Algorithms are developed for generating FIR filter banks for triadic, biorthogonal decompositions. The basis for the algorithms is a set of simultaneous quadratic equations in the decomposition and reconstruction filter coefficients which arise as necessary conditions for biorthogonality. The solution is obtained either by nonlinear optimization or by iterative pseudoinverse solution of linear equations.

The problem of designing a mother wavelet such that this wavelet is as close as possible to a desired signal while it generates, along with a dual, a biorthogonal wavelet decomposition is addressed. It is shown that when the problem is addressed in the frequency domain with bandlimited scaling functions and wavelets, a simple formulation is possible. Further, a closed form sub-optimal solution is possible when the problem is broken down into a series of optimization problems. In the process of the problem formulation, a generalization of the Meyer class of wavelets to the biorthogonal case is obtained.

The paper provides a closed form solution to least squares matching of the energy spectrum density of an orthonormal MRA-generating wavelet with that of a specified arbitrary signal. The authors had previously proposed algorithms for matching with samples of energy spectrum densities. An approach is also proposed to use matched wavelets in detecting an object in an image. Furthermore, the approach captures size, orientation and position information.

A scheme that generates a pair of dual wavelets where one is compactly supported in time and the other is bandlimited is proposed. The wavelets and associated scaling functions give rise to a biorthogonal decomposition of the input. An example is provided to show that, with this scheme, it is possible to generate duals for scaling functions constructed form linear combinations of simpler scaling functions and their dilations.

In many communications systems, spread spectrum techniques are used to spread the original message data over a large bandwidth in order to improve system performance in the presence of narrowband interference. The extent to which such interference can be tolerated depends on the system's processing gain and may be augmented using adaptive filtering techniques. In order to mitigate narrowband interference, spread spectrum receivers can incorporate filtering techniques that suppress interference and make bit decisions on the remaining signal energy. In this paper, block transforms and multirate filterbanks based on hierarchical subband trees are used to transform the received data signal to the transform domain wherein adaptive filtering is performed. Pre- and post-correlation transform domain least-mean-squared (LMS) algorithms are employed on a block-by-block basis to suppress the narrowband interference while simultaneously minimizing the mean-squared error between the received signal and the original data message. Convergence and misadjustment noise are evaluated as functions of the underlying subband transform and various system parameters. Subsequent performance analysis of these algorithms is presented in terms of the overall system bit-error-rate. Analytical and simulation results obtained in the presence of single-tone interference sources are presented. Although not considered, the analytical expressions discussed herein can be easily extended to handle other narrowband sources such as multitone jammers and narrowband Gaussian interference.

This paper generalizes the dyadic wavelet packet bases (DWP), developed by Coifman and Wickerhauser, to time/scale-adjusted DWP bases. These generalized DWP bases provide more flexibility in matching the time-scale characteristics of the input signal. Development of these generalized bases is achieved by combining the previously defined time-invariant DWP bases of Pesquet, Krim, Carfantan, and Proakis with a generalized scale sampling. The generalized scale sampling extends the usual dyadic sampling by adding a real-valued offset parameter to the integer power of two in the scale parameter. This offset parameter value is taken between zero and one. By combining both scale and translation generalizations, signal components existing between consecutive dyadic scales, or consecutive time translations, may be captured. It is shown how these DWP coefficients may be generated from a two step process; first projecting the input signal onto an appropriate space. Then, performing the usual wavelet low and highpass filtering operations, followed by downsampling. The projection operation is shown to be equivalent to a filtering operation. An expression for the filter taps is derived, and basic properties are proven. A translation-invariant transform defined on these scale-adjusted wavelet packets, is developed. An application to transient detection is presented, by developing a transient detector based on this transform. ROC curves, generated by Monte- Carlo simulation, are presented demonstrating detector performance. Detector performance is shown to be independent of the signal translation. It is further shown how matching the basis functions to the time-scale-frequency characteristics of the transient can provide improved detection performance.

An approach to impulsive noise suppression and background normalization of digitized bandlimited electrovcardiogram signals is presented. This approach uses adaptive wavelet filters that incorporate the band-limited a priori information and the shape information of a signal to decompose the data. Empirical results show that the new algorithm has good performance in wideband impulsive noise suppression and background normalization for subsequent wave detection, when compared with subband coding using Daubechie's D4 wavelet, without the bandlimited adaptive wavelet transform.

Appropriately designed wavelets possess great potential for simplifying the information processing requirements of pattern recognition or digital audio and video systems. In order to be effective, it is crucial to match the properties of the wavelets with those of the signals being processed. We have developed a genetic algorithm to design wavelets adaptively by exploiting a wavelet parameterization to search for wavelets that are suited to specific applications. In particular, we have shown how a genetic algorithm can be used to find wavelets whose transforms possess desirable properties and we have demonstrated that genetic algorithm can be used to match a wavelet to signals of interest.

This paper describes the continuing work on a new class of wavelets called abc wavelets, first introduced in 1995. The name is drawn from an extra parameter, c, in addition to the conventional ones of a for dilation and b for translation: (Psi) (a,b,c). The method uses discrete wavelets, where explicit solutions exist for 4 and 6 taps. The wavelets are orthogonal and each system of order N has N/2 - 1 vanishing moments, compared with Daubechies which has N/2 vanishing moments. This means that each wavelet class (4 taps, 6 taps, etc.) is not optimally smooth but the advantage is that another degree of freedom exists which can be used as a special lock condition to mimic any particular form, and thus best model the underlying signal. As an additional advantage, each class does include the Daubechies wavelet of the respective order, so nothing is lost; rather the abc wavelets describe supersets of the conventional discrete wavelets. In this paper, the aspect of speed is addressed, exploiting the free form to derive taps supporting a faster transform than the standard fast Mallet transform.

