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- Mathematical Developments I
- Mathematical Developments II
- Transient Detection in Biological Systems
- Frames and Overcomplete Representations
- Image Coding
- Wavelet Design and Construction
- Receiver Design and Demodulation
- Gabor Transforms
- High-Speed Processing
- Detecting Chaotic Signals and Denoising
- Image Compression
- Multiscale Representations
- Medical Imaging
- Posters
- Medical Imaging
- Segmentation and Classification
- Posters
- Wavelet Design and Construction
- Mathematical Developments II

Mathematical Developments I

Local discriminant bases

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We describe an extension to the `best-basis' method to construct an orthonormal basis which maximizes a class separability for signal classification problems. This algorithm reduces the dimensionality of these problems by using basis functions which are well localized in time- frequency plane as feature extractors. We tested our method using two synthetic datasets: extracted features (expansion coefficients of input signals in these basis functions), supplied them to the conventional pattern classifiers, then computed the misclassification rates. These examples show the superiority of our method over the direct application of these classifiers on the input signals. As a further application, we also describe a method to extract signal component from data consisting of signal and textured background.

Asymptotic singular value decay of time-frequency localization operators

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The Weyl correspondence is a convenient way to define a broad class of time-frequency localization operators. Given a region (Omega) in the time-frequency plane R

^{2}and given an appropriate (mu) , the Weyl correspondence can be used to construct an operator L((Omega) ,(mu) ) which essentially localizes the time-frequency content of a signal on (Omega) . Different choices of (mu) provide different interpretations of localization. Empirically, each such localization operator has the following singular value structure: there are several singular values close to 1, followed by a sharp plunge in values, with a final asymptotic decay to zero. The exact quantification of these qualitative observations is known only for a few specific choices of (Omega) and (mu) . In this paper we announce a general result which bounds the asymptotic decay rate of the singular values of any L((Omega) ,(mu) ) in terms of integrals of (chi)_{(Omega}) *^{-}(mu)^{2}and ((chi)_{(Omega}) *^{-}(mu) )^{^}^{2}outside squares of increasing radius, where^{-}(mu) (a,b) equals (mu) (-a, -b). More generally, this result applies to all operators L((sigma) ,(mu) ) allowing window function (sigma) in place of the characteristic functions (chi)_{(Omega}). We discuss the motivation and implications of this result. We also sketch the philosophy of proof, which involves the construction of an approximating operator through the technology of Gabor frames--overcomplete systems which allow basis-like expansions and Plancherel-like formulas, but which are not bases and are not orthogonal systems.
Spline harmonic analysis and wavelets

Valery A. Zheludev

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We present here a new approach to wavelet transforms based on techniques of SHA. It is a version of HA which operates in spline spaces. SHA enjoys various applications but here we outline a recent application of SHA techniques, namely, accomplishing on the base of periodic splines the wavelet transforms of periodic signals as well as the remarkable informative digital representation of the signals. SHA approach to wavelets yields a tool just as for the constructing a diversity of spline wavelet bases, so for a fast implementation of a decomposition of a function into a fitting wavelet representation and its reconstruction. This approach allows to construct WP bases for refined frequency resolution of signals.

Mathematical Developments II

Oblique projections in discrete signal subspaces of l2 and the wavelet transform

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We study the general problem of oblique projections in discrete shift-invariant spaces of l

_{2}and we give error bounds on the approximation. We define the concept of discrete multiresolutions and wavelet spaces and show that the oblique projections on certain subclasses of discrete multiresolutions and their associated wavelet spaces can be obtained using perfect reconstruction filter banks. Therefore we obtain a discrete analog of the Cohen-Daubechies- Feauveau results on biorthogonal wavelets.
Linear phase paraunitary finite impulse response (FIR) filter banks: a complete characterization .

Radka Turcajova

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A complete characterization of linear phase paraunitary FIR filter banks is given. It is based on factorization into the product of linear factors. For filter banks with an even number of filters centrosymmetric linear factors are used. When the number of filters is odd, the centrosymmetric linear factors do not exist. However, the linear factors in the factorization can be coupled and centrosymmetric quadratic factors can be used. The described factorizations make possible to solve the completion problem--the algorithm for obtaining all possible completions of given filters to a linear phase paraunitary filter bank is derived.

