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Volume 4119 Wavelet Applications in Signal and Image Processing VIII

Akram Aldroubi, Andrew F. Laine, Michael A. Unser December 2000

Conference Location: San Diego, CA, USA Conference Date: Monday 31 July 2000

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Sampling and Approximation

Select this Article		Curvelets, multiresolution representation, and scaling laws Emmanuel J. Candes and David L. Donoho Proc. SPIE 4119, 1 (2000); http://dx.doi.org/10.1117/12.408568 \| Cited 4 times Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Curvelets provide a new multiresolution representation with several features that set them apart from existing representations such as wavelets, multiwavelets, steerable pyramids, and so on. They are based on an anisotropic notion of scaling. The frame elements exhibit very high direction sensitivity and are highly anisotropic. In this paper we describe these properties and indicate why they may be important for both theory and applications.

Select this Article		New Fourier reconstruction algorithms for computerized tomography Daniel Potts and Gabriele Steidl Proc. SPIE 4119, 13 (2000); http://dx.doi.org/10.1117/12.408606 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper we propose two new algorithms for high quality Fourier reconstructions of digital N by N images from their Radon transform. Both algorithms are based on fast Fourier transforms for nonequispaced data (NFFT) and require only O(N² log N) arithmetic operations. While the first algorithm includes a bivariate NFFT on the polar grid, the second algorithm consists of several univariate NFFTs on the so-called linogram.

Select this Article		Irregular sampling algorithm for general subspaces Shidong Li Proc. SPIE 4119, 24 (2000); http://dx.doi.org/10.1117/12.408616 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We provide a sufficient condition for irregular sampling for general subspaces. It is also shown that all existing results on sampling conditions can be directly applied to the sampling formula derived. The iterative implementation of the irregular sampling algorithm has the favorable convergence property of related frames. Examples on case studies and numerical results are also given, which shows that the condition and algorithm we derived are perhaps useful.

Select this Article		Sampling for shift-invariant and wavelet subspaces Jeffrey A. Hogan and Joseph D. Lakey Proc. SPIE 4119, 36 (2000); http://dx.doi.org/10.1117/12.408622 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We investigate several issues surrounding the general question of when a function in a finitely generated shift invariant subspace of L²(R) can be determined by certain of its sample values just as a function band limited to (-1/2, ) can be expressed in terms of its integer samples. The main theme here is how answers to this question depend on general properties of the generators of the shift invariant space, such as orthogonality properties, scaling relations, smoothness and so forth. One of the main issues that we address is the question of how to control aliasing error.

Select this Article		OFDM, Laurent operators, and time-frequency localization Thomas Strohmer Proc. SPIE 4119, 48 (2000); http://dx.doi.org/10.1117/12.408631 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Orthogonal frequency division multiplexing (OFDM) has gained considerable interest as an efficient technology for high- date-data transmission over wireless channels. The design of pulse shapes that are well-localized in the time-frequency plane is of great importance in order to combat intersymbol interference and interchannel interference caused by the mobile radio channel. Recently proposed methods to construct such well-localized functions are utilizing the link between OFDM and Gabor systems. We derive a theoretical framework that shows why and under which conditions these methods will yield well-localized pulse shapes. In our analysis we exploit the connection between Gabor systems, Laurent operators and the classical work of Gelfand, Raikov, and Shilov on commutative Banach algebras. In the language of Gabor analysis we derive a general condition under which the dual window and the canonical tight window inherit the decay properties of the analysis window.

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Continuous Wavelet Transform and Feature Detection

Select this Article		Analysis of random cascades using the wavelet transform: from theoretical concepts to experimental applications Alain Arneodo, J. Delour, and Jean F. Muzy Proc. SPIE 4119, 58 (2000); http://dx.doi.org/10.1117/12.408640 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We describe a formalism that allows us to study space (or time)-scale correlations in multiscale processes. This method, based on the continuous wavelet transform, is particularly well suited to study multiplicative random cascades for which the correlation functions take very simple expressions. This two-point space-scale statistical analysis is illustrated on synthetic multifractal signals and then applied to financial time series and fully developed turbulence data.

