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- Restoration
- Multiwavelets
- Fractals
- Frames and Gabor
- Approximation Theory
- Wavelet Filtering
- Wavelet Design
- Feature Extraction
- Implementation
- Splines
- Communication
- Time-Scale Analysis
- Differential Equations
- Denoising
- Motion/Registration
- Compression
- Biomedical Applications
- Coding
- Denoising
- Frames and Gabor
- Fractals
- Biomedical Applications
- New Applications
- Pattern Recognition
- Fractals

Restoration

Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window

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On the base of local criteria of processing quality, a class of local adaptive linear filters for image restoration and enhancement is introduced. The filters work in a running window in the domain of DFT of DCT and have O (size of the window) computational complexity thanks to recursive algorithms of running DFT and DCT. The filter design and the recursive computation of running DCT are outlined and filtering for edge preserved noise suppression, blind image restoration and enhancement is demonstrated.

Wavelet-based interpolation method for nonuniformly sampled signals

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Most digital signal processing methods have an underlying assumption of regularly-spaced data samples. However, many real-world data collection techniques generate data sets which are not sampled at evenly-spaced intervals, or which may have significant data dropout problems. Therefore, a method of interpolation is needed to model the signal on an even grid of arbitrary granularity. We propose the interpolation of nonuniformly sampled fields using a least- square fit of the data to a wavelet basis in a multiresolution setting.

High-resolution satellite image restoration with frames

Jerome Kalifa,
Stephane G. Mallat,
Frederic Falzon,
et al.

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The optics of satellite cameras as well as the satellite movement create a blurring effect to which is added a noise due to the electronics of the CCD captors. We introduce a deconvolution algorithm which is regularized with a hard thresholding technic, in a wavelet frame. The restoration procedure is fast and provides good metrical and perceptual results.

Regularization constraints in lossy compressed astronomical image restoration

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Astronomical images currently provide large amounts of data. Lossy compression algorithms have recently been developed for high compression ratios. These compression techniques introduce distortion in the compressed images and for high compression ratios, a blocking effect appears. A new algorithm based on the regularization theory is proposed for the restoration of such lossy compressed astronomical images. The image is restored scale by scale in a multiresolution scheme and the information lost during the compression is recovered by applying a regularization constraint. The experimental results show that the blocking effect is reduced and some measurements made on a simulated image show that the astrometic and photometric properties of the restored images are improved.

Multiwavelets

Lattice structure for multifilters derived from complex-valued scalar filter banks

Kurt A. Johnson,
Truong Q. Nguyen

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Multiwavelet-based filter banks, unlike filter banks based on scalar wavelets, are able to provide simultaneously orthogonality, linear phase, and short support. However, a general lattice structure for multifilters, analogous to that available for scalar filter banks has yet to be determined.. Such lattice structures have considerable advantages for both theory and design. This paper derives a complete and minimal lattice structure for a class of 2- wavelet multifilters which are based on complex-valued orthogonal scalar filter banks. An example derived from the Daubechies D

_{6}wavelet is presented, along with considerations of how requiring symmetry and higher approximation order restricts the lattice coefficients.
Oblique multiwavelet bases: examples

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Orthogonal, semiorthogonal and biorthogonal wavelet bases are special cases of oblique multiwavelet bases. One of the advantage of oblique multiwavelets is the flexibility they provide for constructing bases with certain desired shapes and/or properties. The decomposition of a signal in terms of oblique wavelet bases is still a perfect reconstruction filter bank. In this paper, we present several examples that show the similarity and differences between the oblique and other types of wavelet bases. We start with the Haar multiresolution to illustrate several examples of oblique wavelet bases, and then use the Cohen-Daubechies-Plonka multiscaling function to construct several oblique multiwavelets.

Multiwavelet-transform-based image compression techniques

Sathyanarayana S. Rao,
Sung H. Yoon,
Deepak Shenoy

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Multiwavelet transforms are a new class of wavelet transforms that use more than one prototype scaling function and wavelet in the multiresolution analysis/synthesis. The popular Geronimo-Hardin-Massopust multiwavelet basis functions have properties of compact support, orthogonality, and symmetry which cannot be obtained simultaneously in scalar wavelets. The performance of multiwavelets in still image compression is studied using vector quantization of multiwavelet subbands with a multiresolution codebook. The coding gain of multiwavelets is compared with that of other well-known wavelet families using performance measures such as unified coding gain. Implementation aspects of multiwavelet transforms such as pre-filtering/post-filtering and symmetric extension are also considered in the context of image compression.

Nonlinear operator related to refinable function vectors

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We study a certain nonlinear operator T from L

^{2}(R, C^{N}) to itself under which every refinable function vector is a fixed point. The iterations T^{n}f of T on any f (epsilon) L^{2}(R, C^{N}) with the Riesz basis property are investigated; they turn out to be the 'cascade algorithm' iterates of f with weights depending on f only. The paper also gives conditions for convergence of T^{n}f to a limit in different topologies.Fractals

Implicit image models in image fractal compression

Geoffrey M. Davis

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Why does fractal image compression work. What properties must an image have for fractal block coders to work well. What is the implicit image model underlying fractal image compression. The behavior of fractal block coders is clear for deterministically self-similar structures. In this paper we examine the behavior of these coders on statistically self-similar structures. Specifically, we examine their behavior for fractional Brownian motion, a simple texture model. Our analysis suggests that the properties necessary for fractal block coders to work well are not so dissimilar from those required by DCT and wavelet transform based coders. Fractal block coders work well for images consisting of ensembles of locally self-similar regions together with locally stationary regions with decaying power spectra, local statistical similarity, and local isotropy. Our analysis motivates a generalization of fractal block coders that leads to substantial improvements in coding performance and also illuminates some of the fundamental limitations of current fractal compression schemes.

Wavelet and fractal analysis of ground-vehicle images

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A large number of terrain images were taken at Aberdeen Proving Grounds, some containing ground vehicles. Is it possible to screen the images for possible targets in a short amount of time using the fractal dimension to detect texture variations. The fractal dimension is determined using the wavelet transform for these visual images. The vehicles are positioned within the grass and in different locations. Since it has been established that natural terrain exhibits a statistical l/f self-similarity property and the psychophysical perception of roughness can be quantified by the same self-similarity, fractal dimensions estimates should vary only at texture boundaries and breaks in the tree and grass patterns. Breaks in the patterns are found using contour plots of the dimension estimates and are considered as perceptual texture variations. Variation in the dimension estimate is considered more important than the accuracy of the actual dimensions number. Accurate variation estimates are found even with low resolution images.

Frames and Gabor

Scaled Gabor representation: a refined time-frequency decomposition

Shidong Li

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A discrete scaled Gabor representation (SRG) is developed to meet the requirements of localized and refined time- frequency representation of signals. SGR generalizes the metaplectic structure by using windows' translation, modulation and dilation as synthesis waveforms. Fundamental features and importance of SGRs are discussed. We derive fast algorithms for the computation of related analysis sequences at different scales. An example of using SGRs for refined time-frequency representation is also demonstrated. A significant feature of SGRs also lies in the fact that they can be realized in a parallel FFT-based implementation structure.

Constructing near-tight wavelet frames by neural networks

Xin Li

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Suppose that (sigma) is a sigmoidal function which is the activation function of a neural network. Under certain assumptions on the derivatives of (sigma) , we show that a simple linear combination of dilates and translates of (sigma) generates a near tight wavelet frame for L

^{2}(R), which is then used in constructing approximation to multivariate functions by neural networks with one hidden layer.
Multidimensional nonseparable Gabor expansions

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Various generalizations of the classical Gabor expansion are considered. Studying the frame operator via the Kohn- Nirenberg correspondence allows to obtain straightforward structural results for the situation of (i) nonseparable prototypes, and/or (ii) nonseparable time-frequency sampling lattices, and/or (iii) multi-prototypes. For such general Weyl-Heisenberg frames, it is shown how to reformulate the Janssen representation of the frame operator and the Wexler- Raz result. Moreover, an analysis of the analysis operator is performed that leads to quantitative results about the variety of admissible analysis/synthesis prototypes.

Riesz frames and finite-dimensional approaches to problems in frame theory

Ole Christensen

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The inverse frame operator plays a big role in frame theory. For example we need this operator if we want to calculate the frame coefficients or solve a moment problem. For practical purposes it can be a problem that the frame operator is an operator on a Hilbert space, which usually is infinite dimensional. Our purpose here is to find approximative solutions to the problems above, using finite subsets. We find conditions implying that the approximative solutions converge to the correct solutions. Most of the results concern Riesz frames are to be defined in the paper.