A Daubechies wavelet-based bootstrap detection strategy based on the research of Carmona was applied to a set of test signals. The detector was a function of the d-scales. The adaptive detection statistics were derived using Efron's bootstrap methodology, which relieved us from having to make parametric assumptions about the underlying noise and offered a method of overcoming the constraints of modeling the detector statistics. The test set of signals used to evaluate the Daubechies/bootstrap pulse detector were generated with a Hewlett-Packard Fast Agile Signal Simulator (FASS). These video pulses, with varying signal-to-noise ratios (SNRs), included unmodulated, linear chirp, and Barker phase-code modulations baseband (IF) video pulses mixed with additive white Gaussian noise. Simulated examples illustrating the bootstrap methodology are presented, along with a complete set of constant false alarm rate (CFAR) detection statistics for the test signals. The CFAR curves clearly show that the wavelet bootstrap can adaptively detect transient pulses at low SNRs.

Quasi-Monte Carlo methods are an extremely effective approach for computing high dimensional integrals. In this paper we present a concept based on generalized Haar functions systems that allow us to estimate the integration error for practically relevant classes of functions. The local structure of the Haar functions yields interesting new aspects in proofs and results. The results are supplemented by concrete computer calculations.

In this paper a new approach for adaptive wavelet image coding is introduced. Based on an adaptive quadtree image-block partition (similar to fractal coding) the different image blocks may be compressed independently in an adaptive manner. In order to adapt to local image statistics and features we present several possibilities of how to optimize the transform part of a wavelet image block-coder. Additionally we present a parallel algorithm suitable for MIMD architectures for efficient implementation of the proposed method. Finally experimental results concerning coding efficiency and execution time on a cluster of workstations are described.

In the field of signal processing, there is a need to quickly and efficiently detect and extract information from signals. One type of signal feature that is difficult to process is a discontinuous singularity and chaotic structure. In this paper, we conduct an experiment that employs adaptive wavelet transforms to efficiently detect signals from simulated data. The adaptive wavelet transformations create super-mother wavelets, which are used to extract a singularity signal from a noisy transmission.

A linear superposition of Haar transform is given to design an adaptive biorthogonal subband coding. Given Haar scaling function, rect(x), a new symmetric staircase scaling function may be desirable in order to match a specific compression or recognition goal. Then, the associated lowpass filter coefficients are solved from the roots on a unit circle in the z-transform domain. From which the bi-orthogonal and lossless high pass filter is derived for both the forward analysis and the backward synthesis stages. An explicit construction of super-Haar system produces a lossless filter bank which can match an infrared image for achieving pattern recognition and compression simultaneously.

An embedded image coding scheme using the multiwavelet transform and inter-subband prediction is proposed in this research. The new proposed coding scheme consists of the following building components: GHM multiwavelet transform, prediction across subbands, successive approximation quantization, and adaptive binary arithmetic coding. Our major contribution is the introduction of a set of prediction rules to fully exploit the correlations between multiwavelet coefficients in different frequency bands. The performance of the proposed new method is comparable to that of state-of-the-art wavelet compression methods.

The work described in this paper addresses the use of the four-dimensional continuous wavelet transform (CWT) for automatic target recognition (ATR) and detection. This transform is an overcomplete representation with four coordinates: two spatial, t₁ and t₂; a rotational coordinate, (theta) ; and a scale coordinate, a. Two central ideas are discussed in connection with the transform's application to target recognition. The first is cross-scale reconstruction, which refers to exploiting the dominate presence of target features across scales. The second is utilizing the non-spatial coordinate space as a working environment for feature extraction and classification. This aspect is unique to the multidimensional wavelet transform, emanating from the inherent redundancy in the transform representation. Some conclusions are drawn in the last section regarding the utility of the CWT for ATR, and the transform's potential as an analysis tool.

An image data compression system using discrete wavelets is described and implemented. The two dimensional discrete wavelet transform is computed via subband coding by repeating a three step procedure explained in this paper. The procedure essentially decomposes an image at some resolution and produces four new images which are characterized as low-low, low- high, high-low, and high-high spatial frequency components along the x and y directions. The first component is passed as the input for the next iteration; the second and third components were stored in memory for later processing; and, the fourth component is truncated and reset to zero for a simple codec design. In essence, the compression is accomplished by discarding roughly 25% of the wavelet coefficients at every iteration. The inverse wavelet transform has been applied in a similar fashion, by inserting zeros in place of the discarded values. The results are illustrated using the simplest wavelets called Haar(1910), which is known to be not optimal for image compression but is available in hardware and is inexpensive. The nature of the artifacts introduced by the truncation of high frequency coefficients of Haar transform is also discussed.

The ability of wavelet decomposition to reduce signals to a relatively small number of components can be exploited in pattern recognition applications. Several recent studies have shown that wavelet decomposition extracts salient signal features which can lead to improved pattern classification by a neural network. The performance of the neural network classifier is heavily dependent upon the ability of wavelet processing to yield discriminatory features. This paper considers the combination of wavelet and neural processing for classifying 1- dimensional signals embedded in noise. Noisy signals were decomposed using the Haar wavelet basis and feedforward neural networks were trained on wavelet series coefficients at various scales. The experiment was repeated using the 4-coefficient Daubechies wavelet basis. The classification accuracy for both wavelet bases is compared over multiple scales, several signal-to-noise ratios, and varying numbers of training epochs.