Select this Article		Application of two-dimensional continuous wavelet transform for pose estimation Lance M. Kaplan and Romain Murenzi Proc. SPIE 4119, 70 (2000); http://dx.doi.org/10.1117/12.408650 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We introduce a pose estimation method for SAR imagery using the 2D continuous wavelet transform (CWT). The computational complexity of the new approach is comparable to other image- based approaches such as ones that incorporate principle component analysis (PCA). Using the public domain MSTAR database, we show that the CWT-based method provides a better pose estimate than the PCA method.

Select this Article		Wavelet-based contrast computation and application to digital image watermarking Pierre Vandergheynst, Martin Kutter, and Stefan Winkler Proc. SPIE 4119, 82 (2000); http://dx.doi.org/10.1117/12.408658 \| Cited 2 times Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We introduce an isotopic measure of local contrast for natural images that is based on analytic filters and present the design of directional wavelet frames suitable for its computation. We show how this contrast measure can be used within a masking model to facilitate the insertion of a watermark in an image while minimizing visual distortion.

Select this Article		Shape from texture using continuous wavelet transform Hartmut Fuehr, Wen-Liang Hwang, and Bruno Torresani Proc. SPIE 4119, 93 (2000); http://dx.doi.org/10.1117/12.408668 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We give an algorithm for the estimation of the orientation of a planar surface with a homogeneous texture, viewed under perspective projection. We follow the two step procedure which is usually employed for this type of problem: First, a set of local distortion matrices is estimated - here we use wavelets -, then we determine the surface orientation which best fits the local distortions. In both parts the techniques we use are original.

Select this Article		Curvelets and reconstruction of images from noisy radon data Emmanuel J. Candes and David L. Donoho Proc. SPIE 4119, 108 (2000); http://dx.doi.org/10.1117/12.408569 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The problem of recovering an input signal form noisy and linearly distorted data arises in many different areas of scientific investigation; e.g., noisy Radon inversion is a problem of special interest and considerable practical relevance in medical imaging. We will argue that traditional methods for solving inverse problems - damping of the singular value decomposition or cognate methods - behave poorly when the object to recover has edges. We apply a new system of representation, namely the curvelets in this setting. Curvelets provide near-optimal representations of otherwise smooth objects with discontinuities along smooth C² edges. Inspired by some recent work on nonlinear estimation, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build a reconstruction based on the shrinkage of the noisy curvelet coefficients. This novel approach is shown to give a new theoretical understanding of the problem of edges in the Radon inversion problem.

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Functional Analysis of Wavelets I

Select this Article		Multiplexing of signals using superframes Radu V. Balan Proc. SPIE 4119, 118 (2000); http://dx.doi.org/10.1117/12.408578 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper we intend to present the concept of superframes and its use, primarily, in multiplexing techniques. The signal are supposed band limited and three multiplexing schemes are considered: time division multiple access, frequency division multiple access and frequency hoping multiple access (FHMA). The first two schemes give rise to tight superframes, whereas for FHMA, the associated superframes are more complex. For some such superframes the dual superframe is obtained in closed form. An example of a FHMA scheme is also presented.

Select this Article		Nonuniform sampling and spiral MRI reconstruction John J. Benedetto and Hui C. Wu Proc. SPIE 4119, 130 (2000); http://dx.doi.org/10.1117/12.408588 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract There is a natural formulation of the Classical Uniform Sampling Theorem in the setting of Euclidean space, and in the context of lattices, as sampling sets, and unit cells E, e.g., the Voronoi cell. For sampling at the Nyquist rate, the sampling function corresponds to the since function, and it is an integral over E. The set E is a tile for Euclidean space under translation by elements of the reciprocal lattice. We have a constructive, implementable non-uniform sampling theorem in the context of uniformly discrete sampling sets and sets E, corresponding to the unit cells of the uniform sampling result. The set E has the property that the translates by the sampling set of the polar set of E is a covering of Euclidean space. The theorem depends on the theory of frames, and can be viewed as a modest generalization of a theorem of Beurling. The application herein is to fast magnetic resonance imagine by direct signal reconstruction from spectral data on spirals.