Approximation Theory

Integration-free projection of a sampled signal on a multiresolution analysis ladder space via approximation theory

Dumitru Mihai Ionescu,
Mark A. Wickert,
Jacqueline J. Hine

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Let H

^{s}be the Sobolev space of exponent s >= 0. Let V_{i}, i (epsilon) Z,...V_{i}is contained in V_{i-1}is contained in ..., be the ladder spaces of a multiresolution analysis (MRA) in L^{2}(R) associated with a compactly supported scaling function (phi) (epsilon) H^{(sigma}) which verifies supp(phi) equals [ 0, L-1 ], L (epsilon) N, (sigma) > 1/2. Let f (epsilon) L^{2}(R) be compactly supported with countable many discontinuity points and verifying suppf equals [ t_{1}, t_{2}], where 0 < (mu) (suppf) < (infinity) and, for i equals 1,2 and some integer m$_0), t_{i}equals n_{i}(m_{0})2^{mo}with n_{i}(m_{0}) (epsilon) Z. The goal is to approximate f by a function in V_{m}which approaches f in some sense as the scale becomes finer, reproduces the values of f at certain equally spaced, arbitrarily close points and is not necessarily the orthogonal projection of f on V_{m}. Such non-orthogonal projections can be used to start the pyramid algorithm or to approximate f, as an alternative which is better than using the samples themselves in place of the true inner products and faster than integrating for the inner products. The almost uniform convergence feature allows interactive control on the selection of an initial sampling rate that is capable of preserving details down to a desired resolution.
High-accuracy reconstruction from wavelet coefficients

Fritz Keinert,
Soon-Geol Kwon

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The accuracy of the wavelet approximation at resolution h equals 2

^{-n}to a smooth function f is limited by O(h^{n}), where N is the number of vanishing moments of the mother wavelet (psi) . For any positive integer p, we derive basis functions which allow us to recover a smooth f from its wavelet coefficients with accuracy O(h^{p}). Related formulas for recovering derivatives of f are also given.
Uniform approximation of wavelet coefficients

T. D. Svenson,
Jo A. Ward,
K. J. Harrison

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Mallat's pyramid algorithm relates the scaling coefficients of a function at one level to the scaling and wavelet coefficients at lower levels. In practice, the scaling coefficients are estimated at some level m, and the algorithm is used to produce estimates of the scaling and wavelet coefficients at lower levels. Initial errors propagate to lower level estimates. this paper descries conditions under which this process generates estimates which are uniformly reliable at a particular level, and under which the errors at that level tend uniformly to zero as m increases.

Upper bounds of wavelet spectra on the class of discrete Lipschitzian signals

Arthur Ashot Petrosian

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Application of fast discrete orthogonal transforms with various basis functions for data compression and efficient signal coding occupies a special place in the evolution of spectral representations. This has become more apparent with the development of different wavelet and wavelet-packet transforms. Two basic compression procedures, known as zonal and threshold coding, are commonly being applied to the spectral vector. The optimal zonal coding method provides a minimum error of reconstruction for certain compression ratio. In order to determine optimal zonal coding method for the chosen transform one has to obtain the estimates of its spectra on a given class of signals. This task was considered on a general class of input vectors for classical discrete orthogonal transforms, including Fourier, Hartley, cosine, sine, as well as Walsh and Haar transforms. In this paper, we expand those results on various wavelet transforms by evaluating the upper bounds of their spectra. These estimates allow not only to a priori select the wavelet coefficient packets that have minimum input in signal reconstruction, but also to compute the maximum mean-square errors of reconstruction for a particular compression ratio and to analyze efficacy of different wavelets based on that criterion.

Wavelet Filtering

Wavelet filtering in the scale domain

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For a given basic wavelet (psi) (t), two distinct correspondences (called C1 and C2) are established between frequency filters, defined in the frequency domain through multiplication by a transfer function W(f), and scale filters, defined in the wavelet domain through multiplication by a scale transfer function w((sigma) ). W(f) is obtained by performing a scaling convolution of w((sigma) ) with (psi) (f)* (for C1) or its spectral energy density (psi) (f)

^{2}(for C2). For a large class of transfer functions W(f), this relation can be solved for w((sigma) ) by applying the Mellin transform. We call such frequency filters and their associated time-domain convolution operators C1- or C2-admissible with respect to (psi) . In particular, the identity operator (W(f) equalsV 1) is C2-admissible if and only if the wavelet (psi) is admissible in the conventional sense. The implementation of the correspondence C1 is computationally simpler than C2, but C2 can be generalized to time-dependent filters. Applications are proposed to the analysis of atmospheric turbulence data and wideband Doppler filtering.
Multiscale suboctave wavelet transform for denoising and enhancement

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This paper describes an approach for accomplishing sub- octave wavelet analysis and its discrete implementation for noise reduction and feature enhancement. Sub-octave wavelet transforms allow us to more closely characterize features within distinct frequency bands. By dividing each octave into sub-octave components, we demonstrate a superior ability to capture transient activities in a signal or image more reliably. De-noising and enhancement are accomplished through techniques of minimizing noise energy and nonlinear processing of transform coefficient energy by gain.

Fast approximate Fourier transform via wavelets transform

Haitao Guo,
C. Sidney Burrus

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We propose an algorithm that uses the discrete wavelet transform as a tool to compute the discrete Fourier transform (DFT). The Cooley-Tukey FFT is shown to be a special case of the proposed algorithm when the wavelets in use are trivial. If no intermediate coefficients are dropped and no approximations are made, the proposed algorithm computes the exact results, and its computational complexity is on the same order of the FFT. The main advantage of the proposed algorithm is that the good time and frequency localization of wavelets can be exploited to approximate the Fourier transform for many classes of signals resulting in much less computation. Thus the new algorithm provides an efficient complexity vs accuracy tradeoff. When approximations are allowed, under certain sparsity conditions, the algorithm can achieve linear complexity. It has been shown that the thresholding of the wavelet coefficients has near optimal noise reduction property for many classes of signals. We show that for the same reason, the proposed algorithm also reduces the noise while doing the approximation. If we need to compute the DFT of noisy signals, the proposed algorithm not only can reduce the numerical complexity, but also can produce cleaner results. In summary, we propose a novel fast approximate Fourier transform algorithm using the wavelet transform. Since wavelets are the conditional basis of many classes of signals, the algorithm is very efficient and has built-in de-noising capacity.

Wavelet-based decompositions for nonlinear signal processing

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Nonlinearities are often encountered in the analysis and processing of real-world signals. This paper develops new signal decompositions for nonlinear analysis and processing. The theory of tensor norms is employed to show that wavelets provide an optimal basis for the nonlinear signal decompositions. The nonlinear signal decompositions are also applied to signal processing problems.

Wavelet Design

Malvar wavelets on sphere

Xiang-Gen Xia,
Lei Zhang

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In this paper, Malvar wavelets defined on a unit sphere are introduced. They are constructed by using a similar technique of constructing 2D Malvar wavelets, where one dimension (theta) is periodic on (0, 2(pi) ) and the other dimension (phi) is not periodic on (0, (pi) ).

Robust detection via subspace design

Chuang He,
Jose M. F. Moura

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The goal of this paper is to describe a simple receiver with good performance for multipath communications channels. The new receiver is 'almost' invariant or robust to the multipath channel distortion. It preserves the simplicity of a traditional receiver-correlator receiver and at the same time approximates the computational intensive subspace receiver which is maximally invariant. The maximally invariant test statistic is the energy of the orthogonal projection of the received signal on the multipath signal subspace S. Calculating this orthogonal projection directly is a difficult multi-dimensional nonlinear optimization problem. Instead, we design a representation subspace G to approximate S. The gap metric is used as the measure between subspaces. The gap metric is closely related to the principle angles between subspaces. Wavelet multiresolution tools are called upon to facilitate the subspace design. Once we have designed the representation subspace G, we use the energy of the orthogonal projection of the received signal on G as the new test statistic. Simulation results demonstrate the robustness of the new receiver.

Theory and design of two-dimensional nonseparable shift-invariant filter banks

Y. Hui,
C. W. Kok,
Truong Q. Nguyen

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Shift-variance limits the usage of 2D maximally-decimated filter banks in many image processing applications where shift-invariance is desired. Several approaches have been proposed to overcome this problem for both 1D and 2D multirate systems. All of the existing methods are 1D-based algorithms, i.e. dealing with 2D image by separable processing, and thus do not have the advantages that a non- separable 2D system offers. In this paper, a framework for the theory and the design of 2D shift-invariant filter banks with non-separable sampling and/or non-separable filters are presented. There are three major advantages that the proposed 2D shift-invariant filter banks have over the separable 2D systems using the existing methods. First of all, it is a non-separable 2D approach and therefore has the advantages that a 'true' 2D approach offers. Secondly, the resulting 2D filter banks and wavelets have the conventional dyadic structure while possessing better SI property. Finally, the proposed filter bank is independent of input images. Design examples are presented.

Lifting scheme and finite-precision error-free filter banks

Carolyn Pfeffer Johnston

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A means of extending wavelets and multiresolution decompositions to the finite field case is demonstrated, which uses the lifting scheme developed by W. Sweldens. The advantages and disadvantages of using finite field discrete wavelet transforms in coding schemes are discussed, and the binary case is examined in detail. The use of the new scheme is expected to pay off in practice by permitting fast computation in small memory on crude platforms; in particular, possible applications to downlinking satellite- borne SAR systems are discussed.

