in the primary auditory cortex exhibit distinctive selectivities in responses to various acoustic features. Recent physiological studies suggest three spatial dimensions along which the neural response patterns can be systematically organized: the tuning frequencies of the neurons are logarithmically mapped on the tonotopic axis, and the shapes of the tuning curves, in terms of symmetry and bandwidth, vary gradually along two other spatial dimensions. In this report, it is shown that these variations can be effectively modeled by a complex wavelet transform. With such a tie, one can employ well- established wavelet theories into analyzing and understanding how acoustic signals are processed in the auditory system, and thereby design novel engineering applications that are perceptually oriented.

One of the primary visual properties used by radiologists in classifying masses is the sharpness of the edge of the mass. Wavelet transforms can be thought of as multiscale edge detectors. We report using the edge detection and classification properties of wavelet transforms to help classify masses on mammograms. We digitized six masses from mammograms: three benign and three malignant. Our preliminary results indicate that edge properties of masses in mammograms can be obtained from features in the wavelet transform domain. These edge properties can be used to help classify masses prior to biopsy. In particular, the change in the direction of the edge gradient at intermediate scales is indicative of malignancy. This work must be extended to a much larger sample size. The larger sample size will allow other measures to be used. More importantly the interaction between measures can then be observed. Undoubtedly a combination of measures will be required to classify masses accurately.

We demonstrate a simple and effective method for image contrast enhancement based on the multiscale edge representation of images. The contrast of an image can be enhanced simply by stretching or scaling the multiscale gradient maxima of the image. This method offers flexibility to selectively enhance features of different sizes and ability to control noise magnification. Experimental results from enhanced medical images are presented.

In this paper we introduce an algorithm for imaging a time- varying object from its projections at different fixed times. We show that the reconstruction of coarse features, corresponding to low spatial-frequency data, can be made nearly instantaneously in time from the evolving data. A temporal sequence of these low spatial-frequency reconstructions can be used to estimate the motion of the object. Once the motion is estimated, we may use the estimate to compensate for some of the motion of fine scale features. This enables accurate reconstructions of the time varying fine structure in several cases. The algorithm is demonstrated for a selection of phantoms and actual MRI studies. In general, this technique shows promise for a wide variety of applications in MRI, as well as for heart imaging using X-ray CT. Clinical applications should include both functional MRI such as dynamic imaging of oxygen usage and blood flow in the brain, and motion imaging of joints, angiography in the lungs, and heart imaging.

Electromyographic (EMG) signals are pulse-based signals with the high-energy components located in the pulses, also called envelopes. These pulses contain information that is vital for EMG signal analysis. In a person with a spinal cord injury, the envelopes are cluttered with noise and are difficult to detect. In this paper, we will show that the simultaneous use of a pico filter (FatBear) and wavelets is a robust method for the detection of the signal in a cluttered environment. The FatBear, a nonarithmetic, piecewise continuous filter, can be used as a filter for pulse-width filtering, impulse rejection, and edge enhancement. The FatBear will be used as a preliminary step to eliminate the impulsive noise present in the signal. Wavelet techniques will then be applied to process the signal. As a result, we will obtain the information in the pulse interval without the noise.

Since wavelet analysis is an effective tool for analyzing transient signals, we studied its feature extraction and representation properties for events in electrocardiogram (EKG) data. Significant features of the EKG include the P-wave, the QRS complex, and the T-wave. For this paper the feature that we chose to focus on was the P-wave. Wavelet analysis was used as a preprocessor for a backpropagation neural network with conjugate gradient learning. The inputs to the neural network were the wavelet transforms of EKGs at a particular scale. The desired output was the location of the P-wave. The results were compared to results obtained without using the wavelet transform as a preprocessor.

We propose the design of a hearing aid based on the wavelet transform. The fast wavelet transform is used to decompose speech into different frequency components. This paper presents the difficulties in the use of wavelet transforms for speech processing and shows how the careful selection of wavelet coefficients can enable the four major categories of speech - voiced speech, plosives, fricatives, and silence - to be identified. With knowledge of these four categories, it is shown how speech can be easily and effectively segmented.