Discrete biorthogonal wavelets and PR finite impulse response (FIR) filter banks via block circulant matrices

Jaroslav Kautsky,
Radka Turcajova

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Factorization of orthogonal block circulant matrices can not be generalized in a straightforward way for block circulant matrices which are merely invertible. However, they can be decomposed into an orthogonal matrix and an atom that represents the `nonorthogonal' part of the matrix. Atoms can be characterized by nilpotent block-companion matrices. This characterization permits, for example, to derive bounds for the width of the band of the inverse of a banded block circulant matrix.

Transient Detection in Biological Systems

Detecting ionic currents in single channels using wavelet analysis, part I: zero mean Gaussian noise

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The patch clamp technique opened a new field in biological research and shed light on membrane permittivity for ionic currents. The key element in patch clamp measurements is the detection of the ionic currents in a single biological channel. It is known that the channels open and close at random times, thus modulating the ionic currents. The measured current switches between two levels corresponding to the open and close states of the channel. Determining the statistics of the open and closed periods is of crucial importance to the experimenter, because it reflects the response of channel protein to drugs and other factors. The detected signal is strongly corrupted by instrumentation and other noises, rendering the detection of the opening and closing moments extremely difficult. We describe the use of the wavelet transform and its associated multiresolution (multiscale) analysis to detect the currents through single ionic channels corrupted with noise.

Wavelet denoising of EEG signals and identification of evoked response potentials

Rene A. Carmona,
Lonnie H. Hudgins

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The purpose of this study is to apply a recently developed wavelet based de-noising filter to the analysis of human electroencephalogram (EEG) signals, and measure its performance. The data used contained subject EEG responses to two different stimuli using the `odd-ball' paradigm. Electrical signals measured at standard locations on the scalp were processed to detect and identify the Evoked Response Potentials (ERP's). First, electrical artifacts emitting from the eyes were identified and removed. Second, the mean signature for each type of response was extracted and used as a matched filter to define baseline detector performance for the noisy data. Third, a nonlinear filtering procedure based on the wavelet extrema representation was used to de-noise the signals. Overall detection rates for the de-noised signals were then compared to the baseline performance. It was found that while the filtered signals have significantly lower noise than the raw signals, detector performance remains comparable. We therefore conclude that all of the information that is important to matched filter detection is preserved by the filter. The implication is that the wavelet based filter eliminates much of the noise while retaining ERP's.

Wavelet-based detection of transients in biological signals

Tahsin Mzaik,
Jogikal M. Jagadeesh

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This paper presents two multiresolution algorithms for detection and separation of mixed signals using the wavelet transform. The first algorithm allows one to design a mother wavelet and its associated wavelet grid that guarantees the separation of signal components if information about the expected minimum signal time and frequency separation of the individual components is known. The second algorithm expands this idea to design two mother wavelets which are then combined to achieve the required separation otherwise impossible with a single wavelet. Potential applications include many biological signals such as ECG, EKG, and retinal signals.

Frames and Overcomplete Representations

Extrapolation in multiresolutions

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An extrapolation problem for multiresolution is studied. Solutions involve Hankel type operators defined in terms of the generating sequence of the scaling function of the multiresolution.

Frame and special types of extended pseudobiorthogonal bases

Nasr-Eddine Berrached,
Hidemitsu Ogawa

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As a generalization of the concept of pseudo-biorthogonal bases (PBOB), we already presented the theory of the so-called extended pseudo-biorthogonal bases (EPBOB). We introduce in this paper two special types of EPBOB called EPBOB's of Type O and of type L. Characterizations, construction methods, inherent properties, and mutual relations of these types of EPBOB are discussed.

Frame of wavelets for edge detection

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We present in the following work, a multiscale edge detection algorithm whose aim is to detect edges of any slope. Our work is based on a generalization of the Canny-Deriche filter, modelized by a more realistic edge than the traditional step shape edge. The filter impulse response is used to generate a frame of wavelets. For the merging of the wavelet coefficients, we use a geometrical classifier developed in our laboratory. The segmentation system thus set up and after the training phase does not require any adjustment nor parameter. The main original property of this algorithm is that it leads to a binary edge image without any threshold setting.

Image Coding

Subband coding and noise reduction in multiresolution analysis frames

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We consider the effect of the quantization noise introduced by coding at subbands. We demonstrate that significant noise reduction is achieved by using wavelet frames and their associated filter banks in a subband signal processing system.