Unitary mappings between multiresolution analysis of L2(R) and a parameterization of low-pass filters

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This paper examines classes of unitary operators of L

^{2}(R) contained in the commutant of the shift operator, such that for any pari of multiresolution analyses of L^{2}(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions. We show that the Dai-Larson unitary parameterization of orthonormal wavelets is not suitable for the study of scaling functions. These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary operators acting on L^{2}([ -(pi) , (pi) ]), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets by passing the spectral factorization process.Feature Extraction

2D multirate Bayesian framework for multiscale feature detection

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This paper presents a novel methodology for designing a 2D multiscale feature detector, which consists of a filter bank and a maximum a posteriori (MAP) classifier. The framework assumes the availability of a one-scale filter with a particular indicator response to the desired feature. This filter is used to generate a multiscale set of discrete filters by sampling on a rectangular lattice to preserve the indicator responses at all the scales. The net step in the framework consists of designing the filter bank to approximate the generated filters. A 2D MAP detector is then designed to minimize detection errors. With the assumption of known feature, the resulting detector depends only on the filter bank, and not on the noise. Relaxing this assumption yields a detection algorithm that is noise dependent and computationally intensive. The framework is applied to edge detection in a noisy environment, and the results indicate efficient detection. Moreover the 2D MAP can find feature end-points by direct processing of the image. This is unlike conventional methods where edges need to be first detected and then processed to locate the corners. Examples are presented to demonstrate the algorithm.

Detection and classification of acoustic transients using wavelet transforms

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The search for an optimum mother wavelet for the detection of transient signals can be made tractable if one realizes that all transients are generated by a physical system. The canonical model for many transients can be derived from a simple parallel resonant circuit. It is shown that the impulse response from this circuit acts as a mother wavelet and parameterizes the circuit via its resonant frequency and quality factor. Once the family of transients is defined, a wavelet-based estimator correlator is used to detect and classify transient signals with unknown frequency and arrival times.

Hypercomplete circular harmonic pyramids

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In this contribution we present a steerable pyramid based on a particular set of complex wavelets named circular harmonic wavelets (CHW). The proposed CHWs set constitutes a generalization of the smoothed edge wavelets introduced by Mallat, consisting of extending the local differential representation of a signal image from the first order to a generic n-th order. The key feature of the proposed representation is the use of complex operators leading to an expansion in series of polar separable complex functions, which are shown to possess the space-scale representability of the wavelets. The resulting tool is highly redundant, and for this reason is called hypercomplete circular harmonic pyramid (HCHP), but presents some interesting aspects in terms of flexibility, being suited for many image processing applications. In the present contribution the main theoretical aspects of the HCHPs are discussed along with some introductory applications.

Multiscale merging by a mixed approach for edge detection

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We present an edge detection algorithm from a multiscale analysis, for the real images with which we are confronted in our laboratory. Our work is based on a generalization of the Canny-Deriche filter. This filter is used to generate a frame of wavelets. We impose a minimal degree of freedom in the decomposition of the image on the frame by controlling the lower bound A and upper bound B of the frame. This minimum, obtained with A equals B, ensures the stability of the numerical analysis of the image. We propose a new criterium, the separating power, which allows to choose in a deterministic way the intrinsic parameters of the filters. This criterium introduces the concept of detection resolution which is a new concept in edge detection. Also, we evaluate the maximal delocalization obtained at the limit of the separating power. Our merging algorithm is based on human vision which zooms back and forth on the image in order to identify global structures or details in the image. The algorithm searches for candidate edge points in a certain neighborhood through the scales, in an independent way in the direction of increasing scale and in the direction of decreasing scales. The mapping points are selected and classified in order of priority. The merging method can be followed by a segmentation method selected from existing ones thus enabling a great adaptability of the complete system in the edge sense. Examples of results obtained with a (beta) -version of the method are presented.

Pyramidal vision model applied to astronomical images

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A multiscale vision model based on a pyramidal wavelet transform is described in the present paper. The pyramidal wavelet algorithm is modified in order to satisfy a correct sampling at each scale. Objects are defined by trees of statistically significant coefficients in the wavelet transform space. The object images are then restored using the conjugate gradient method. By comparing with the model based on the a trous algorithm, tests on simulated and real images show that this model presents a good compromise between analysis quality and the memory space and computation time needed.

Character recognition using a biorthogonal discrete wavelet transform

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We present an approach to off-line optical character recognition for hand-written or printed characters using for feature extraction and classification biorthogonal discrete wavelet transform. Our aim is to optimize character recognition methods independently of printing styles, writing styles and fonts used. Characters are identified with their contours, thus characterized from their curvature function. Curvature function is used for feature extraction while classification is accomplished by LVQ algorithms. This method achieves great recognition accuracy and font insensitivity requiring only a small training set of characters.

Implementation

LIFTPACK: a software package for wavelet transforms using lifting

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We present LIFTPACK: a software package written in C for fast calculation of 2D biorthogonal wavelet transforms using the lifting scheme. The lifting scheme is a new approach for the construction of biorthogonal wavelets entirely in the spatial domain, i.e., independent of the Fourier transform. Constructing wavelets using lifting consists of three simple phases: the first step or lazy wavelets splits the data into two subsets, even and odd, the second step calculates the wavelet coefficients as the failure to predict the odd set based on the even, and finally the third step updates the even set using the wavelet coefficients to compute the scaling function coefficients. The predict phase ensures polynomial cancelation in the high pass and the update phase ensures preservation of moments in the low pass. By varying the order, an entire family of transforms can be built. The lifting scheme ensures fast calculation of the forward and inverse wavelet transforms that only involve FIR filters. The transform works for images of arbitrary size with correct treatment of the boundaries. Also, all computations can be done in-place.

Fast parallel implementation of the wavelet-packet best-basis algorithm on the MP-2 for real-time MRI

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Adaptive signal representations, such as those determined by best-basis type algorithms, have found extensive application in image processing, although their use in real time applications may be limited by the complexity of the algorithm. In contrast to the wavelet transform which can be computed in O(n) time, the full wavelet packet expansion required for the standard best basis search takes O(n log n) time to compute. In the parallel work, however, the latter transform becomes attractive to implement, due to a theoretical speedup of O(log n) when the number of processors equal the number of data elements. This note describes near real-time performance obtained with a parallel implementation of best basis algorithms for wavelet packet bases. The platform for our implementation is a DECmpp 12000/Sx 2000, a parallel machine identical to the MasPar MP-2. The DECmpp is a single instruction, multiple data system; such systems support a data parallel programming model, a model well suited to the task at hand. We have implemented the 1D and the 2D WPT on this machine and our results show a significant speedup over the sequential counterparts. In the 1D case we almost attain the theoretical speedup, while in the 2D case we increase execution speed by about two orders of magnitude. The current implementation of the 1D transform is limited to signals of length 2048, and the 2D transform is limited to images of size: 32 X 32, 64 X 64, and 128 X 128. We are currently working on extending our transform to handle signals and images of larger size.

Comparison of algorithms for the fast computation of the continuous wavelet transform

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We introduce a general framework for computing the continuous wavelet transform (CWT). Included in this framework is an FFT implementation as well as fast algorithms which achieve O(1) complexity per wavelet coefficient. The general approach that we present allows a straight forward comparison among a large variety of implementations. In our framework, computation of the CWT is viewed as convolving the input signal with wavelet templates that are the oblique projection of the ideal wavelets into one subspace orthogonal to a second subspace. We present this idea and discuss and compare particular implementations.

Fast-wavelet compass edge detector

D. J. Hebert,
HyungJun Kim

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As an approach to the wavelet detection of local scale and orientation in 2D images we make use of a well known, computationally efficient triangulation of the image domain and some of its lesser-known properties. We choose wavelets supported by cartesian and quincunx lattice squares, antisymmetric about the diagonal. Computational algorithms are based on properties of the triangulations such as the following: the cells of the triangulation form the leaves of a binary tree and the nodes of a directed graph consisting of a simple cycle; the cells are also identified with blocks of interlaced quadtrees consisting of cartesian and quincunx lattice points which also form the vertices of the cells. Pyramid algorithms based on hierarchical triangular scanning of the pixels and half-wavelet-supporting triangles provide efficient encoding and decoding based on local triangle data and stacks. As examples we introduce a Haar wavelet which detects diagonals of squares and we construct a triangular version of the TS wavelet transform which has been recently proposed as an efficient approach to lossless and lossy image compression. We render an edge-enhanced image by reconstruction from significant coefficients of edge- detecting wavelets.

Methods for regular VLSI implementations of wavelet filters

Andreas Klappenecker,
Volker Baumgarte,
Armin Nueckel,
et al.