We describe a wavelet-based technique for identifying aircraft from acoustic emissions during takeoff and landing. Tests show that the sensor can be a single, inexpensive hearing-aid microphone placed close to the ground. The paper describes data collection, analysis by various techniques, methods of event classification, and extraction of certain physical parameters from wavelet subspace projections. The primary goal of this paper is to show that wavelet analysis can be used as a divide-and- conquer first step in signal processing, providing simplification and noise filtering. The idea is to project the original signal onto the orthogonal wavelet subspaces, both details and approximations. Subsequent analysis, such as system identification, nonlinear systems analysis, and feature extraction, is then carried out on the various signal subspaces.

This paper presents a wavepacket-based transient signal detector for detecting unknown deterministic signals in Gaussian white noise. The detector consists of the best wavepacket basis algorithm of Coifman and Wickerhauser, together with the recently developed translation invariant wavelet transform (TI) of Weiss. The uniqueness of this approach concerns the use of the TI which provides two advantages over nominal wavepacket methods. One advantage is that the TI detector performance is independent of sample shifts of the input signal. The second advantage concerns energy distribution in the wavepacket domain. The adaptability of wavepackets to the input signal, provides a distinct advantage over wavelet methods. Use of the TI with wavepacket methods provides the further advantage of a sharper energy concentration in the wavepacket domain. That is, more energy is concentrated into a fewer number of coefficients, thereby providing larger peak energy values. We exploit this higher peak energy, by using the maximum energy as a detection statistic. A numerical investigation is conducted by sweeping over frequency, phase and damping constant for an exponentially damped sinusoid. Detector performance is evaluated through ROC curve comparison, generated by Monte-Carlo simulation. TI detector performance is compared to the Mallat wavelet transform detector and the nominal wavepacket based detector. Results show on average, a performance improvement in the TI based detector over the nominal wavepacket and wavelet detectors.

We evaluate via numerical simulation the performance of a detection algorithm applicable to unknown transients in acoustic signals. This is a filter-then-detect scheme in which the filtering is accomplished by thresholding in the wavelet domain. The incoming time series is separated into 'signal' and 'noise' in the wavelet transform domain. The set of coefficients representing the 'signal' is inverse-transformed back to the time domain. An energy threshold is applied to the recovered signal time series. The performance of the wavelet filtering as a part of the detection process is determined by constructing the receiver-operating-characteristic curve, which displays the dependence of the probability of detection on the probability of false alarm as functions of an energy threshold.

Comparative results are presented for the classification of underwater transient sounds. Comparisons are made using the short-time Fourier transform (STFT) and the Morlet continuous wavelet transform (CWT). Performance as a function of signal-to- noise ratio are presented for synthetic chirp transients. Classification results for three classes of biological signals are also presented. Adaptive energy windows and moments of the transformed signals were used as features for classification. Certain classes of transient signals were found for which one of the transforms was superior. In general, wavelets were better for broadband signals, while the STFT was optimal on narrowband signals.

Artificial neural networks (ANN) can be used to process digital image screen halftoning (DISH), designed to be adaptive to the local variation of image intensity based on the wavelet transform (WT) preprocessing of the local gradient at each pixel. Our preliminary digital simulation results have shown an improved multiresolution visual effect of the bilevel representation of a gray-scale image. An interesting device concept is to build a fast 'WT chip' of order (N) with a smart 'neurochip' for DISH applications, in order to achieve an nonuniformly enhanced dot matrix printing.

This paper provides a simple parametrization of perfect reconstruction filter banks with arbitrary regularity. The parametrization shows that a perfect reconstruction filter bank with regularity N can be constructed by using an overlapping discrete 2N point Chebyshev transform followed by an orthogonal 2N X 2 transform and an arbitrary perfect reconstruction filter bank.