We describe a local projection transform that uses the discrete and continuous wavelet transforms to represent edge features over multiple scales. High order aspect invariant features are generated from chords in the transform space. A novel noise coding technique binds these features into a coherent pattern. This approach permits a reduction of the dimensionality of the information giving rise to lower processing requirements and thus permitting implementation of automatic target recognition (ATR) on conventional computer architectures.

We consider time-frequency detection and range-Doppler signal processing applications for wavelet processing; these differ considerably from the standard data representation and compression applications. In these cases, new wavelet functions and processing are necessary. We show that shift-invariance is essential (standard DWTs do not provide this), that dyadic scales are not sufficient, that wideband mother wavelet functions are necessary for some high noise cases and that small scale changes are generally required. A new Gabor wavelet implementation is described that is fast [O(N)], shift-invariant, flexible, and is not restricted to dyadic scales or to one basis function. It is compared to conventional dyadic DWTs, block shift-invariant DWTs, wavelet packets and other wavelet transform filter realizations requiring long filter sizes, large latency and long data buffers.

We note the use of Gabor wavelet filters to locate objects and edges (for detections of regions of interest in scenes), to locate clutter regions (to reduce false alarms) and to produce distortion invariant features (for object classification). We describe new ways to select Gabor filter parameters, new nonlinear ways to combine Gabor function outputs for improved performance and to reduce the number of filters necessary.

An increase in accuracy and reduction in computational complexity of the common wavelet- based target recognition techniques can be achieved by using interpretable features for recognition. In this work, the Best Discrimination Basis Algorithm (BDBA) is applied to select the most discriminant complete orthonormal wavelet basis for recognition purposes. The BDBA uses a relative entropy criterion as a discriminant measure. Then, interpretable features are selected from the most discriminant basis by utilizing symbolic knowledge about the domain. The domain theory that contains this symbolic knowledge is implemented in a backpropagation neural network. The output of the backpropagation neural network gives a final recognition decision. The results of our simulations show that the recognition accuracy of the proposed Automatic Feature Based Recognition System (AFBRS) is better than the recognition accuracy of a system that performs recognition using the Most Discriminant Wavelet Coefficients (MDWC).

The definition of a spatial filter using wavelets is presented. The technique is of the same order as a frequency domain beamformer, O(N log₂ N). The array's beam response to the wavelet is defined and the beam response for a linear array of sensors to a wavelet is presented.

Several methods have been proposed to detect chirp signals buried in white noise, including wavelet shrinkage and matching pursuits. However, most of these methods perform poorly when the signal-to-noise ratio (SNR) becomes very low (e.g. lower than 0 dB). In this work, we present a new time-frequency technique using the multi-window Fourier transform (MWFT) and local maximum indexing. The new method detects the location and frequency range of a chirp signal with SNR up to minus 6 dB.

Detection and extraction of unknown signal in noise is an important issue in radar. When signal is an unknown transient, the representation in terms of basis functions which are localized in both time and frequency, such as Gabor representation, is very useful for signal detection. By taking time-frequency decomposition, the noise tends to spread its energy into entire time-frequency domain, while the signal often concentrates its energy within a small region with a limited time interval and frequency band. Therefore, the signal embedded in noise is much easier to be recognized in the time-frequency domain than that in either time or frequency domain. Constant false alarm rate (CFAR) processing is an optimal way to set up a threshold for detecting signals in noise environment. In this paper, we extend the CFAR processing to the time-frequency domain. By setting a CFAR threshold for the time-frequency Gabor coefficients, we can examine the Gabor coefficients and determine whether there is a signal. Then, the signal can be extracted by using the detected signal's Gabor coefficients. Therefore, the time location, the time duration, the frequency range, and other parameters of the unknown signal can be measured. The SNR of the extracted signal is improved about 10 - 12 dB over the observed noisy signal.

A new technique for efficient speckle reduction in synthetic aperture radar (SAR) images using wavelet packets is proposed. A phenomenological approach is used in adapting the wavelet packet transform to minimize the speckle coefficients. With thresholding and averaging, the resulting non-linear, multirate filtering can be used to preprocess images for reduction of speckle while retaining contrast in key objects necessary for target classification. An example of the algorithm using a SAR image is provided.

High resolution radar image is always demanded. To achieve high resolution, wideband signal and longer imaging time are required. However, due to time-varying behavior of returned radar signals and due to multiple backscattering behavior of targets, radar image resolution can be significantly degraded and images become blurred. The conventional radar processor uses the Fourier transform to retrieve Doppler information. In order to use the Fourier transform adequately, some restrictions must be applied: the scatterers must remain in their range cells and their Doppler frequency contents should be stationary during the entire imaging time duration. However, due to the target's complex motion, the Doppler frequency contents are actually time-varying. Therefore, the Doppler spectrum obtained from the Fourier transform becomes smeared, and, thus, the resolution of the radar image is degraded. However, the restrictions of the Fourier processing can be lifted if the Doppler information can be retrieved with a method which does not require stationary Doppler spectrum. Therefore, the image blurring caused by the time-varying Doppler spectrum can be resolved without applying sophisticated motion compensation. By replacing the conventional Fourier transform with a time-frequency transform, a 2-D range-Doppler Fourier frame becomes a 3- D time-range-Doppler cube. By sampling in time, a time sequence of 2-D range-Doppler images can be viewed. Each individual time-sampled image from the cube provides superior image resolution and also enhanced signal-to-noise ratio. When the target contains cavities or duct-type structures, these scattering mechanisms appear in radar images as blurred clouds extended in range dimension. It is very useful to combine adaptive time-frequency wavelet transform with the radar imaging technique so that the 'clouds' can be removed and their resonance frequencies can be identified. By applying time-frequency processing for each cross-range lines of radar image, a 3-D range-Doppler-frequency cube is generated. The frequency slices of the cube provide information for identifying scattering centers as well as resonance frequencies.