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We investigate three approaches to VLSI implementation of wavelet filters. The direct form structure, the lattice form structure, and an algebraic structure are used to derive different architectures for wavelet filters. The algebraic structure exploits conjugacy properties in number fields. All approaches are explained in detail for the Daubechies 4- tab filters. We outline the philosophy of a design method for integrated circuits.

Splines

Wavelets based on splines: an application

Pramila Srinivasan,
Leah H. Jamieson

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In this paper, we describe the theory and implementation of a variable rate speech coder using the cubic spline wavelet decomposition. In the discrete time wavelet extrema representation, Cvetkovic, et. al. implement an iterative projection algorithm to reconstruct the wavelet decomposition from the extrema representation. Based on this model, prior to this work, we have described a technique for speech coding using the extrema representation which suggests that the non-decimated extrema representation allows us to exploit the pitch redundancy in speech. A drawback of the above scheme is the audible perceptual distortion due to the iterative algorithm which fails to converge on some speech frames. This paper attempts to alleviate the problem by showing that for a particular class of wavelets that implements the ladder of spaces consisting of the splines, the iterative algorithm can be replaced by an interpolation procedure. Conditions under which the interpolation reconstructs the transform exactly are identified. One of the advantages of the extrema representation is the 'denoising' effect. A least squares technique to reconstruct the signal is constructed. The effectiveness of the scheme in reproducing significant details of the speech signal is illustrated using an example.

Construction of shift-orthogonal wavelets using splines

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We present examples of a new type of wavelet basis functions that are orthogonal across shifts, but not across scales. The analysis functions are low order splines while the synthesis functions are polynomial splines of higher degree n

_{2}. The approximation power of these representations is essentially as good as that of the corresponding Battle- Lemarie orthogonal wavelet transform, with the difference that the present wavelet synthesis filters have a much faster decay. This last property, together with the fact that these transformation s are almost orthogonal, may be useful for image coding and data compression.
Bivariate box splines for image processing

Mingjun Lai

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Bivariate box splines for image interpolation, enhancement, digital filter design, subband coding bank, hexagonal filtering will be discussed. Some existing and new results will be presented. A computational method for box spline image interpolation and box spline digital filters are included.

Spline wavelet transforms II

Tian-Xiao He

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In this paper, we will discuss integral wavelet transforms and orthogonal multiresolution analysis associated with spline functions in shift-invariant spaces of B-splines. A recurrence relation formula and the corresponding algorithm about the B-wavelets will also be given.

Communication

Applications of nonmaximally decimated multirate filter banks in intersymbol interference cancellation

Xiang-Gen Xia

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In this paper, we study applications of nonmaximally decimated multirate filterbanks (NMDMF) in a communication channel with intersymbol interference (ISI), where the ISI transfer function may not have inverses, such as frequency- selective fading channels. The NMDMF plays the preprocessing role. We present a necessary and sufficient condition on an FIR ISI transfer function in terms of its zero set such that an FIR NMDMF with N channels and decimation by K, 0 < K < N, for the encoder exists so that the decoder is able to recover the original signal with an FIR synthesis system, where the integers K and N are arbitrarily fixed. The condition is easy to check and satisfy. In addition to the conditions on the channel transfer functions, the practical design of the FIR NMDMF precoders and their FIR synthesis banks for the reconstruction is also given. A numerical example is presented to illustrate the theory.

Wavelet-packet hopping

Tim N. Davidson,
Kon Max Wong

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Wavelet packet division multiplexing (WPDM) is a multiplexing scheme in which the message signals are coded onto wavelet packet basis functions for transmission. A feature of WPDM is that the coding waveforms overlap in time and frequency, which can provide greater capacity and greater immunity to certain channel imperfections that time division and frequency division multiplexing. By analogy with frequency-hopped communication schemes, the WPDM scheme can be further enhanced by providing a framework for hopping the transmission parameters of the scheme in a pattern which is known by the receiver. In this paper, we exploit the underlying filter back structure of WPDM and the recent development of time-varying filter banks for signal analysis to provide a framework for slow wavelet packet hopping which does not compromise the data rate.

Performance of wavelet-packet division multiplexing in impulsive and Gaussian noise channels

Jiangfeng Wu,
Kon Max Wong,
Qu Jin,
et al.

Show abstract

Wavelet packet division multiplexing (WPDM) is a multiple signal transmission scheme based on wavelet packets. Due to its bandwidth efficiency, channel assignment flexibility, and inherent security features, it has attracted considerable interest. However, before it can be employed as a practical multiplexing technique, the performance of WPDM in various channels has to be fully investigated. In this paper, we explore the system performance when the channel includes impulsive noise as well as Gaussian noise. After briefly reviewing the system model of WPDM, we derive an expression for the probability of error for WPDM in the presence of both impulsive and Gaussian noise. The derivation extends previous work on the performance of digital communication systems when impulsive noise and Gaussian noise are the causes of error to WPDM. For the purpose of comparison, the performance of a TDM system in the same channel environment in also evaluated. The calculations and corresponding simulation results illustrate that WPDM can provide greater immunity to impulsive noise than TDM. This result is intuitively pleasing since one noise burst, which can destroy one bit in TDM, will be spread over all the channels of a WPDM system. In each channel, the spread effects of the impulsive noise may not be strong enough to cause an error.

Time-Scale Analysis

Psychoacoustic frequency scales versus frequency warping in scale cepstrum

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In this paper, we derive a frequency-warping function by analyzing speech data obtained from the TIMIT database. Until now, numerous frequency scales have been proposed, based purely on psychoacoustic studies. Many speech recognition algorithms have been using such frequency scales for the spectral analysis at the signal processing front- end. The motivation for the use of such psychoacoustic frequency scales, is that, since these are based on the properties of the human auditory perception, they may provide accurate representation of the relevant spectral information in speech. Since the preference of one scale over another is ad hoc, and since the goal is to achieve better recognition, experiments are conducted to determine if better recognition rates are indeed obtained using any one such scale. In this paper, we analyze actual speech data, and present evidence of the kind of frequency-warping that may be necessary to achieve speaker-independent recognition of vowels. This provides us with the motivation to use such frequency-warping functions in speech recognition. Surprisingly, the frequency-warping obtained is similar to the Mel-scale obtained from psychoacoustic studies. This suggests that the ear may be using such a frequency-warping to remove extraneous speaker-specific information, while identifying and recognizing phonemes.

Wavelet quadrature methods for detecting coherent structures in fluid turbulence

Jon Harald Kaspersen,
Lonnie H. Hudgins

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The energy cascade found in fully developed fluid turbulence is known to originate as large-scale organized motions called coherent structures. The process of detecting, locating, and tracking these coherent structures is therefore of central importance to the continued study of turbulence. For certain types of flow, these structures can be associated with quadrature events between the streamwise- and transverse-instantaneous velocities. A number of researchers have begun applying wavelet-based methods to the problem of coherent structure detection. Significant performance improvements over other existing methods have already been reported. In this paper, we describe several variations of the standard wavelet cross-transform suitable for use as quadrature detectors. One adaptation simultaneously employs two different analyzing wavelets: one with odd symmetry and one with even symmetry. With these wavelets, the resulting cross-transform can be made to detect quadrature events in the time-scale plane, and thus isolate coherent structures very effectively. The method is described, and demonstrated for cylinder wake flow data. Its performance is then quantitatively compared to some of the popular alternative techniques for detecting coherent structures.

Analysis of chirp signals by time-frequency localization frames

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A theory of frames that extend Gabor analysis by including chirping is discussed. The chirping parameter in these 'time-frequency localization frames' depends on time and/or frequency shift parameters that can be adapted to analyze and detect chirps in noisy signals. Radar/sonar applications are outlined. The frame theory is motivated by a generalized notion of square-integrable group representation developed by Ali, Antoine, and Gazeau, together with ideas in Baraniuk's thesis on a metaplectic extension of Cohen's class.

Wavelet-based higher-order time-scale analysis

Sapna Mehta,
Edward J. Powers,
In-Seung Park

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The objective of this paper is to investigate and provide insight into wavelet-based higher-order time-scale analysis, with an emphasis on the third-order statistic, the bispectrum. We also use a wavelet-based higher-order time- scale approach to detect and quantify intermittent phase- locking of first- and second-order components in laboratory generated random sea waves.

Wavelet analysis of observed geopotential and wind: blocking and local energy coupling across scales

Aime Fournier

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Atmospheric blocking during three unusual winter months is studied by multiresolution analysis and a wavelet based adaptation of traditional Fourier series based energetics. We demonstrate that blocking, in part a large and localized meteorological phenomenon, is largely described by just the largest scale part of the multiresolution analysis. New forms of the transfer functions of kinetic energy with the mean and eddy parts of the atmospheric circulation are introduced. These quantify the spatially localized conversion of energy between scales. A new accounting method for wavelet indexed transfers permits the introduction of a physically meaningful localized scale flux function. These techniques are applied to the data, and some support is found for the hypothesis that blocking is partially maintained by an inverse cascade.