Wavelet-based approaches versus subband coding techniques for still-image coding

Manuel A. Sola,
Sebastia Sallent-Ribes

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In this paper, the performance of filters implied in the discrete wavelet transform approach is compared with a set of filters of common application in image processing. A multidimensional multirate codec is designed in which the best subband decomposition and coding is jointly obtained by minimization of a rate-distortion criterion. We also introduce the idea of directional sensitivity based on the initial sampling lattice. This allows the codec to adapt better to the spectral content of the signal. Finally, taking as a starting point the continuous wavelet transform, we propose a discrete multiresolution analysis that would lead to new discrete transforms of special interest when analyzing discrete scalings and shiftings: the completely discrete wavelet transform.

Wavelets and waveform coding: extensions and new results

Michael J. Lyall,
Sathyanarayana S. Rao,
Prashant P. Gandhi

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Recently, we demonstrated the use of orthonormal wavelets as desirable waveforms for baseband waveform coding application in digital communications. In this paper, we examine the use of generalized biorthogonal wavelets for waveform coding. Though the transmit and receive waveforms have different supports in the biorthogonal case, it is shown that binary symbols can be extracted easily at the receiver. Spectral characteristics of several types of biorthogonal wavelets are examined and resulting codecs' bandwidth efficiencies (in bits/sec/Hz) are provided. An M-band wavelet codec is also proposed. Preliminary results indicate that the M-band codec (used in this study) will be out-performed by a two-band codec of the same order.

Error correction in images using overcomplete pyramid representations

Subrata Rakshit,
Charles H. Anderson

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The Burt pyramid was developed as a means of decomposing images into multiscale representations. Unlike the wavelet transforms, which are complete orthonormal transformations, the pyramid transforms are overcomplete like frames. In addition, the Burt pyramid has an exact reconstruction rule. This paper first examines the consequences of that redundancy and exact reconstruction. Even though the pyramids are not strictly frames, the pyramids are characterized in much the same way as frames. The issue of error detection is addressed next, followed by the development of an error correction algorithm. This leads to a 10 dB noise reduction in addition to the reduction inherent to reconstruction of frames. Examples are given to show the performance for different types of noise.

Wavelet Design and Construction

Finite element multiwavelets

Vasily Strela,
Gilbert Strang

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Finite elements with support on two intervals span the space of piecewise polynomomials with degree 2 n - 1 and n - 1 continuous derivatives. Function values and n - 1 derivatives at each meshpoint determine these `Hermite finite elements'. The n basis functions satisfy a dilation equation with n by n matrix coefficients. Orthogonal to this scaling subspace is a wavelet subspace. It is spanned by the translates of n wavelets W

_{i}(t), each supported on three intervals. The wavelets are orthogonal to all rescalings W_{i}(2^{j}t-k), but not to translates at the same level (j equals 0). These new multiwavelets achieve 2 n vanishing moments and high regularity with symmetry and short support.
Constructions of two-dimensional nonseparable Malvar wavelets

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Malvar wavelets or lapped orthogonal transform has been recognized as a useful tool in eliminating block effects in transform coding. Suter and Oxley extended the Malvar wavelets to more general forms, which enable one to construct an arbitrary orthonormal basis on different intervals. In this paper, we generalize the idea in Suter and Oxley from 1D to 2D cases and construct nonseparable Malvar wavelets, which is potentially important in multidimensional signal analysis. With nonseparable Malvar wavelets, we then construct nonseparable Lemarie-Meyer wavelets which are band-limited.

Design of regular 2D nonseparable wavelets/filter banks using transformations of variables

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In this paper we are concerned with the design of 2D biorthogonal, 2-channel filter banks where the sampling is on the quincunx lattice. Such systems can be used to implement the nonseparable Discrete Wavelet Transform and also to construct the nonseparable scaling and wavelet functions. One important consideration of such systems (brought into attention by wavelet theory) is the regularity or smoothness of the scaling and wavelet functions. The regularity is related to the zero-property--the number of zeros of the filter transfer function at the aliasing frequency (((omega)

_{1},(omega)_{2}) equals ((pi) ,(pi) ) for the quincunx lattice). In general the greater the number of zeros, the greater the regularity. It has been shown previously by the authors that the transformation of variables is an effective and flexible way of designing multidimensional filter banks. However the wavelet aspects of the filter banks (i.e., regularity) were not considered. In this paper we shall show how the zero- property can be easily imposed through the transformation of variables technique. A large number of zeros can be imposed with ease. Arbitrarily smooth scaling and wavelet functions can be constructed. Several design examples will be given to illustrate this.Receiver Design and Demodulation

Wavelet-packet-based multiple-access communication

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We explore the role that the wavelet packet transforms can play an increasing multiple access communication throughput via waveform design and receiver design. Wavelet packets offer a flexible framework which allows us to select and refine signature waveforms at intermediate time-frequency levels leading to the development of efficient methods for near optimal receiver implementation. The optimal receiver will jointly demodulate all of the received waveforms and has a complexity which is exponential in the number of users, while a suboptimal version will attempt to do the same with a greatly reduced computational complexity. We believe that our wavelet packet recursive joint detection system described in this paper offers a good alternative approach to existing comparable methods. Specifically, our preliminary analysis and simulation results indicate the potential for a marked computational improvement over the optimal detector and additional flexibility and efficiency over other proposed joint detection schemes for multiple access communication.