Recently there has been a great deal of interest in the use of wavelets and implementation of wavelet transforms. We discuss applications of the wavelet transform to image analysis, including target recognition, rotation and scale invariance, and pattern recognition in the presence of background noise. We propose a method for both scale and rotation invariant pattern recognition based on wavelet features of an image. Wavelets offer advantages in these applications because of their improved ability to discriminate signals in the presence of noise. Performance can be improved by careful selection of the mother wavelet for a given application; we have chosen a two-dimensional Gaussian mother wavelet. Computer simulations of wavelet transform based pattern recognition are discussed, which illustrate scale and rotation invariant target recognition in the presence of noise. Because the wavelet transform is essentially a correlation between the input signal and the family of daughter wavelets, it may be implemented by any type of programmable correlator. In particular, acousto-optic devices offer several advantages; they are programmable and capable of changing their input functions in real time, and commercial devices are available which offer large time-bandwidth products. Acousto-optic image correlators are discussed as a potential implementation of the wavelet transform; by encoding a 1 dimensional wavelet filter bank as a 2 dimensional image, we can implement the wavelet transform image processor without requiring a 2 dimensional spatial light modulator. An alternative implementation utilizes a 2 dimensional array of acousto-optic correlators for a hybrid implementation of a quadrature mirror filter bank.

In this paper we make a comparison between wavelet transforms and the local cosine transform of various types of images. This builds on our previous work involving acoustic signals, where we found that the local cosine transform gave a more compact representation for certain types of signals and performed as well as wavelets for others. This held even for signals that were transient in nature, where one might expect the wavelets to do better. We are interested in determining if the same holds true for images, which tend to include many transients, such as edges. We are also investigating the extent to which the rms error can be used to evaluate the perceptual quality of the reconstructed images.

Two processors are considered for image quadrature mirror filtering (QMF). The neuromorphic infrared focal-plane sensor (NIFS) is an existing prototype analog processor offering high speed spatio-temporal Gaussian filtering, which could be used for the QMF low- pass function, and difference of Gaussian filtering, which could be used for the QMF high- pass function. Although not designed specifically for wavelet analysis, the biologically- inspired system accomplishes the most computationally intensive part of QMF processing. The Texas Instruments (TI) TMS320C80 Multimedia Video Processor (MVP) is a 32-bit RISC master processor with four advanced digital signal processors (DSPs) on a single chip. Algorithm partitioning, memory management and other issues are considered for optimal performance. This paper presents these considerations with simulated results leading to processor implementation of high-speed QMF analysis and synthesis.

In this work, we describe an approach to texture discrimination which builds statistical models based on the orthogonal wavelet representation of the image. These results compared with results from already known texture recognition methods seem encouraging. The method proposed in this paper has been tested in images of different structure. It is executed in two stages. In the first, a multiresolution approximation, up to a given resolution, is used for the decomposition of the image on a wavelet orthogonal basis. In the second stage, the set of the coefficients produced during the previous stage is modeled using a set of statistical measures. We used a set of four statistical measures that contribute to the most accurate texture description.

A rotation-invariant optical correlation filter using wavelet transform to produce easily detectable correlation peaks in the presence of noise and to provide better intraclass recognition is proposed. The proposed filter is designed by using the energy spectra of the wavelet transformed reference image and random noise. Because the energy spectrum of the wavelet transformed reference image is higher than that of the random noise in the specified band, the proposed filter has good discrimination ratio (DR), high signal to noise ratio (SNR), and low distortion sensitivity (DS). The wavelet function used in this paper is the Mexican-hat function, and it is chosen by investigating the relation between the energy spectra of the reference image and the various wavelet functions. The optimal dilation parameters of the wavelet function are also achieved with varying the dilation parameters of the wavelet function.

We present comparative results of texture analysis performed with wavelet transform, exact Karhunen-Loeve (KL) transform and approximate KL transform. Two well-known methods of approximate KL basis construction are discussed, and a new algorithm for optimized orthogonal transform of image ensemble is proposed. The last one is based on the evaluation of factored approximation of 2D-images correlation matrix, the eigenfunctions of which permit us to calculate a suitable approximate KL basis. Experimental results are obtained for texture ensemble with controlled correlation properties. It is shown that in some cases the energy accumulation for wavelet transform is much worse than for approximate 2D-algorithm with the comparable complexity.

An automated method for detecting microcalcification clusters is presented. The algorithm begins with a digitized mammogram and outputs the center coordinates of regions of interest (ROIs). The method presented uses a non-linear function and a 12-tap least asymmetric Daubechies (LAD12) wavelet in a tree structured filter bank to increase the signal to noise level by 10.26 dB. The signal to noise level gain achieved by the filtering allows subsequent thresholding to eliminate on average 90% of the image from further consideration without eliminating actual microcalcification clusters 95% of the time. Morphological filtering and texture analysis are then used to identify individual microcalcifications. Altogether, the method successfully detected 44 of 53 microcalcification clusters (83%) with an average of 2.3 false positive clusters per image. A cluster is considered detected if it contains 3 or more microcalcifications within a 6.4 mm by 6.4 mm area. The method successfully detected 13 of the 14 malignant cases (93%).