We present a new approach to studying a discrete Gabor expansion (DGE). We show that, in general, DGE is not the usual biorthogonal decomposition, but belongs to a larger and looser decomposition scheme which we call pseudo frame decomposition. It includes the DGE scheme proposed as a special case. The standard dual frame decomposition is also a special case. We derive algorithms using techniques for Gabor sequences to compute 'biorthogonal' sequences through proper matrix representation. Our algorithms involve solutions to a linear system to obtain the 'biorthogonal' windows. This approach provides a much broader mathematical view of the DGE, and therefore, establishes a wider mathematical foundation towards the theory of DGE. The general algorithm derived also provides a whole class of discrete Gabor expansions, among which 'good' ones can be generated. Simulation results are also provided.

The covariance matrix in Kalman filter is reduced using compactly supported orthonormal wavelet transform and is parameterized by only O(N) coefficients, where N is the dimension of the state vector. An approximate filtering algorithm, in which the covariances remain in such a transformed and compressed form throughout the time recursion, is designed. For estimation of space-time processes characteristic of geophysical flows, the proposed algorithm performs near optimally, while reducing computational and storage requirements of Kalman filter.

The use of generalized self-similar tilings and nonseparable Haar wavelets for characterization of image statistics over irregularly shaped regions is detailed. Requirements for uniqueness of the image representation are presented and richness of the class of tiles is explored. New results include the uniqueness of the representation on the toroidal integer lattice. The resulting classification of a scalar valued image using the estimated statistics is presented over an irregular tiling.

This paper develops a steerable multiscale analysis theory. A number of models based on the wavelet theory are proposed for multiscale TV scene analysis and image processing, including image representation, image edge detection, and noise analysis and removal. A multiscale interpretation method is discussed that makes full use of multiresolution images and edge feature. In order to employ multiple information and all relative information reasonably and effectively, information fusion has been investigated. The idea of geometric reasoning has been also proposed for interpreting objects in TV scenes. Finally, some experiments have illustrated that the proposals described above are feasible.

Two parametrizations are presented for the Daubechies wavelets. The first one is based on the correspondence between the set of multiresolution analyses with compact support orthonormal basis and a group developed by Pollen. In the second parametrization, emphasis is put on the regularity conditions of the Daubechies wavelets. The orthonormality conditions characterizing the Daubechies wavelets are solved and general complex solutions are described.

The signal decomposition techniques are an important tool for analyzing nonstationary signals. The time-frequency resolution of the decomposition basis functionals is essential to a variety of signal processing applications. The recently introduced wavelet transform is a very promising tool for signal analysis, but little attention has been paid to the time-frequency resolution property of wavelets. This paper describes a procedure to design wavelets with better time-frequency resolution. Some design examples and comparisons with traditional wavelets are also presented.

A complex wavelet is used in the continuous wavelet transform to obtain edges from a digital image in two orthogonal directions. The square root of the sum of the squares of the real and imaginary parts of the wavelet transform are used to find the edges in an arbitrary direction. Theoretical models are derived for the cases of a step edge and ramp edge. The computer implementation of the wavelet edge detector involves the creation of two masks; one for the real part and one for the imaginary part. An expression is derived for the size of the edge masks using the inflection points of a curve derived from the wavelet. Finally, the edge detector is applied to two synthetic aperture radar images, and the resulting images are shown.

The edge of an image is one of the important features in image analysis and processing. The edge often corresponds to the sharp variation points of intensities that carry most of the image information. In practice, the implementation of edge detection is often performed by some multiscale operators. Wavelet transform is closely related to multiscale edge detection. This paper suggests a regularization method for determining scales for wavelet transform adaptively for each site in image plane. An energy function was introduced to obtain a set of optimal scales by minimizing this function. The multiscale wavelet transform filter was derived from this energy function. The maxima of wavelet transform modulus at these scales acquired from the local site of the image were detected to form the edge map. Experiments for real image shows that both step and differ edges can be detected by this method.