Modulation classification using wavelet transform

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The wavelet transform has found applications in singularity classification/detection of signal waveforms. In this research, we apply the Morlet wavelet to detect the phase changes, and use the phase change rate as a feature for the classification of PSK modulation schemes. The likelihood function of the alphabet size with respect to the number of symbol change, which corresponds to the phase change of PSK signals, is derived by assuming that the transmitted symbol sequence is i.i.d. and equally likely distributed in an alphabet set. The classification problem can then be formulated as a likelihood ratio test by using the hypothesis testing technique. We show the performance of BPSK/QPSK and CW/BPSK classifiers in numerical experiments.

Gabor Transforms

General theory of discrete Gabor expansion

Shidong Li

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We present a new and more general theory of discrete Gabor expansions for arbitrary dimensional spaces. We show that a discrete Gabor expansion is in fact a general frame decomposition. We provide a complete characterization of all possible discrete Gabor expansions. We reveal an intrinsic dimension invariance property of the (discrete) Gabor expansion. We derive a parametric algorithm for computing all analysis waveforms that are dimension independent. We shall also consider the issue of optimum Gabor expansion and the construction of non-separable 2D discrete Gabor expansions.

Inexpensive Gabor decompositions

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This is a paper on the discrete Gabor transform. We discuss the calculation of dual and tight Gabor atoms, for Gabor atoms which satisfy certain support restrictions related to the relevant time- and frequency lattice constants. These conditions imply that the Gabor frame operator is just a pointwise multiplication operator and therefore computing the inverse or the square root of the inverse frame operator are computationally inexpensive.

High-Speed Processing

Synthesis of regular very large scale integration (VLSI) architectures for the 1D discrete wavelet transform

Jose Fridman,
Elias S. Manolakos

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A methodology for synthesizing parallel computational structures has been applied to the Discrete Wavelet Transform algorithm. It is based on linear space-time mapping with constraint driven localization. The data dependence analysis, localization of global variables, and space-time mapping is presented, as well as one realization of a 3-octave systolic array. The DWT algorithm may not be described by a set of Uniform or Affine Recurrence Equations (UREs, AREs), thus it may not be efficiently mapped onto a regular array. However it is still possible to map the DWT algorithm to a systolic array with local communication links by using first a non-linear index space transformation. The array derived here has latency of 3M/2, where M is the input sequence length, and similar area requirements as solutions proposed elsewhere. In the general case of an arbitrary number of octaves, linear space-time mapping leads to inefficient arrays of long latency due to problems associated with multiprojection.

Fast algorithms for running wavelet analyses

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We present a general framework for the design and efficient implementation of various types of running (or over-sampled) wavelet transforms (RWT) using polynomial splines. Unlike previous techniques, the proposed algorithms are not necessarily restricted to scales that are powers of two; yet they all achieve the lowest possible complexity: O(N) per scale, where N is signal length. In particular, we propose a new algorithm that can handle any integer dilation factor and use wavelets with a variety of shapes (including Mexican-Hat and cosine-Gabor). A similar technique is also developed for the computation of Gabor-like complex RWTs. We also indicate how the localization of the analysis templates (real or complex B-spline wavelets) can be improved arbitrarily (up to the limit specified by the uncertainty principle) by increasing the order of the splines. These algorithms are then applied to the analysis of EEG signals and yield several orders of magnitude speed improvement over a standard implementation.

Detecting Chaotic Signals and Denoising

Self-similar set identification in the time-scale domain

Sam Heidari,
George A. Tsihrintzis,
Chrysostomos L. Nikias,
et al.

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This paper presents an extensive study of the wavelet transform of self-similar signals and its properties. In particular, theorems are derived for the wavelet transform of deterministic self- similar signals and are used to identify and characterize them. Applications of interest, such as characterization and analysis of real chaotic signals in the presence of additive noise, are included.