This paper presents a novel and computationally efficient approach to an adaptive mammographic image feature enhancement using wavelet-based multiresolution analysis. Upon wavelet decomposition applied to a given mammographic image, we integrate the information of the tree-structured zerocrossings of wavelet coefficients and the information of the low-pass filtered subimage to enhance the desired image features. A discrete wavelet transform with pyramidal structure has been employed to speed up the computation for wavelet decomposition and reconstruction. The spatio-frequency localization property of the wavelet transform is exploited based on the spatial coherence of image and the principle of human psychovisual mechanism. Preliminary results show that the proposed approach is able to adaptively enhance local edge features, suppress noise, and improve global visualization of mammographic image features. This wavelet-based multiresolution analysis is therefore promising for computerized mass screening of mammograms.

The wavelet transform (WT) has been used to study the nonstationary information in the electroencephalograph (EEG) as an aid in determining the anesthetic depth. A complex analytic mother wavelet is utilized to obtain the time evolution of the various spectral components of the EEG signal. The technique is utilized for the detection and spectral analysis of transient and background processes in the awake and asleep states. It can be observed that the response of both states before the application of the stimulus is similar in amplitude but not in spectral contents, which suggests a background activity of the brain. The brain reacts to the external stimulus in two different modes depending on the state of consciousness of the subject. In the case of awake state, there is an evident increase in response, while for the sleep state a reduction in this activity is observed. This analysis seems to suggest that the brain has an ongoing background process that monitors external stimulus in both the sleep and awake states.

We present, in this paper, a wavelet-based acoustic signal analysis to remotely recognize military vehicles using their sound intercepted by acoustic sensors. Since expedited signal recognition is imperative in many military and industrial situations, we developed an algorithm that provides an automated, fast signal recognition once implemented in a real-time hardware system. This algorithm consists of wavelet preprocessing, feature extraction and compact signal representation, and a simple but effective statistical pattern matching. The current status of the algorithm does not require any training. The training is replaced by human selection of reference signals (e.g., squeak or engine exhaust sound) distinctive to each individual vehicle based on human perception. This allows a fast archiving of any new vehicle type in the database once the signal is collected. The wavelet preprocessing provides time-frequency multiresolution analysis using discrete wavelet transform (DWT). Within each resolution level, feature vectors are generated from statistical parameters and energy content of the wavelet coefficients. After applying our algorithm on the intercepted acoustic signals, the resultant feature vectors are compared with the reference vehicle feature vectors in the database using statistical pattern matching to determine the type of vehicle from where the signal originated. Certainly, statistical pattern matching can be replaced by an artificial neural network (ANN); however, the ANN would require training data sets and time to train the net. Unfortunately, this is not always possible for many real world situations, especially collecting data sets from unfriendly ground vehicles to train the ANN. Our methodology using wavelet preprocessing and statistical pattern matching provides robust acoustic signal recognition. We also present an example of vehicle recognition using acoustic signals collected from two different military ground vehicles. In this paper, we will not present the mathematics involved in this research. Instead, the focus of this paper is on the application of various techniques used to achieve our goal of successful recognition.

In this research we compare general harmonic wavelet transforms (GHWT), constant Q transforms (CQT) and the Cone kernel time-frequency distribution (CKTFD) for the analysis of musical signals. The first two consist of a series of band pass filters that have a constant Q (quality), each of which correspond to a semitone interval (or better) in the musical scale. The CKTFD is not a constant Q type transform but belongs to the more general class of bilinear time-frequency distributions with the special property of reducing the (usually) undesirable interference terms common in these types of distributions. Their computation schemes are compared and the advantages of each discussed. All three have a structure that may be easily parallelized. We used the three methods to analyze a musical note (middle C) played on an electric piano. There are subtle differences in the results of the methods. All three quite clearly show the first four harmonics of the musical note (C₄). However, the CQT reveals that harmonics higher than four extend in time for almost the entire duration of the note. Neither the GHWT nor the CKTFD show harmonics higher than four for the entire length of the signal, though they do reveal them (i.e., frequencies higher than 1024 Hz.) at the initiation of the note. The power spectrum of the signal does reveal harmonics from one through four as having most of the power but harmonics five through ten are also revealed. Unfortunately, the time variation of the signal cannot be extracted from the power spectrum hence the use of time-frequency or time-scale diagrams. Aside from musical applications, such methods would be useful for t-f analysis of vibrating machinery.

In wireless image communication, image compression is necessary because of the limited channel bandwidth. The associated channel fading, multipath distortion and various channel noises demand that the applicable image compression technique be amenable to noise combating and error correction techniques designed for wireless communication environment. In this study, we adopt a wavelet-based compression scheme for wireless image communication applications. The scheme includes a novel scene adaptive and signal adaptive quantization which results in coherent scene representation. Such representation can be integrated with the inherent layered structure of the wavelet-based approach to provide possibilities for robust protection of bit stream against impulsive and bursty error conditions frequently encountered in wireless communications. To implement the simulation of wireless image communication, we suggest a scheme of error sources modeling based on the analysis of the general characteristics of the wireless channels. This error source model is based on Markov chain process and is used to generate binary bit error patterns to simulate the bursty nature of the wireless channel errors. Once the compressed image bit stream is passed through the simulated channel, errors will occur according to this bit error pattern. Preliminary comparison between JPEG-based wireless image communication and wavelet-based wireless image communication has been made without application of error control and error resilience to either case. The assessment of the performance based on image quality evaluation shows that the wavelet-based approach is promising for wireless communication with the bursty channel characteristics.

Within the past few years, wavelet transforms and filter banks have received considerable attention in the technical literature, prompting applications in a variety of disciplines including applied mathematics, speech and image processing and compression, medical imaging, geophysics, signal processing, and information theory. More recently, several researchers in the field of communications have developed theoretical foundations for applications of wavelets as well. The objective of this paper is to survey the connections of wavelets and filter banks to communication theory and summarize current research efforts.