Differential Equations

Adaptive pseudowavelet algorithms for solving nonlinear partial differential equations

James M. Keiser,
Gregory Beylkin

Show abstract

We numerically solve nonlinear partial differential equations of the form u

_{t}equals Lu + Nf(u) where L nd N are linear differential operators and f(u) is a nonlinear function. Equations of this form arise in the mathematical description of a number of phenomena including, for example, signal processing schemes based on solving partial differential equations or integral equations, fluid dynamical problems, and general combustion problems. A generic feature of the solutions of these problems is that they can possess smooth, non-oscillatory and/or shock-like behavior. In our approach we project the solution u(x,t) and the operators L and N into a wavelet basis. The vanishing moments of the basis functions permit a sparse representation of both the solution and operators, which has led us to develop fast, adaptive algorithms for applying operators to functions, e.g. Lu, and computing functions, e.g. f(u) equals u^{2}, in the wavelet basis. These algorithms use the fact that wavelet expansions may be viewed as a localized Fourier analysis with multiresolution structure that is automatically adaptive to both smooth and shock-like behavior of the solution. In smooth regions few wavelet coefficients are needed and in singular regions large variations in the solution require more wavelet coefficients. Our new approach allows us to combine many of the desirable features of finite-difference, (pseudo) spectral and front-tracking or adaptive grid methods into a collection of efficient, generic algorithms. It is for this reason that we term our algorithms as adaptive pseudo- wavelet algorithms. We have applied our approach to a number of example problems and present numerical results.
Multiscale nonlinear system modeling using wavelet networks

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Using wavelet networks, it is possible to capture the characteristics of non-linear dynamic systems in a multi- scale modeling strategy. Starting from the coarsest approximation we go step-wise to the finer scales. At each step the error signal or the residue of the system is modeled. This procedure is repeated until the residual drops below some modeling error bound. The modeling is carried out using compactly supported biorthogonal wavelets. By choosing appropriate wavelet basis, it is possible to obtain a near optimal model.

Denoising

Translation- and direction-invariant denoising of 2D and 3D images: experience and algorithms

Thomas P.-Y. Yu,
Arne Stoschek,
David L. Donoho

Show abstract

Removal of noise from 2D and 3D datasets is a task frequently needed in applications typical examples including in magnetic resonance imaging, in seismic exploration, and in video processing. We are currently interested in visualizing macromolecular structures of biological specimen, in which slices of very noisy electron microscopy (EM) images are volume rendered. Volume rendering of those datasets without any denoising normally gives very 'foggy' results that are not very informative. The wavelet and image processing communities have proposed in the past decade various multiscale image representations, many of which are of potential use for image de-noising. One of our goals here is to explore the importance of translation and direction invariance to the quality of reconstruction, which leads us to study the use of various tight frames for image reconstruction. We have developed 2D translation invariant transforms for both the isotropic and anisotropic wavelet bases. These allow us to develop a 2D analog of the 1D translation invariant de-noising algorithm proposed by Coifman and Donoho. We have also developed algorithms for implementing directionally-invariant de-noising for digital images. We have experiments to measure the relative importance of translation- and direction-invariance for both isotropic and anisotropic transforms. We also are exploring how to apply tight frames for linear inversion of noisy indirect data, which is what ultimately is needed in EM tomography.

Application of the wavelet transform in seismic data processing for the development of new noise reduction techniques

Fotios P. Kourouniotis,
Robert F. Kubichek,
Nicholas G. K. Boyd III,
et al.

Show abstract

New techniques for developing more efficient noise reduction schemes are presented and implemented by applying the wavelet transform (WT) to a series of stationary and non- stationary signals. Their effectiveness is illustrated with specific applications to both real and synthetic seismic data, and the superiority over Fourier transform (FT) based methods is demonstrated. These methods aim at the efficient reduction of the effects that surface waves, airwaves, and direct waves can have on the interpretation of a seismic record. We first apply the WT on each trace in a common- depth-point gather and then perform stacking in the WT domain and compute both the mean and median transforms. Then, the signal-to-noise ratio of the stacked transforms is estimated and used as a criterion to improve the quality of the transformed data, and finally the total energy in the stacked WT plane is computed and redistributed in order to boost weak events. The advantage of stacking in the WT domain is that it allows for detection of weak reflections overpowered by high amplitude surface and air waves. Additionally, it is shown that by frequency modulating a mother wavelet, further attenuation of surface waves, airwaves, and first breaks may be achieved.

Multiresolution adaptive noise filtering based on Laplacian pyramids

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A recently investigated approach to noise filtering in digital images consists of considering a multiresolution decomposition of the input image, and applying a different adaptive filter to each resolution layer. The wavelet decomposition has been employed for multiresolution noise- reduction, thanks to its capability to capture spatial features within frequency subbands. Conversely, Laplacian pyramids (LP) look attractive because of their full band- pass frequency property, which enables connected image structures to be represented on multiple scales. The idea of the present work is to apply an adaptive minimum mean squared error filter to the connectivity-preserving different resolution layers into which the noisy image is decomposed. For natural images, each layer of the LP is characterized by a signal-to-noise ratio (SNR) that decreases for increasing spatial resolution. Therefore, each filter may be tuned to the SNR of the related layer, so as to preserve the spatial details of the less noisy layers to a larger extent. Once all the resolutions, including the base-band, have been adaptively smoothed, a noise-filtered image version is achieved by recombining the layers of the LP. Theoretical frameworks are developed for both additive and multiplicative noise models. Experimental results of de- noising carried out on images with simulated noise and on true synthetic aperture radar images validate the potentiality of the approach in terms of both SNR improvement and visual quality.

Speckle filtering by wavelet analysis and synthesis

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In speckled radar images, filtering must achieve a tradeoff between smoothing of homogeneous areas and edge and texture preservation. Multiscale analysis splits up the image information content, such as edges and texture, according to a scale factor by successive lowpass and highpass filterings followed by downsampling. The speckle noise is present on each downsampled image. Each image level is then filtered in order to reduce the speckle noise. High frequency images are processed by median filtering or spatial filtering, or by using a threshold. On low frequency images a distinction is made between homogeneous areas, textural areas and areas including edges according to the values of the variation coefficient. Each class is processed differently. A Wiener filter including a multiplicative noise hypothesis for the speckle is used for textured areas. For homogeneous areas the pixel value is simply replaced by the mean value. For areas containing edges, the pixel value is let unchanged. The filtered image is finally obtained y synthesis from these images. This algorithm has been applied to an ERS1 image.

Motion/Registration

Motion estimation using complex wavelets

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This paper describes a new wavelet-based approach to the motion estimation problem for digital video. A complex- valued discrete wavelet transform is used to decompose each frame into a subsampled directionally bandpass filtered hierarchy. The transform is defined so that at each level there is an approximate correspondence between local translation and coefficient phase shift. This relationship is used to estimate motion with each orientation subband. The estimates are combined over all orientations and scales using a coarse-to-fine refinement strategy to produce a fractional-pel accurate motion field with a directional confidence measure. The technique is suitable for video compression schemes and can also be used for stereo vision and image registration.

Hierarchical multiresolution technique for image registration

Radka Turcajova,
Jaroslav Kautsky

Show abstract

Our aim is to derive a fully automatic method for highly accurate registration of large low quality images of objects in a distance. A global geometric transformation is used to model the displacement; the parameters of the transformation are then determined by fitting to a field of local displacement estimates which are computed by normalized local cross-correlation matching. Because the displacements can be large, direct matching would be computationally too expensive. This can be resolved by using image pyramids and hierarchical refining of the estimate of transformation describing the displacement. Replacing the standard image pyramids by wavelet decompositions of images is studied in this paper. The proposed approach is based on merging the local displacement estimates from different subbands and applying an iterative algorithm which fits the transformation only to those local estimates that are more likely to be correct. The result of tests on both genuine and artificially created pairs of misaligned images are presented and different possible strategies and suitability of particular wavelet bases are discussed.

General stereo image matching using symmetric complex wavelets

HePing Pan

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General stereo image matching provides an adequate but hard problem with sufficient complexity, with which the potential of wavelets may be exploited to a full extend. An ideal stereo image matching algorithm is supposed to be invariant to the scale, translation, rotation, and partial correspondence between two given stereo images. While the multi-resolution of wavelets is good at scale adaptivity, we also require the wavelet transform and pyramids to be translation- and rotation-invariant. This paper is intended to serve for three purposes: (1) To present the general problem of stereo image matching in a sufficient depth and extent, so that pure wavelet mathematicians could think on adequate and efficient solutions, (2) To present a complete algorithm for top-down image matching including surface reconstruction by using wavelet pyramids, (3) To search for a wavelet family optimal for image matching. It is expected that a family of adequately designed wavelets could provide a generic and robust solution to the stereo image matching problem, which could be an important breakthrough in computer vision, photogrammetry, and pattern recognition.