Speckle reduction via wavelet shrinkage with application to synthetic-aperture-radar-based automatic target detection/reduction (ATD/R)

Haitao Guo,
Jan Erik Odegard,
Markus Lang,
et al.

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We propose a novel speckle reduction method based on shrinking the wavelet coefficients of the logarithmically transformed image. The method is computational efficient and can significantly reduce the speckle while preserving the resolution of the original image. Wavelet processed imagery is shown to provide better detection performance for synthetic-aperture radar based automatic target detection/recognition problem.

Random wavelet transformation and its properties

Tomasz Bielecki,
Jie Chen,
Stephen S. T. Yau

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A discrete random wavelet transformation is defined as a random field of stochastic integrals of translates and dilates of a compactly supported wavelet function with respect to a fractional Brownian motion. It is shown that the transformation exhibits several properties analogous to the ones demonstrated previously for the `classical' wavelet transformation of a fractional Brownian motion. Moreover, some important advantages of the transformation defined here, over its `classical' counterpart, are demonstrated.

Image Compression

Restoring shapes of noisy curves

Harvey Diamond,
Louise A. Raphael,
Daniel Williams

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We present a quasi-interpolant spline method to reduce noise and preserve shape of closed curves. Quasi-interpolation provides a natural way to construct a compact filter for generating approximate orthogonal projections.

Enhancement of decompressed images at low bit rates

Ramesh A. Gopinath,
Markus Lang,
Haitao Guo,
et al.

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Transform coding at low bit rates introduces artifacts associated with the basis functions of the transform. For example, decompressed images based on the DCT (discrete cosine transform)-- like JPEG--exhibit blocking artifacts at low bit rates. This paper proposes a post-processing scheme to enhance decompressed images that is potentially applicable in several situations. In particular, the method works remarkably well in `deblocking' of DCT compressed images. The method is non-linear, computationally efficient, and spatially adaptive--and has the distinct feature that it removes artifacts while yet retaining sharp features in the images. An important implication of this result is that images coded using the JPEG standard can be efficiently post-processed to give significantly improved visual quality in the images.

Digital image compression using wavelets and wavelet packets based on nonstationary and inhomogeneous multiresolution analyses

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Unlike the classical wavelet decomposition scheme it is possible to have different scaling and wavelet functions at every scale by using non-stationary multiresolution analyses. For the bidimensional case inhomogeneous multiresolution analyses using different scaling and wavelet functions for the two variables are introduced. Beyond it, these two methods are combined. All this freedom is used for compact image coding. The idea is to build out of the functions in a library that special non-stationary and/or inhomogeneous multiresolution analysis, that is best suited for a given image in the context of compact coding (in the sense of optimizing certain cost-functions).

Coupling of human visual system model and wavelet transform for image compression

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In this work we study a combination between wavelet transform and a model of the human visual system both from the mathematical and the computational point of view. We will combine the two procedures in such a way that the computational complexity of the whole procedure is reduced for the maximum possible amount. The gaol is to improve the quality compression, by modelling the human visual system in the compression-decompression tasks. As a result, new filters for image compression are provided for any given multiresolution analysis, independently of the coding method adopted. The proposed algorithm has been applied to grey level images and compared to more traditional approaches which do not comprehend a modelization of the human visual system.

Multiscale Representations

Multiplicative and zero-crossing representations of signals

Anca Deliu,
Michael L. Hilton,
Bjorn D. Jawerth,
et al.

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The implicit sampling theorem of Bar-David gives a representation of band limited functions using their crossings with a cosine function. This cosine function is chosen such that its difference with the original function has sufficient zero crossings for a unique representation. We show how, on an interval, this leads to a multiplicative representation involving a Riesz product. This provides an alternative to the classic additive Fourier series. We discuss stability and implementation issues. Since we have an explicit reconstruction formula, there is no need for an iterative algorithm.

Scale and harmonic-type signals

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We show that signals which are harmonically related to a fundamental frequency which has a finite bandwidth is modeled by the sum of scaled functions. We present a method to study such signals and give an example which illustrates the method.