Both filterbank theory and error correction codes have been studied extensively during recent decades. In this paper, we first build a bridge between these two theories. We restate some results in error correction coding theory by using filterbank terminologies over finite fields. We then apply filterbank theory to error correction codes for partial response channels (PRC), such as magnetic recording systems. We present necessary and sufficient conditions for uniquely decodable convolutional codes for PRC. We also extend the results to multihead and multitrack recording systems. Examples of uniquely decodable codes are given. With these conditions one is able to check whether a convolutional code in a PRC is uniquely decodable.

Modulation in which binary data is mapped by a discrete wavelet transform and then communicated as pulse amplitude modulation (PAM) is studied for its immunity to pulsed interference. The presentation is in terms of two specific examples in which the input mappers are a Hadamard transform and a two-stage Haar multirate synthesis filter bank, respectively. The input data stream is segmented into contiguous blocks of four bits (plus or minus 1 s) that are linearly block-transformed into four PAM coefficients whose dependencies can be used to correct some single and multiple erasures resulting from pulsed interference within the block. The encoding transformations are rate 1, nonredundant mappings describable by trellis diagrams and the language of partial response signaling. For pedagogical purposes the example is carried through in the noiseless case where erasures are the sole channel impairment. Noise-fee results for the identity mapping are also given; a comparison reveals that the Haar mapping is the best of the three. Some results for the additive white Gaussian noise case follow, in which case no one method proves uniformly superior to the others.

Spread spectrum communications is a technique in which the transmission bandwidth is much larger than that normally required to transmit at the given data rate. The use of this excess bandwidth provides the system with advantages in the areas of anti-jam communications, high resolution ranging, resistance to multipath fading, and low probability of intercept/detection of the transmissions. The traditional spreading sequences used in spread spectrum communications are the maximal length sequences, due to their good randomness properties as well as their ease of generation. Recently a new class of spreading codes has been developed based upon the time-frequency duality of multirate filter bank structures. Unlike the maximal length sequences, these new codes are not limited to being binary valued. Instead, the elements of the sequences are determined by an optimization process which emphasizes certain desirable code properties. In this paper, spreading codes based upon multirate filter banks are developed for use in a channel characterized by both multiuser and multipath interference. Several sets of codes are designed for various channel conditions and analytical bit error rate results are generated. These results are then compared to those for conventional maximal length sequences.

The Gabor transform (GT) is applied to the super-resolution of noisy dot image on the infrared focal plane array (FPA) for the remote surveillance of aircraft or missiles. A unique solution of this kind of ill-posed problem is possible because we have incorporated the measured or a priori known size information of the engine/nozzle. Yet noise makes the super-resolution ill- conditioned. We surmount this difficulty by incorporating the GT into a modified Papoulis- Gerchberg iteration algorithm. This is possible because the GT is a local Fourier transform (FT), it matches the localized object signal (object size one unit) but mismatches the global nature of noise. In a practical case of a photon-limited signal having a signal to noise ratio as low as 1.3, our approach recognizes a simulated missile plume. We also show additional resolution can be gained if the radar backscattering from the nozzle and other scatterers is fused with the spatially resolved single image pixel.

Multi-sensor data such as visual/IR imagery are difficult to register due to the discrepancy in their grayscale characteristics. This paper presents a wavelet-based point feature extraction algorithm to select distinct and consistent point features across images in order to overcome this difficulty. It incorporates local statistical information of the image intensity to locate point features in multi-resolution contour maps generated using the wavelet transform. Experimental results using real multi-sensor data are presented.

With the new trend of smaller missions in which sensors will be carried on separate platforms, the amount of remote sensing data to be combined will increase tremendously, and will require fast and automatic image registration and fusion. Image registration techniques will help develop 'ready to use' global datasets from multi-instrument/multi-platform/multi-temporal observations, while image fusion will provide new image products summarizing some basic understanding of the original data. These methods will find applications in numerous domains such as Earth science data analysis, map updating, and space exploration. Our work on image registration and fusion focuses on the speed of such methods and on their ability to handle multi-sensor data. These two requirements brought us to the utilization of multi-resolution wavelet transforms to perform such tasks. Our registration algorithm utilizes a wavelet-based multi-resolution search to determine the best transformation between two or more images to be registered. As of now, the algorithm searches for rotation, translation or a composition of both. This algorithm has been tested successfully on uni-sensor images -- landsat-thematic mapper (TM), advanced very high resolution radiometer (AVHRR), and geostationary operational environmental satellite (GOES) data, as well as multi-sensor data such as modis airborne simulator (MAS) with landsat-TM data. The second step in the combination of the data deals with the fusion of the data. This fusion can be considered at two levels; either the fusion occurs after registration of the original data and before any further analysis, or each individual dataset is analyzed independently and then a composite image is created. Both approaches may be considered utilizing a wavelet-based approach. Some preliminary results on image fusion are presented.

A system is presented for compression of hyperspectral imagery. Specifically, DPCM is used for spectral decorrelation, while an adaptive 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 exceeding 41 dB.

The local spectral information of the continuous wavelet transform with Morlet wavelets can, with slight modification, be used to perform 'local' cross spectral analysis with very good time resolution. This 'phase correction' absolutely references the phase of the wavelet transform to the zero time point, thus assuring that the amplitude peaks are regions of stationary phase. Thus phase differences between the 'local spectra' of time series from two spaced receivers can be used to infer time lags as a function of frequency. This method is used to determine apparent phase velocities of atmospheric gravity waves from hydroxyl airglow data. The instrument used, UWOSCR, is a scanning radiometer in the near infra-red, taking a 256 pixel image of the OH airglow every minute. Pixel-to-pixel time lags are used to determine the phase velocity as a function of frequency and time.