Select this Article		Classifying characteristic functions giving Weyl-Heisenberg frames Peter G. Casazza and M. C. Lammers Proc. SPIE 4119, 142 (2000); http://dx.doi.org/10.1117/12.408596 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We examine the question of which characteristic functions yield Weyl-Heisenberg frames for various values of the parameters. We also give numerous applications of frames of characteristic functions to the general case (g, a, b).

Select this Article		Density of Gabor Schauder bases Baiqiao Deng and Christopher E. Heil Proc. SPIE 4119, 153 (2000); http://dx.doi.org/10.1117/12.408600 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract A Gabor system is a fixed set of time-frequency shifts G(g, (Lambda) ) equals [e^2(pi ib x)g(x-a)] (a,b) (epsilon) (Lambda) of a function g (epsilon) L²(R^d). We prove that if G(g, (Lambda) ) forms a Schauder basis for L²(R^d) then the upper Beurling density of (Lambda) satisfies D⁺((Lambda) ) <EQ 1. We also prove that if G(g, (Lambda) ) forms a Schauder basis for L²(R^d) and if g lies in a the modulation space M^1,1(R^d), which is a dense subset of L²(R^d), or if G(g, (Lambda) ) possesses at least a lower frame bound, then (Lambda) has uniform Beurling density D((Lambda) ) equals 1. We use related techniques to show that if g (epsilon) L¹(R^d) then no collection [ g(x-a)]_a(epsilon (Gamma) ) of pure translates of g can form a Schauder basis for L²(R^d). We also extend these results to the case of finitely many generating functions g_l,...,g_r.

Select this Article		Generalized frame multiresolution analysis of abstract Hilbert spaces and their applications Manos Papadakis Proc. SPIE 4119, 165 (2000); http://dx.doi.org/10.1117/12.408601 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We define a very generic class of multiresolution analysis of abstract Hilbert spaces. Their core subspaces have a frame produced by the action of an abelian unitary group on a perhaps infinite subset of the core subspace, which we call frame multi scaling vector set. We characterize the associated frame multi wavelet vector sets by generalizing the concept of the low and high pass filters and the Quadrature Mirror filter condition. We include an extensive overview of related work of other and we conclude with some examples.

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Natural Image Modeling

Select this Article		Hierarchical wavelet-based image model for pattern analysis and synthesis Clayton D. Scott and Robert D. Nowak Proc. SPIE 4119, 176 (2000); http://dx.doi.org/10.1117/12.408602 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Despite their success in other areas of statistical signal processing, current wavelet-based image models are inadequate for modeling patterns in images, due to the presence of unknown transformations inherent in most pattern observations. In this paper we introduce a hierarchical wavelet-based framework for modeling patterns in digital images. This framework takes advantage of the efficient image representations afforded by wavelets, while accounting for unknown pattern transformations. Given a trained model, we can use this framework to synthesize pattern observations. If the model parameters are unknown, we can infer them from labeled training data using TEMPLAR, a novel template learning algorithm with linear complexity. TEMPLAR employs minimum description length complexity regularization to learn a template with a sparse representation in the wavelet domain. We illustrate template learning with examples, and discuss how TEMPLAR applies to pattern classification and denoising from multiple, unaligned observations.

Select this Article		Learning optimal codes for natural images and sounds Michael S. Lewicki Proc. SPIE 4119, 185 (2000); http://dx.doi.org/10.1117/12.408603 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract A Bayesian method for inferring an optimal basis is applied to the problem of finding efficient codes for natural images. The key to the algorithm is multivariate non- Gaussian density estimation. This is equivalent, in various forms, to sparse coding or independent component analysis. The basis functions learned by the algorithm are oriented and localized in both space and frequency, bearing a resemblance to the spatial receptive fields of neurons in the primary visual cortex and to Gabor wavelet functions. An important advantage of the probabilistics framework is that it provides a method for comparing the coding efficiency of different bases objectively. The learned bases are shown to have better coding efficiency compared to traditional Fourier and wavelet bases. This framework can also be used to learn efficient codes of natural sound and the learned codes share many of the coding properties of the cochlear nerve. Time-frequency analysis is used to show that it is possible to derive both Fourier-like and wavelet-like representations by learning efficient codes for different classes of natural sounds.