Multiscale shape simplification for object recognition

Peter Forte,
Darrel Greenhill

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The outline of an imaged object is usually obtained as a linked list of edge elements (`edgels'). When these edgels are connected, the resulting shape is hardly ever smooth. This is because even when edgels are detected with subpixel accuracy, the spatial and gray level quantization of the original image mans that consecutive edges show random fluctuations in position and orientation. Fluctuations may also occur as a result of noise or natural variation in the object's boundary. Hence to recognize an object it is necessary to represent the boundary at varying scales of resolution in order to extract the underlying shape. High frequencies may be discarded using smoothing filters or by thresholding wavelet transforms. In this paper these approaches are described and contrasted with an alternative approach of the authors' based on term rewriting. In the latter approach the object outline is represented by a sparse array of edgels. Between any two consecutive edgels the path of the object boundary can be reconstructed (by a contour completion algorithm) to within a tolerance given by the current value of the scale space parameter. As this parameter increases, the number of edgels required to define the outline decreases--hence the shape becomes simpler at the cost of increasing approximation.

Medical Imaging

Detection of microcalcifications in mammograms using wavelets

Robin N. Strickland,
Hee Il Hahn

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Clusters of fine, granular microcalcifications in mammograms may be an early sign of disease. Individual grains are difficult to detect and segment due to size and shape variability and because the background mammogram texture is typically inhomogeneous. We present a two- stage method based on wavelet transforms for detecting and segmenting calcifications. The first stage consists of a full resolution wavelet transform, which is simply the conventional filter bank implementation without downsampling, so that all sub-bands remain at full size. Four octaves are computed with two inter-octave voices for finer scale resolution. By appropriate selection of the wavelet basis the detection of microcalcifications in the relevant size range can be nearly optimized in the details sub-bands. In fact, the separable 2D filters which transform the input image into the HH details sub-bands are closely related to pre- whitening matched filters for detecting Gaussian objects (idealized microcalcifications) in Markov noise (background noise). The second stage is designed to overcome the limitations of the simplistic Gaussian assumption and provides a useful segmentation of calcifications boundaries. Detected pixel sites in the LH, HL, and HH sub-bands are heavily weighted before computing the inverse wavelet transform. The LL component is omitted since gross spatial variations are of little interest. Individual microcalcifications are often greatly enhanced in the output image, to the point where straightforward thresholding can be applied to segment them. FROC curves are computed from tests using a well-known database of digitized mammograms. A true positive fraction of 85% is achieved at 0.5 false positives per image.

Reduced motion artifacts in magnetic resonance imaging by adaptive spatio-temporal multiresolution reconstruction

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

Posters

Concise and scale-specific extraction of biomedically relevant information from visual evoked potential signals: combining factor analysis with wavelet decomposition

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The nervous system possesses an intrinsic multiscale organization of processing systems. Evoked potentials (EPs) and other neurometric signals contain corresponding multiscale information about the normal and disordered functioning of the nervous system. The discrete wavelet transform (DWT) explicitly distinguishes among multiple scales of waveform structure, and can be used to decompose EPs in a manner that respects this intrinsic organization. In this paper we provide evidence for the multiscale structure of EPs. We demonstrate that EPs contain scale-specific information of biomedical, neurophysiological, and neuropsychological relevance. Finally, we show that the DWT provides information about small-scale phenomena that is inaccessible by standard neurometric waveform analysis techniques.

Medical Imaging

Discrete dyadic wavelets for contrast enhancement

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In this paper, we establish a mathematical connection between dyadic-wavelet-based contrast enhancement and traditional unsharp masking. Our derivation is completely based in the discrete domain. These findings may provide a better theoretical understanding of these algorithms, and facilitate the acceptance of multiscale enhancement techniques applied to medical imaging.

Segmentation and Classification

Smooth image segmentation via multiresolution analysis

Jing Zhou,
Xiang Fang,
Bijoy K. Ghosh

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In this paper, the problem of segmentation of smooth images has been studied using multiresolution analysis. The approximated image intensity function is modeled as a quadratic polynomial with additive noise within local windows. The analysis has been carried out with the aid of a new orthonormal wavelet basis introduced in this paper. A procedure has been developed to approximate an image at a coarse resolution by dropping the components of the image in such a way that small bumps at finer resolutions are suppressed. An image segmentation scheme is proposed. It performs initial segmentation on a coarse approximation of the image, and then updates the segments of the image at a finer resolution. The proposed algorithm has been tested on a variety of real images such as human faces, natural scenes, and medical images.

Image segmentation using phase coding and wavelet transform

Daniel G. Hennequin,
Jacques Lewandowski

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The phase-coded image of a scene results from the projection of a grid on the objects of the scene. We show how the wavelet transform can be used as an oriented pattern detector for the segmentation of phase-coded images. The detector coupled with a rotation of the digital camera enables detection of 3D planar surfaces. A digital image rotation technique using a scaling function from multiresolution analysis is also presented. Discussion of Fourier transform methods of segmentation of phase-coded images motivates the oriented pattern detection approach.