This paper presents an approach which addresses both de-noising and contrast enhancement. In a multiscale wavelet analysis framework, we take advantage of both soft thresholding and hard thresholding wavelet shrinkage techniques to reduce noise. In addition, we carry out nonlinear processing to enhance contrast within structures and along boundaries. Feature restoration and enhancement are accomplished by modifying the gain of a signal's variational energy. The multiscale discrete dyadic wavelet transform adapted in this paper is treated as a process for the diffusion of variational energy from a signal stored as the power (scaled variational energy) of wavelet coefficients. We show that a discrete dyadic wavelet transform has the capability to separate feature variational energy from noise variational energy. De- noising and feature enhancement are achieved by simultaneously lowering noise variational energy and raising feature variational energy in the transform domain. We present methods for achieving this objective, including regulated soft thresholding and adaptive nonlinear processing combined with hard thresholding. We have applied this algorithm to synthetic and real signals as well as images with additive Gaussian white noise. Experimental results show that de-noised as well as enhanced signals and images are free from artifacts. Sample analysis and experimental results are presented.

Wavelets have been widely utilized for image analysis and image/video coding. In this paper, we describe how wavelet transform (WT) can facilitate the important computer vision technique -- stereo vision -- by establishing valid constraints in the WT domain and combining some psychophysical knowledge of the human visual system. Because of the displacement presented in the stereo images, continuous WT is suited for stereo image processing and analysis. Four constraints are established and called the smooth component gradient constraints (SCGC) and smooth component Laplacian constraints (SCLC), respectively. To derive them, geometric conditions and psychophysical knowledge for human binocular visual information fusion are combined. These constraints greatly improve the efficiency of stereo matching: matching accuracy and speed is greatly improved, and matching robustness in noisy environment is significantly improved too, compared with traditional algorithms without those constraints. Experiments show encouraging results. The paper discuses the possibility of establishing some further constraints in the WT domain too.

Digital elevation models will be more and more important concerning the generation of geo information systems. Unfortunately, the consumption of image data is very high. To reduce data transfer time and cost high efficient image data compression techniques are required. LuRaTech is under way developing a wavelet based image data compression software for space applications uses, primarily. The question to be answered is how the individual processing steps of digital elevation modeling are influenced by the quality of image data sources. Very important in this scope is the automatic image matching and photogrammetric adjustment process. In this paper we discuss the automatic impact of losses due to data compression on the photogrammetric evaluation of 3-line scanner imagery coming from the modular optoelectronic multispectral scanner (MOMS-02), which has been flown successfully on the German space shuttle mission D2 in April/May 1993. We look at two methods for the quantification of the impact on parallax measurements. The results focus on the comparison of the automatically extracted interest-operator points, the displacement of objects for different compression levels, comparison of mean values of the quality figure, comparison of correlation value, standard deviation of object displacements, number of conjugate points, mean of maximum of correlation coefficient and the histogram statistics of different images for various compression levels.

This paper discusses a method for reconstructing images using a joint transform optical correlator (JTC) architecture. The architecture employs a single liquid crystal television (LCTV) as the spatial light modulator (SLM) operating only on the phase of the incident coherent light. The wavelet transform (WT)-based image decomposition and synthesis is performed in the transform plane instead of the correlation, or wavelet transform plane. While the method presented here does not yield a true reconstruction in the strictest sense, it has many useful characteristics. Experimental results are presented as well.

Frequency-resolved optical gating (FROG) is a technique for measuring ultrashort laser pulses that involves producing a spectrogram of the pulse and then retrieving the intensity and phase of the electric field using a phase-retrieval algorithm. Since noise on experimental FROG traces reduces the performance of the retrieval algorithm, removing the noise is crucial. In previous work we have shown that subtracting the mean of the noise, optimized lowpass filtering, and suppression of the corners of the trace provides an efficient tool for denoising FROG traces. The recent development of wavelet noise-reduction techniques for signal and image processing now provides a new method for attacking this problem. We apply a two- dimensional discrete wavelet transform to the noisy FROG trace, threshold the wavelet coefficients, and perform the inverse wavelet transform to regain the trace. In combination with other noise-filtering methods, this efficiently removes noise from the trace and improves the algorithm's ability to retrieve the intensity and phase of the pulse accurately, especially in fairly low-noise situations, where extremely high accuracy is desired. In addition to wavelet- coefficient thresholding, we also investigate the possibility of using a geometrical scheme for filtering the wavelet coefficients, thus combining data compression and noise reduction.

The wavelet transform is a popular signal processing technique, particularly due to its impressive results in data compression. Its usefulness includes two-dimensional data for use in image processing and three-dimensional data for use in video processing. In image processing, the current trends are for image sizes which require a substantial amount of computing power; an application processing a 1024 by 1024 standard quality image requires many millions of processing steps per image frame. When processing sequences of these images for video, the throughput required is considerable in order to attain even low display rates. Three-based architectures have been proposed to provide this throughput rate by processing pixels in a data parallel fashion. Each level of the wavelet transform is processed using an array or a plane of processing elements operating in parallel on shared or distributed data. The largest of these architectures, the plane-based H-tree design, provides a real-time, pipelineable implementation of the 2DWT, but is costly in terms of VLSI area due to its requirement of O(n²) processors for a n by n data-set. In this paper, we look at methods for improving the practicality of these architectures by reducing the required area for a given problem size. This is achieved by adding extra processors at the root of the tree, which allows processing of larger images with an insignificant addition of hardware in exchange for a detrimental effect on the processing speed. We conclude the paper by presenting area/time trade-offs which can be used to evaluate cost/performance specifications.