Select this Article		Learning sparse wavelet codes for natural images Bruno A. Olshausen, Phil Sallee, and Michael S. Lewicki Proc. SPIE 4119, 200 (2000); http://dx.doi.org/10.1117/12.408604 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We show how a wavelet basis may be adapted to best represent natural images in terms of sparse coefficients. The wavelet basis, which may be either complete or overcomplete, is specified by a small number of spatial functions which are repeated across space and combined in a recursive fashion so as to be self-similar across scale. These functions are adapted to minimize the estimated code length under a model that assumes images are composed as a linear superposition of sparse, independent components. When adapted to natural images, the wavelet bases become selective to different spatial orientations, and they achieve a superior degree of sparsity on natural images as compared with traditional wavelet bases.

Select this Article		Morphological statistics of natural images and consequences Yann Gousseau Proc. SPIE 4119, 208 (2000); http://dx.doi.org/10.1117/12.408605 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We present statistics on natural images dealing with homogeneous and connected regions, defined as connected components of differences of level sets. We show that their area is distributed according to a power law. From these experimental results, we show precisely how the modeling of images as functions of bounded variation, widely used in image restoration methods, is incompletely adapted to natural images. For this purpose we use a model of convolution and sampling for the formation of images. We also use our experimental results to confirm the scale invariance of natural images.

Select this Article		Multiscale image segmentation using joint texture and shape analysis Ramesh Neelamani, Justin K. Romberg, Hyeokho Choi, Rudolf H. Riedi, and Richard G. Baraniuk Proc. SPIE 4119, 215 (2000); http://dx.doi.org/10.1117/12.408607 \| Cited 1 time Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We develop a general framework to simultaneously exploit texture and shape characterization in multiscale image segmentation. By posing multiscale segmentation as a model selection problem, we invoke the powerful framework offered by minimum description length (MDL). This framework dictates that multiscale segmentation comprises multiscale texture characterization and multiscale shape coding. Analysis of current multiscale maximum a posteriori segmentation algorithms reveals that these algorithms implicitly use a shape coder with the aim to estimate the optimal MDL solution, but find only an approximate solution.

Select this Article		Random cascades on wavelet trees and their use in analyzing and modeling natural images Martin J. Wainwright, Eero P. Simoncelli, and Alan S. Willsky Proc. SPIE 4119, 229 (2000); http://dx.doi.org/10.1117/12.408598 \| Cited 1 time Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We develop a new class of non-Gaussian multiscale stochastic processes defined by random cascades on trees of wavelet or other multiresolution coefficients. These cascades reproduce a rich semi-parametric class of random variables known as Gaussian scale mixtures. We demonstrate that this model class can accurately capture the remarkably regular and non- Gaussian features of natural images in a parsimonious fashion, involving only a small set of parameters. In addition, this model structure leads to efficient algorithms for image processing. In particular, we develop a Newton- like algorithm for MAP estimation that exploits very fast algorithm for linear-Gaussian estimation on trees, and hence is efficient. On the basis of this MAP estimator, we develop and illustrate a denoising technique that is based on a global prior model, and preserves the structure of natural images.

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Pattern Recognition/Extraction

Select this Article		Dyadic wavelet-based affine invariants for planar object recognition Mahmoud I. Khalil and Mohamed M. Bayoumi Proc. SPIE 4119, 241 (2000); http://dx.doi.org/10.1117/12.408608 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Dyadic wavelet transform has been used to derive affine invariant functions. The invariant functions are based on the dyadic wavelet transform of the object boundary. Two invariant functions have been calculated using different numbers of dyadic levels. Experimental results show that these invariant functions outperform some traditional invariant functions. The stability of these invariant functions have been tested for a large perspective transformation.