Local and global multiscale image classification

Jean-Marc Boucher,
Goze Benie,
Regis Fau,
et al.

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Different unsupervised Bayesian classification algorithms can be associated to a multiscale image analysis procedure leading to improvements, both in computation time and classification performances. Two kinds of algorithms are used for the classification itself: (1) local methods on a pixel-by-pixel basis and (2) global methods, which require a Markov random field model for the whole class image. Unsupervised Bayesian classification requires two steps, one for the parameter estimation of each local or global mode and one for the Bayesian classification itself. A Gaussian density with parameters depending on the class is assumed for the pixels. In a multiscale analysis scheme, the image is decomposed by successive filtering and downsampling, which allows to separate homogeneous areas and edges according to a pyramidal structure. One scale pyramid containing smaller and smaller smoothed images and one wavelet pyramid with the complementary information concerning details are built. Unsupervised Bayesian classification is done at each level of the scale pyramid, from top to bottom, by taking into account pixels which are assumed well classified at the previous level. The wavelet pyramid can be used to help the classification by defining if a classified pixel belongs to an homogeneous area or not. The homogeneity criterion consists in a variance comparison at each stage and a thresholding. A comparison has been made on very noisy synthetic images, which permits to measure the improvements and drawbacks brought by the multiscale analysis in local and global classification.

Posters

Wavelet transform adaptive filtering

Laurence M. Dang

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An LMS adaptive filtering algorithm is presented utilizing wavelet transforms. Its performance is compared to DCT and Walsh-Hadamard transform-based adaptive filtering. The experimental analysis is performed in the case of the system identification of an unknown system or filter for stationary input signals. The results show some improvement in the weight modelling of the filter with comparable convergence rates. A new performance criteria, the diagonality factor, is introduced in order to show the specific effect of the wavelet transform on a signal. A Mean Average Difference is also utilized to compare the weight modelling performance of the various transform-based LMS adaptive filterings studied in this paper.

Audio signal compression using circular wavelet packets

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We have successfully compressed audio signals using wavelet packets based on a recently developed fast wavelet transform (FWT) scheme using circular convolution with an adaptive hybrid filter/basis system. This algorithm gives perfect reconstruction of the data; edge effects are removed entirely. As a result, the quality of audio signal compression is much improved. To illustrate this, we present results from our comparison study where we compressed a test signal using these `circular wavelet packets' and wavelet packets based on the standard FWT.

Information theoretic approach to space-frequency localization

Subrata Rakshit,
Charles H. Anderson

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The design of spatio-temporal filters to estimate velocity has been dominated by the frequency domain approach. Filters are designed to be as compact as possible in frequency domain for velocity resolution and compact in the spatial domain for good spatial resolution. An alternative is to design filters such that their outputs are maximally informative. An information theoretic analysis highlights the importance of the input prior in filter design. Under the assumption of a simple input or that only the dominant component of a complex input needs to be estimated, it is shown that a filter bank of broadly tuned filters can be used to accurately estimate the input frequency. For such inputs there is an incremental gain in information at a huge increase in cost for filter banks using sets of filters tuned narrower than a certain width. The issues of resolution and complex inputs are also addressed. The output of a filter bank encodes information in a redundant manner making it robust to noise in the system. By developing a probability measure it is possible to make the output of the system generate probability distributions of parameter. Examples for orientation, spatial and temporal frequency are given.

Multiresolution estimation of motion using the wavelet transform

Enrique Barria,
Jogikal M. Jagadeesh

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Many applications in research, in manufacturing and in every day life require measurement of velocity. Generally the velocity is estimated from a sequence of images. There are many methods reported in the literature using heuristics and transform techniques. In general these methods are tailored to specific applications. We propose a wavelet transform based technique to estimate velocity. We assume that the scene can be piecewise modeled by a multi variable polynomial. We have derived wavelet functions that can be used to estimate velocity from components of the wavelet transform. The technique has been demonstrated for 1D analytical functions and the error is less than 0.5% for polynomials of finite degree, and less than 5% for sinusoids. For 2D sequence of images, the image space is represented by a spatio temporal cube, and partitioned into subcubes of different dimensions. Wavelet transform techniques can then be applied to these variable size segments at multiple resolution. To estimate the two velocity components V

_{x}and V_{y}in this spatio temporal cube requires the solution of a system of 8 non linear equations. We are in the process of solving these non linear equations.
Wavelet transform approach to the feature extraction for incoherent radar ship target

Jun Lu,
Wenxian Yu,
Yue Jin,
et al.