Filter bank and multiresolution analysis have been widely used for wavelets. However, for multi-dimensional signals, the convolution algorithm needed for filter bank and multiresolution analysis is too complicated. In this paper, we propose a basic wavelet matrix, which can have either perfect reconstruction or desired result according to the chosen filter properties. The basic wavelet matrix method can be applied to pyramid wavelet decomposition, visual-based wavelet decomposition, tensor product, wavelet packets and adaptive tree-structured decomposition. Edge effects, the choice of filters are also discussed.

A wavelet multiresolution analysis is performed on atmospheric fields simulated by a multilevel 3-dimensional atmospheric boundary layer model. Wavelet cospectra of the vertical wind and potential temperature are calculated and compared with radial Fourier cospectra. The former indicate most of the field variance to have horizontal scales roughly equal to the vertical scale, as should be the case for convectively driven turbulence. Fourier spectra exhibit a minus 3 power law, suggesting that the statistics may depend only on a quantity with units of time. Observations of time- and scale-dependent structures suggest certain physical mechanisms at work. The multiresolution analysis analogue of turbulent energy equations is formulated. This framework supports the proposed physical mechanisms.

Here we present an FFT based architecture and algorithm for computing discrete wavelet transform of one dimensional discrete signals. The presented architecture is non recursive unlike dyadic subband decomposition and discrete wavelet transform coefficients at all resolutions can be generated simultaneously without waiting for generation of coefficients at a higher resolution. For long wavelet filters, this architecture is faster than architectures proposed so far, for DWT computation based on time domain convolvers. This architecture can be fully pipelined and complexity of control circuits for this architecture is much lower as compared to time domain convolvers and their systolic array implementations (which involve complex routing of data) proposed before. In time-domain convolution based architectures, with a single set of convolver, computation of DWT for an N-point signal takes a minimum of N cycles, whereas the proposed architecture, with full hardware implementation (no multiplexing of hardware) and when fully pipelined takes only a fraction of N cycles. This architecture will be suitable for analysis of signals like EEG, seismic data, etc., which are quasi-infinite, one dimensional signal streams. The speed advantage comes by using FFT based frequency domain convolutions and also due to consequent introduction of more parallelism.

It is shown that any convolution operator in the time domain can be represented exactly as a multiplication operator in the time-scale (wavelet) domain. The Mellin transform establishes a one-to-one correspondence between frequency filters (system or transfer functions) and scale filters, which are defined as multiplication operators in the scale domain, subject to the convergence of the defining integrals. Applications to the denoising of random signals are proposed. We argue that the present method is more suitable for removing the effects of atmospheric turbulence than the conventional procedures based on Fourier analysis because it is ideally suited for resolving spectral power laws.

We present a multiresolution dynamic predictor (MDP) based on neural networks for multi- step prediction of a time series. The MDP utilizes the discrete biorthogonal wavelet transform to compute wavelet coefficients at several scale levels and recurrent neural networks (RNNs) to form a set of dynamic nonlinear models for prediction of the time series. By employing RNNs in wavelet coefficient space, the MDP is capable of predicting a time series for both the long-term (with coarse resolution) and short-term (with fine resolution). Experimental results have demonstrated the effectiveness of the MDP for multi-step prediction of intracranial pressure (ICP) recorded from head-trauma patients. This approach has applicability to quasi- stationary signals and is suitable for on-line computation.

This paper presents some closed-form expressions for a continuous wavelet transform (CWT) of a sinusoidal signal using a single-cycle sinusoidal wavelet. The derivations of the forward and inverse transforms are shown explicitly. A plot of the transform for a real sinusoid is also shown.

Understanding of the EKG components P, QRS (R wave), and T is essential in recognizing cardiac disorders and arrhythmias. An estimation method is presented that models the R wave component of the EKG by adaptively computing wavelet parameters using fuzzy logic. The parameters are adaptively adjusted to minimize the difference between the original EKG waveform and the wavelet. The R wave estimate is derived from minimizing the combination of mean squared error (MSE), amplitude difference, spread difference, and shift difference. We show that the MSE in both non-noise and additive noise environment is less using an adaptive wavelet than a static wavelet. Research to date has focused on the R wave component of the EKG signal. Extensions of this method to model P and T waves are discussed.

Wavelet functions have been successfully used in many problems as the activation function of feedforward neural networks [ZB92], [STK92], [PK93]. In this paper, a family of polynomial wavelets generated from powers of sigmoids is described which provides a robust way for designing neural network architectures. It is shown, through experimentation, that function members of this family can present a very good adaptation capability which make them attractive for applications of function approximation. In the experiments carried out, it is observed that only a small number of daughter wavelets is usually necessary to provide good approximation characteristics.

After relating the adaptive wavelet transform to the human visual and hearing systems, we exploit the synergism between such a smart sensor processing with brain-style neural network computing. The freedom of choosing an appropriate kernel of a linear transform, which is given to us by the recent mathematical foundation of the wavelet transform, is exploited fully and is generally called the adaptive wavelet transform (WT). However, there are several levels of adaptivity: (1) optimum coefficients: adjustable transform coefficients chosen with respect to a fixed mother kernel for better invariant signal representation, (2) super-mother: grouping different scales of daughter wavelets of same or different mother wavelets at different shift location into a new family called a superposition mother kernel for better speech signal classification, (3) variational calculus to determine ab initio a constraint optimization mother for a specific task. The tradeoff between the mathematical rigor of the complete orthonormality and the speed of order (N) with the adaptive flexibility is finally up to the user's needs. Then, to illustrate (1), a new invariant optoelectronic architecture of a wedge- shape filter in the WT domain is given for scale-invariant signal classification by neural networks.