Compression

Region-based wavelet image compression

Kelly R. Debure,
Michael L. Hilton

Show abstract

Transform-based image coders exploit the information packing ability of some mathematical transforms in order to reduce the number of significant transform coefficients needed to accurately represent an image. Large coefficients are often associated with those regions where an image changes a lot, such as the boundaries between objects with differing visual characteristics. One way to reduce the number of significant transform coefficients is to segment an image into regions of similarity and then apply the transform to each region separately. We propose a novel image compression technique which first segments an image into arbitrary regions and then applies a region-adapted wavelet transform to each region.

Space-frequency adaptive trellis-coded wavelet image coding

Veyis Nuri

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In this paper, a high-quality, space-frequency adaptive, trellis coded wavelet image codec is presented. The algorithm seeks jointly optimal space and frequency segmentation of an image. For this, first, wavelet packets are formed to localize the high-energy frequency bands. Then, the wavelet coefficients are further classified to maximize the coding gain. Design target is set to minimize Lagrangian cost functions which are formed with distortion in l

^{2}norm being the cost and coding rate as constraint. Resultant mapping is used to quantize the wavelet coefficients with trellis code quantization. This is followed by adaptive arithmetic coding producing the final compressed bitstream. Described approach is tested on several ISO standard images and results are compared to other compression techniques. Representative examples are included.
Image deblocking via multiscale edge processing

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Blocking artifacts are the most objectionable drawback of block-based image and video coders. We describe a novel technique for removing blocking artifacts via multiscale edge processing. The new technique exploits the advantages of an invertible multiscale edge representation from which the block edges can be easily identified and removed. By virtue of the multiscale edge processing one is able to deblock images effectively without blurring perceptually important features or introducing new artifacts. We present the deblocking algorithm with experimental results and a discussion.

Volume compression of MRI data using zerotrees of wavelet coefficients

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Volume data such as those acquired by magnetic resonance imaging techniques can be compressed efficiently using the wavelet transform. Wavelet compression methods need to retain both the value and the location of the significant coefficients. We present experimental results demonstrating the use of zerotree encoding methods in wavelet compression can enhance the ability to further compress volume data.

Adaptive multiresolution control for video coding

Licia Capodiferro,
A. Manca,
Giovanni Jacovitti

Show abstract

Multiresolution representation of images provides a flexible tool for information rate control in video coding for transmission and storage purposes. In this work, the different spatial resolution capability of the human visual system with the angular displacement from the foveal line of sight is exploited. When the region of interest of the observer can be determined with sufficient reliability, the resolution is regulated according to the expected receptive resolution. This leads to significant bitrate savings in some typical applications.

Biomedical Applications

Matched filter reconstruction from tomographic projections

Christian Kappeler

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3D data acquisition is increasingly used in positron emission tomography (PET) to collect a larger fraction of the emitted radiation. Major practical difficulties with 3D- PET arise from large sizes of the data sets and reconstruction times. To take advantage of lossy compression we have developed an algorithm with possibility of compression built into. iT is based on a complete and orthonormal system of 2D wavelets and uses the technique of matched filters. These matched filters are obtained by mapping each 2D wavelet component with the model of the data acquisition geometry. The resulting projection images are convolved with a Ramachandran-Lakshminarayanan filter and multiplied with the projection data. The values of these scalar products in projection space are equal to the contribution of the given 2D wavelet component in the unknown object. This allows for reducing the number of matched filters used by application of a simple thresholding criterion in frequency space.

Extraction of frequency domain kurtosis from non-Gaussian and nonstationary signals using the wavelet transform

Sheau-Fang Lei,
Roger P. Hamernik

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Conventional metrics used to quantify signals in noise/hearing research are primarily derived from time- averaged energy and spectral analyses. Such metrics, while appropriate for Gaussian signals, are of limited value in more complex sound environments. Many of the sounds encountered in industrial/military environments have non- Gaussian and nonstationary distributed waveforms. These signals may have the same energy and spectra as those of a continuous Gaussian signal, yet they can produce very different effects on the auditory system. This result has led to efforts to develop additional metrics, incorporating the temporal characteristics of a signal, that could be useful in evaluating hazardous acoustic environments. Previous research suggests that frequency domain kurtosis (FDK) may be useful in such an application. This paper shows that good estimates of FDK can be obtained from an application of the wavelet transform. The wavelet transform, which has features in common with the cochlear micromechanical analysis of a signal, will reflect the temporal variations of the frequency components in a signal. A signal is decomposed by the wavelet transform on a logarithmic scale, and then the fourth-order kurtosis estimates are computed across the different octave bands from the wavelet transform results. Complex signals whose effects on hearing are known, and which are similar to realistic industrial noises, are used as model signals from which the FDK metric is extracted using the wavelet transform. Animal model experiments have shown that FDK is highly correlated with both the frequency specificity of hearing loss and the severity of trauma. Use of the wavelet transform to obtain an FDK metric lends itself to incorporation into digital analysis systems that may be useful in the assessment of complex noises for hearing conservation purposes.

Multiscale image enhancement of chromosome banding patterns

Qiang Wu,
Kenneth R. Castleman

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Visual examination of chromosome banding patterns is an important means of chromosome analysis. Cytogeneticists compare their patient's chromosome image against the prototype normal/abnormal human chromosome banding patterns. Automated chromosome analysis instruments facilitate this by digitally enhancing the chromosome images. Currently available systems employing traditional highpass/bandpass filtering and/or histogram equalization are approximately equivalent to photomicroscopy in their ability to support the detection of band pattern alterations. Improvements in chromosome image display quality, particularly in the detail of the banding pattern, would significantly increase the cost-effectiveness of these systems. In this paper we present our work on the use of multiscale transform and derivative filtering for image enhancement of chromosome banding patterns. A steerable pyramid representation of the chromosome image is generated by a multiscale transform. The derivative filters are designed to detect the bands of a chromosome, and the steerable pyramid transform is chosen based on its desirable properties of shift and rotation invariance. By processing the transform coefficients that correspond to the bands of the chromosome in the pyramid representation, contrast enhancement of the chromosome bands can be achieved with designed flexibility in scale, orientation and location. Compared with existing chromosome image enhancement techniques, this new approach offers the advantage of selective chromosome banding pattern enhancement that allows designated detail analysis. Experimental results indicate improved enhancement capabilities and promise more effective visual aid to comparison of chromosomes to the prototypes and to each other. This will increase the ability of automated chromosome analysis instruments to assist the evaluation of chromosome abnormalities in clinical samples.

Coding

Integrated wavelet compression and restoration

Zia-ur Rahman

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The performance of wavelet compression algorithms is generally judged solely as a function of the compression ratio and the vidual artifacts which are perceivable in the reconstructed image. The problem then becomes one of obtaining the best compression with fewest visible artifacts--a very subjective measure. Our wavelet compression algorithm uses an information theoretic analysis for the design of the compression maps. We have previously shown that maximizing the information for a given visual communication channel also maximizes the visual quality of the restored image. We utilize this to design quantization maps which maximize information for a given compression ratio. Hence we are able to design quantization maps which maximize the restorability of an image--i.e. the information content, the image quality, and the mean-square difference fidelity--for a given compression ratio.

Edge-compensated wavelet analysis of irregularly supported fields

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Though wavelets have been used extensively for image coding, compression,and denoising, they are also gaining popularity in the geophysical sciences as an analysis tool. Capitalizing on the wavelet's relatively tight localization in both the time and frequency domains, the wavelet transform of a data field can yield significant information about the localized frequency content of the underlying process. As an analysis technique, though, standard wavelet transforms suffer from some of the real work constraints that data sets often impose. Primary among these is the fact that the data set is not typically supported on a standard rectangular grid. We investigate the application of a boundary-compensated wavelet transform supported on an arbitrarily-shaped region. Applications to satellite-based altimetry of ocean basins are presented.

Lossless compression of medical images based on an enhanced generalized multidimensional S-Transform

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A variant of the S-transform (ST), which is a multiresolution Walsh-Hadamard transform having the structure of a dyadic wavelet decomposition, is proposed for both speeding up computation,and enabling extension to 3D data, when reversible coding of medical images and image sequences is concerned. It is derived by exploiting the same parity of the sum and the difference of two integers in a separable fashion, and thereby it has been easily extended to decorrelate volumetric data. Also, the spatial structure of the ST is considered by modelling the statistics of the different subbands of integer coefficients as generalized Gaussian probability density functions (PDF), and by fitting individual codebooks for variable length coding. The estimate of the shape factor of the PDF is based on a novel criterion matching the entropy of the theoretical and actual distributions. Coding performance comparisons are made with a similar algorithm, like the reduced-difference pyramid (RDP), designed for the purpose of hierarchical lossless image compression,as well as with lossless JPEG. Tests carried out on medical images and tomographic sequences show improvements of the proposed scheme over both the RDP and the 2D ST. Archival/retrieval are feasible on-line, still with the benefits of multiresolution coding for telebrowsing.