A framework of physically based shape morphing with the multiscale wavelet descriptor is proposed. We formulate the problem of shape metamorphosis with the Lagrangian dynamic equation which simulates the deformation as a process driven by a certain force as the result of being released from strain energy. Then we show the discretization of Lagrange's equation with respect to the wavelet representation and derive the corresponding mass and stiffness matrices. We show the computation of entries of the stiffness matrix by solving a system of linear algebraic equations. Due to the multiscale representation capability of the wavelet descriptor, the graph in intermediate frames can be generated via multiresolution rendering. Experiments are conducted to demonstrate the performance of the proposed physically based morphing algorithm.

An airborne test platform known as the Airborne Visible and Infrared Imaging Spectrometer (AVIRIS) has been in operation since 1989. The data gathered form AVIRIS comes in 224 bands covering the 0.4 - 2.4micrometers range of the electromagnetic spectrum. The bands are spaced approximately 10nm apart with a bandwidth of just 10nm. Thus AVIRIS provides a nearly continuous spectral signature for a ground area 20m square. Because of the high dimensionality of such data, previous techniques for extracting information from multispectral imagery become computationally prohibitive. In the technique presented the wavelet transform is used to select features in the spectral signature on which the classification of the AVIRIS data set is carried out. The ability of the wavelet transform to localize both in time and frequency may make the technique able to characterize absorption bands better than many other transform techniques. The results of the technique will be compared against a classification using coefficients of the discrete cosine transform as features and a classification of the original data set.

We present in this paper an architectural design for a wavelet transform chip for use in real-time one-dimensional signal processing applications. Based on the observation that further levels of the wavelet transform require only as much computation as the first level, our architecture requires only one row of processing elements to compute the complete transform. This is compared to previous designs requiring one row of processing elements per level of the transform. Our architecture provides the output of the transform in two forms, one with all levels multiplexed on one line (useful for transmission or compression) and the other as individual levels on separate lines synchronized in time to facilitate real-time analysis. We consider the usefulness of this architecture for real-time analysis of audio signals (typically 40kHz sampling rate) and discuss the design and implementation benefits of the computational simplicity of the presented architecture.

A detection technique based on a synergistic composition of wavelet feature detectors is demonstrated on sonar imaging data. The wavelets are used to preprocess the imagery for enhancing highlights and shadows. A neural network is trained on the preprocessed imagery to weight the output of two filters for underwater object detection. This approach is demonstrated on multiple scales. Results indicate this composite approach is highly effective.

An image representation technique, motivated by Stephane Mallat's matching pursuit algorithm, has been developed for image analysis and decomposition. We've simplified the mechanics of the algorithm to enable an extremely fast implementation via optical processing. Initial computer simulations show that our algorithm is capable of decomposing and representing an 2D image with a linear combination of basis image with high speed and high fidelity.

Wavelet transforms are used to analyze electroencephalographic (EEG) data recorded from individual subdural electrodes (one dimension) and electrode grids (two dimensions) in direct contact with the brain. Structure in the data is resolved and noise filtered using surrogate data techniques. Spike and seizure detection from individual electrodes are compared with event extraction using windowed Fourier transforms. Wavelet transforms are a powerful means to identify epileptiform activity such as spikes in one dimension from such data, and offer a method to localize the foci of epileptic seizures in two dimensions.

The problem of identifying a target from noisy high-range- resolution (HRR) data is addressed for the case of large radar target signature databases. The problem is treated using a general formulation of the problem of M-ary target identification in the presence of additive white Gaussian noise. A multiresolution analysis of the data and signature database relative to a discrete orthonormal wavelet basis is employed and shown to lead to a scale-sequential identification algorithm that has the property of making an identification having prespecified detection probabilities with the minimum number of computations. The general formulation is outlined and a computer-simulated example of target identification using the procedure is presented and discussed.

Wavelet decomposition of data has been used with success for data compression in a large class of signals. The wavelet packets representation is a decomposition that includes as a special case the wavelet transform. In this scheme, wavelet packets are assembled as a representation of the signal. There exist a large number of possible wavelet combinations for each signal. An adaptation scheme based on the genetic optimization method is presented for determining an optimal representation for data compression. Experiment results using real data as well as autoregressive random processes are used to compare the performance of the adapted packets and the wavelet transform.