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In this paper, a new approach to the feature extraction for incoherent radar ship target recognition is proposed, based on the discrete dynamic wavelet transform. Experiments are carried out for the method with practical video-echo data of four kinds. The corresponding results indicate that the extracted features are typical with high data compression.

Multichannel texture segmentation using Talbot effect

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Textured image is considered as the repetition of some primitives with a certain rule of displacement, thus every region in the image has different periodic structure. The segmentation is realized by its self-imaging effect. A series of Fresnel images can be obtained at the fractional Talbot distances depended on the periodicity of the original. All these images are a summation of the Fourier frequency modulated by a phase factor, which related with the fractional Talbot distance. Therefore, these images can be considered as the multichannel Talbot transform of the original image and represent the texture features to a certain extent. By comparing these images, different texture regions are segregated. Theoretical analysis and primitive experimental results are presented.

Quantitative evaluation of wavelet-based image processing algorithms

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Wavelet analysis is currently being investigated as an image enhancement tool for use in mammography. Although this approach to image processing appears to have great promise, there remain major uncertainties regarding an optimal form of wavelet based algorithms. It is, therefore, desirable to have a quantitative method for evaluating a wavelet based image processing algorithm. Optimization of algorithms prior to evaluation using standard Receiver Operating Characteristic method is made possible. A mathematical method has been developed where the input signal is a gaussian with added random noise. An enhancement factor (EF) is obtained from input and output signal-to-noise ratios, SNR

_{i}and SNR_{o}, (EF equals SNR_{o}/SNR_{i}). The development and testing of this method is described, and a practical application in given showing the major features of a wavelet based image processing algorithm based on the Frazier-Jawerth transform.
Spectral radius of sets of matrices

Mohsen Maesumi

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Spectral radius of sets of matrices is a fundamental concept in studying the regularity of compactly supported wavelets. Here we review the basic properties of spectral radius and describe how to increase the efficiency of estimation of a lower bound for it. Spectral radius of sets of matrices can be defined by generalizing appropriate definitions of spectral radius of a single matrix. One definition, referred to as generalized spectral radius, is constructed as follows. Let (Sigma) be a collection of m square matrices of same size. Suppose L

_{n}((Sigma) ) is the set of products of length n of elements (Sigma) . Define p_{n}((Sigma) ) equals max_{A(epsilon}L_{n}[p(A)]^{1/n}where p(A) is the usual spectral radius of a matrix. Then the generalized spectral radius of (Sigma) is p((Sigma) ) equals lim sup_{nyields(infinity})p_{n}((Sigma) ). The standard method for estimating p((Sigma) ), through p_{n}((Sigma) ), involves m^{n}matrix calculations, one per each element of L_{n}((Sigma) ). We will describe a method which reduces this cost to m^{n}/n or less.Wavelet Design and Construction

Fractal function, splines, intertwining multiresolution analysis, and wavelets

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The construction of smooth, orthogonal compactly supported wavelets is accomplished using fractal interpolation functions and splines. These give rise to multiwavelets. In the latter case piecewise polynomial wavelets are exhibited using an intertwining multiresolution analysis.

Mathematical Developments II

Class of smooth nonseparable N-dimensional scaling functions

Mohsen Maesumi

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Wavelets of compact support are an important tool in many areas of signal analysis. A starting point for the construction of such wavelets is the scaling function, the solution of the dilation equation. We study the dilation equation (phi) (X) equals (Sigma)

_{K}C_{K(phi})(2X-K) where K (epsilon) {0...m}^{N}, (phi) :R^{N}yields R, C_{K}(epsilon) R. This paper gives a set of sufficient conditions on C_{K}under which the solution of the dilation equation has a specific degree of regularity. We will construct (phi) through infinite products of 2^{N}associated matrices with entries in terms of C_{K}. The conditions for regularity are based on certain sum rules that triangularize all of the associated matrices and on certain inequalities that control the eigenvalues of the matrices. The net effect of the sum rules is to specify the coefficients C_{K}in terms of a binomial interpolation of their values at the corners of the N-cube, {0,m}^{N}. The inequalities are based on sums of the coefficients at the corners of the various faces of the N-cube. The number of derivatives that (phi) possesses and the Holder exponent of the last derivative can be determined for the sums.