Image compression scheme based on wavelet transform and the human visual system

Jiaming Li,
Jesse S. Jin

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An image compression method based on wavelet transforms and the human visual system (HVS) is presented in this paper. In the proposed method, we emphasize on the issue of designing a quantizer for wavelet transform coefficients, which takes into account the relative importance of different coefficients in human visual perception and minimizes noise introduced in quantization. Since inmost of image processing systems, the final observer of the processed images is human beings, a distortion measurement combining human visual characteristics rather than the usually adopted mean square error to judge the quality of compressed image is presented. The images compressed with out method have less annoying effects under a higher compression ratio than that without incorporating HVS properties.

Family of image compression algorithms which are robust to transmission errors

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In this work, we present a new family of image compression algorithms derived from Shapiro's embedded zerotree wavelet (EZW) coder. These new algorithms introduce robustness to transmission errors into the bit stream while still preserving its embedded structure. This is done by partitioning the wavelet coefficients into groups, coding each group independently, and interleaving the bit streams for transmission, thus if one bit is corrupted, then only one of these bit streams will be truncated in the decoder. If each group of wavelet coefficients uniformly spans the entire image, then the objective and subjective qualities of the reconstructed image are very good. To illustrate the advantages of this new family, we compare it to the conventional EZW coder. For example, one variation has a peak signal to noise ratio (PSNR) slightly lower than that of the conventional algorithm when no errors occur, but when a single error occurs at bit 1000, the PSNR of the new coder is well over 5 dB higher for both test images. Finally, we note that the new algorithms do not increase the complexity of the overall system and, in fact, they are far more easily parallelized than the conventional EZW coder.

Wavelet transcoding of block DCT-based images through block transform domain processing

Uipil Chong,
Seung P. Kim

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In this paper, we propose a method of efficient computation of wavelet coefficients from DCT-based coded image/video signals. Block transform domain filtering is well suited for transcoding of such data. First direct transform domain processing removes the necessary of inverse transform. Second, the number of nonzero elements in the blocks are significantly smaller than spatial domain. Therefore, the amount of computation can be reduced accordingly. Finally, the block processing algorithm provides a parallel processing method. Hence a fast implementation of the algorithm is well suited.

Efficient wavelet compression for images of arbitrary size

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Wavelet compression for arbitrary size images is discussed. So far, wavelet compression has dealt with restricted size images, such as 2

^{n}X 2^{m}. I propose practical and efficient methods of wavelet transform for arbitrary size images, i.e. method of extension to F (DOT) 2^{m}and method of extension to even numbers at each decomposition. I applied them to 'Mona Lisa' with the size of 137 X 180. The two methods showed almost the same calculation time for both encoding and decoding. The encoding times were 0.83 s and 0.79 s, and the decoding times were 0.60 s and 0.57 s, respectively. The difference in bit-rates was attributed to the difference in the interpolation of the edge data of the image.
Wavelets for electronics structure calculations

Mary E. Brewster,
George I. Fann

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We report on wavelet compression of a 1D prototype benchmark problem of sufficient complexity to capture the features of 3D quantum electronics structure calculations. These problems have a multi-scale character through the presence of a set of singularities corresponding to atomic nuclei. Theoretical estimates of asymptotic decay across scales are derived and verified numerically. In addition, we study the effects of the location of singularity on the accuracy of the compression and on scaling.

Fast reversible wavelet image compressor

HyungJun Kim,
Ching-Chung Li

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We present a unified image compressor with spline biorthogonal wavelets and dyadic rational filter coefficients which gives high computational speed and excellent compression performance. Convolutions with these filters can be preformed by using only arithmetic shifting and addition operations. Wavelet coefficients can be encoded with an arithmetic coder which also uses arithmetic shifting and addition operations. Therefore, from the beginning to the end, the while encoding/decoding process can be done within a short period of time. The proposed method naturally extends form the lossless compression to the lossy but high compression range and can be easily adapted to the progressive reconstruction.

Denoising

Multiscale geometric filter based on the wavelet transform

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A viable approach to noise filtering in a spatially heterogeneous environment consists of considering a multiresolution representation of the noisy image, nd of applying a different adaptive filter to each layer. The wavelet decomposition has been widely employed, thanks to its capability to capture spatial features within frequency subbands. Geometric filter is a nonlinear local operator that exploits a morphologic approach to smooth noise using a complementary hull algorithm, which as the effect of gradually reducing the maximum curvature of the boundary of the grey-level profile along all of the 8-neighbor directions. The idea of the present scheme is to apply the complementary-hull algorithm to the different subbands into which the noisy image is decomposed. The hull is applied only on the direction along which the signal is structured. The number of iterations is adjusted to the SNR of the subbands, so as to preserve spatial details to the largest extent. Results and comparisons with the standard geometric filter are presented for images affected by synthetic multiplicative noise.

Wavelet-based denoising methods: a comparative study with applications in microscopy

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This paper describes different methodologies for noise reduction or denoising with applications in the field of microscopy. An in depth study on wavelet- and polynomial based denoising has been performed by considering standard test images and phantom tests with moderate and high levels of Gaussian noise. Different thresholding methods have been tested and evaluated and in particular a novel sigmoidal- type thresholding method has been proposed. In real applications, noise variance estimation problem becomes crucial because most of the thresholding estimators tends to overestimate this value. A comparison with the Hermite polynomial transform (HPT) and a modification of the HPT based in detecting the position and orientation of relevant edges has been accomplished. From this study one can conclude that both wavelet-based and polynomial-based denoising methods perform better than any other nonlinear filtering method both in terms of perceptual quality and edge-preserving characteristics.

Frames and Gabor

Frame bounds for finite dimensional frames

John W. Pierre

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In the study of wavelet theory, frames and the computation of frame bounds play an important role. This short letter details the calculation of frame bounds in the case where the frame is composed of a finite number of elements. The final step in solving for the frame bounds uses the Rayleigh principle.

Fractals

Determination of the scaling parameters of affine fractal interpolation functions with the aid of wavelet analysis

Leonid I. Levkovich-Maslyuk

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Fractal interpolation functions have become popular after the works of M.Barnsley and his co-authors on iterated function systems and their applications to data compression3'4. Here, we consider the following problem: given a set of values of a fractal interpolation function, recover the contractive affine mappings generating this function. The suggested solution is based on the connection, which is established in the work, between the maxima skeleton of wavelet transform of the function and positions of the fixed points of the affine mappings in question.
Keywords: Fractal interpolation,wavelets, data compression.

Detection of chaos based on wavelet transform

Yaojun Wu,
Baoqi Tao

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The method in which the wavelet transform and fractal theories are applied to detect chaotic signals with the additive observed noises is presented in this paper. Smoothing operator is derived from wavelet transform and used to process the chaotic time series. According to its character of no-scale interval in certain scales, a detected example of the noisy chaotic behavior produced by a Lorenz attractor is provided by using the improved G-P algorithm to calculate its fractal dimension. The results shows that the method introduced here expresses a good ability to detect chaos.

Biomedical Applications

Signal/background separation by wavelet packets for detection of microcalcifications in mammograms

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We developed a method of weighted wavelet packets for separation of small, low contrast signals from large, inhomogeneous background. Our method was applied to the enhancement of microcalcifications on digital mammograms, which appear as small bright spots superimposed on the background representing the structure of breast tissue, for improvement of the performance of our computer-aided diagnosis scheme for detection of clustered microcalcifications. Our method first approximates signals if interest by a set of wavelets that are extracted from a wavelet packet dictionary by means of the matching pursuit algorithm. The selected set of wavelets is then subjected to a supervised learning process for optimization of the weights assigned to individual time-frequency tiles for enhancement of the microcalcifications and suppression of the background structures. In an analysis of 82 regions of interest extracted from our mammographic database, our new method showed a sensitivity of 92 percent and a specificity of 75 percent. Our new method is shown to perform better than our previous method based on the fixed-weight, orthogonal wavelet transform.

Improved wavelet-packet compression of electrocardiogram data II: coding of compressor output and QRS clustering

Brian D. Bradie

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Two modifications to a wavelet packet based compression scheme for single lead electrocardiogram data have been tested for their effect upon compression performance. First, differential pulse code modulation and an adaptive arithmetic coder wee implemented to improve the coding of the major components of compressor output. The improved strategies produced a 17 percent reduction in average data rate; since all techniques were lossless, there was no effect on the quality of the reconstructed signals. Second, 'QRS clustering' was incorporated into the preprocessing stage of the algorithm. Beats of similar structure were grouped, and a separate average, vector was maintained for the dominant and non-dominant beat types found in the signal. Beat similarity was quantified using a combination of linear correlation and average, absolute, point-by-point difference. A significant reduction in coefficient data rate was anticipated; however, for the test cases used in this study, a slight increase in coefficient data rate was produced.

Orthogonal wavelets for image transmission and compression schemes: implementation and results

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Diagnostic quality medical images consume vast amounts of network time, system bandwidth and disk storage in current computer architectures. There are many ways in which the use of system and network resources may be optimize without compromising diagnostic image quality. One of these is in the choice of image representation, both for storage and transfer. In this paper, we show how a particularly flexible method of image representation, based on Mallat's algorithm, leads to efficient methods of both lossy image compression and progressive image transmission. We illustrate the application of a progressive transmission scheme to medical images, and provide some examples of image refinement in a multiscale fashion. We show how thumbnail images created by a multiscale orthogonal decomposition can be optimally interpolated, in a minimum square error sense, based on a generalized Moore-Penrose inverse operator. In the final part of this paper, we show that the representation can provide a framework for lossy image compression, with signal/noise ratios far superior to those provided by a standard JPEG algorithm. The approach can also accommodate precision based progressive coding. We show the results of increasing the priority of encoding a selected region of interest in a bit-stream describing a multiresolution image representation.

Classification of epileptic EEG using neural network and wavelet transform

Arthur Ashot Petrosian,
Richard Homan,
Danil Prokhorov,
et al.

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One of the major contributions of electroencephalography has been its application in the diagnosis and clinical evaluation of epilepsy. The interpretation of the EEG is achieved through visual inspection by a trained electroencephalographer. However, descriptions of rules used during the visual analysis of data are often subjective and can vary from one reader to another. Computerized methods are a means to standardize this process. In recent years, much effort has been made to develop such methods that can characterize different interictal, ictal, and postictal stages. the main issue of whether there exists a preictal phenomenon remains unresolved. In the present study we address this issue making use of specifically designed and trained recurrent neural networks in conjunction with signal wavelet decomposition technique. The purpose of this combined consideration was to demonstrate the potential for seizure prediction by up to several minutes prior to its onset.

New Applications

Analyzing engineering surface texture using wavelet filter

Xiangyang Liu,
Jay Raja

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Analysis of surface profile generally involves filtering to separate short wavelengths from medium and long wavelengths. This is usually accomplished using digital filters. The wavelet filter is an ideal means to separate the profile into different bands. The space frequency localization and multiscale presentation of different wavelength components is useful in manufacturing process control and in establishing relationship between surface texture and function. The use of wavelet filter to analyze surface is explored in this paper. This paper deals with the evaluation of suitable wavelet basis for analyzing surface texture of machined surfaces using wavelet filter. The multiscale surface features are analyzed using wavelet filter to explore the potential use of wavelet filter in monitoring of manufacturing process and feature detection in engineering surfaces.

Wavelets in music analysis and synthesis: timbre analysis and perspectives

Regis Rossi Alves Faria,
Ruggero Andrea Ruschioni,
Joao Antonio Zuffo

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Music is a vital element in the process of comprehending the world where we live and interact with. Frequency it exerts a subtle but expressive influence over a society's evolution line. Analysis and synthesis of music and musical instruments has always been associated with forefront technologies available at each period of human history, and there is no surprise in witnessing now the use of digital technologies and sophisticated mathematical tools supporting its development. Fourier techniques have been employed for years as a tool to analyze timbres' spectral characteristics, and re-synthesize them from these extracted parameters. Recently many modern implementations, based on spectral modeling techniques, have been leading to the development of new generations of music synthesizers, capable of reproducing natural sounds with high fidelity, and producing novel timbres as well. Wavelets are a promising tool on the development of new generations of music synthesizers, counting on its advantages over the Fourier techniques in representing non-periodic and transient signals, with complex fine textures, as found in music. In this paper we propose and introduce the use of wavelets addressing its perspectives towards musical applications. The central idea is to investigate the capacities of wavelets in analyzing, extracting features and altering fine timbre components in a multiresolution time- scale, so as to produce high quality synthesized musical sounds.

Application of wavelet transform in characterization of nonlinear optical materials

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Wavelet transform technique is applied to the analysis of data collected in experiments on the characterization of nonlinear optical materials which may be in the form of liquid, thin film or crystal. Many characterization techniques are based on nonlinear optical processes such as higher harmonic generation in which second harmonic or third harmonic signals may be generated by the nonlinear material. When the optical path length of the material is changed, the interference between bound and free waves forms a fringe pattern. Conventional Fourier transform techniques are not suitable for analyzing such fringes when they have a variable periodicity and a low signal-to-noise ratio. However, the wavelet transform method is best suited for such signals because it provides a better resolution in both space and frequency domains. In this study, optical properties of materials are extracted from these fringe patterns by decomposing them into coefficients which are inner products of the signal and a family of wavelets generated from a mother wavelet by dilation and shift operations.

Pattern Recognition

Template basis techniques for pattern recognition

Mark E. Lehr,
Keh-Shin Lii

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The purpose of this paper is to present an approach to pattern recognition which acknowledges theories in the fields of perception, the human visual system and decomposition. It is hoped that by taking a panoramic view of the subject matter that insights into the issue may uncover a possible course of action. At the very least, there should be evidence that certain tools used in the process adequately fit the analysis of the problem. The contemporary perceptual theory to classification recognizes the fundamental concepts of scale and localization. The visual system can solve problems that would be intractable using a single depiction of a scene by having access to representations at different spatial scales. Enhancement, analysis and compression are the areas of image processing most germane to pattern recognition. And the simplest statistical approach to identifying patterns utilizes templates. All these properties can be inherently exploited in the wavelet domain. By unifying the process of noise reduction, segmentation, feature extraction and classification it is possible to develop a general technique which might not be optimal but has the advantage of being computationally efficient. All the desired properties that are required in an analytic task of this nature are captured with multiresolution analysis.

Performance analysis of a wavelet-based hybrid neurosystem for signal classification

Chung T. Nguyen,
Kai F. Gong

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This paper is concerned with the problem of determining performance of a wavelet-based hybrid neurosystem trained to provide efficient feature extraction and signal classification. The hybrid network consists of a parallel array of neurosystems. Each neurosystem is constructed with three single neural networks; two of which are feature extraction networks, and the other is a classification network, are provided with magnitude and location information of the wavelet transform coefficients, respectively, and are trained with self-organizing rules. Their outputs are then presented to the classification network for pattern recognition. Based on the topological maps provided by the feature extraction neural networks, the back-propagation algorithm is used to train the third network for pattern recognition. The combination of wavelet, wavelet transform, and hybrid neural network architecture and advanced training algorithms in the design makes the system unique and provides high classification accuracy. In this paper, system performance is shown to be intrinsically related to basis kernel function used in feature extraction. A method for selecting the optimal basis function and a performance analysis using simulated data under various noise condition are presented and compared against other pattern recognition techniques.

Development of a linear time-scale coherence spectrum

Surya Santoso,
Edward J. Powers

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In spectral analysis of two or more signals, it is desirable to quantify the statistical relationships among them. The linear coherence spectra based on Fourier analysis have been widely used to quantify the linear relationships between two stationary fluctuating signals. However, in the case of nonstationary signals, the Fourier-based linear coherence spectrum has a limited utility, since temporal information is now required. It is the purpose of this paper to extend the notion of the classical linear coherence spectra to the time-scale coherence spectrum. it will be shown that not only is the new time-scale coherence spectrum capable of measuring linear relationships between two nonstationary signals in the time-scale domain but also of detecting their phase differences as well. Using the Cauchy-Schwarz inequality, it can be shown that the time-scale coherence spectrum is bounded by zero and one. The latter number indicates a perfect linear relation between the two nonstationary signals at a particular time and scale. The efficacy of the new method is demonstrated and compared with that of the classical coherence spectrum through numerical examples.

Character extraction from documents using wavelet maxima

Wen-Liang Hwang,
Fu Chang

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The extraction of character image is an important front-end processing for optical character recognition (OCR) and other applications. This process is extremely important because the OCR applications usually extract salient features and process on them. The existence of noise not only destroys features of characters, but also introduces unwanted features. We propose a new algorithm which removes unwanted background noises from a textual image. Our algorithm is based on the observation that the magnitude of the intensity variation of character boundaries differs form that of noises at various scales of their wavelet transform. Therefore, most of the edges corresponding to the character boundaries at each scale can be extracted using a thresholding method. The internal region of characters is determined by a voting procedure, which uses the arguments of the remaining edges. The interior of recovered characters is solid containing no holes. Characters tend to become fattened, because of the smoothness being applied in the calculation of wavelet transform. To obtain a quality restoration of character image, the precise locations of characters at the original image are then estimated using a Bayesian criterion. Detailed algorithm with careful analysis of the free parameters are also conducted in this paper. The method is simple and effective. We also present some experimental results that suggest its effectiveness.

Fractals

Construction of two-dimensional multiwavelets on a triangulation

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A family of continuous, compactly supported, bivariate multi-scaling functions have recently been constructed by Donovan, Geronimo, and Hardin using self-affine fractal surfaces.In this paper we describe a construction of associated multiwavelets that uses the symmetry properties of the multi-scaling functions. Illustrations of a particular set of scaling functions and wavelets are provided.