Select this Article		Multiresolution wedgelet transform for image processing Roger L. Claypoole, Jr. and Richard G. Baraniuk Proc. SPIE 4119, 253 (2000); http://dx.doi.org/10.1117/12.408609 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Edges in images convey a great deal of information, but wavelet transforms do not provide an economical representation. Thus, popular wavelet-based compression and restoration techniques perform poorly in the presence of edges. We present her a new multiresolution wedgelet transform based on the lifting construction. This transform provides an economical edge representation and thus offers the potential for improved image processing. We demonstrate this potential with applications in image denoising.

Select this Article		Dyadic frames of directional wavelets as texture descriptors Gloria Menegaz, Attilio Rivoldini, and Jean-Philippe Thiran Proc. SPIE 4119, 263 (2000); http://dx.doi.org/10.1117/12.408610 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We propose a wavelet-based texture classification system. Texture descriptors are local energy measures within the feature images obtained by projecting the samples on Dyadic Frames of Directional Wavelets. Rotation invariant features are obtained by taking the Fourier expansion of the subsets of components of the original feature vectors concerning each considered scale separately. Three different classification schemes have been compared: the Euclidean, the weighted Euclidean and the KNN classifiers. Performances have been evaluated on a set of 13 Brodatz textures, from which both a training set and a test set have been extracted. Results are present in the form of confusion matrices. The KNN classifier provides the globally best performance, with an average recognition rate around the 96 percent for the original non-rotated test set, and 88 percent when the rotated versions are considered. Its simplicity and accuracy renders the proposed method highly suited for multimedia applications, as content-based image retrieval.

Select this Article		Rotation-invariant texture retrieval using steerable wavelet-domain hidden Markov models Minh N. Do, Aurelie C. Lozano, and Martin Vetterli Proc. SPIE 4119, 274 (2000); http://dx.doi.org/10.1117/12.408611 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract A new statistical model for characterizing texture images base don wavelet-domain hidden Markov models and steerable pyramids is presented. The new model is shown to capture well both the subband marginal distributions and the dependencies across scales and orientations of the wavelet descriptors. Once it is trained for an input texture image, the model can be easily steered to characterize that texture at any other orientations. After a diagonalization operation, one obtains a rotation-invariant description of the texture image. The effectiveness of the new model is demonstrated in large test image databases where significant gains in retrieval performance are shown.

Select this Article		Learning optimal wavelets from overcomplete representations Hamid Eghbalnia and Amir H. Assadi Proc. SPIE 4119, 286 (2000); http://dx.doi.org/10.1117/12.408612 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Efficient and robust representation of signals has been the focus of a number of areas of research. Wavelets represent one such representation scheme that enjoys desirable qualitites such as time-frequency localization. Once the Mother wavelet has been selected, other wavelets can be generated as translated and dilated versions of the Mother wavelet in the 1D case. In the 2D case tensor product of two 1D wavelets is the most often used transform. Over complete representation of wavelets has proved to be of great advantage, both in sparse coding of complex scenes and multi-media data compression. On the other hand over completeness raises a number of technical difficulties for robust computation and systematic generalization of constructions beyond their original application domains.

Select this Article		High-resolution formant estimation and its application in frequency scaling of speech Douglas J. Nelson, S. Umesh, and Leon Cohen Proc. SPIE 4119, 294 (2000); http://dx.doi.org/10.1117/12.408613 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We present a method for accurate estimation of formant frequencies. The method is based on differentiating the phase of the short time Fourier transform. The motivation for the method is its application to the estimation of the recently introduced 'universal warping function' which is aimed at separating the speaker dependence from the phonetic content of a speech utterance. The universal warping function is determined by the nature of the relationship between formants of different speakers for phonetically similar sounds and requires an accurate estimate of formants. The proposed method provides sufficiently accuracy for its estimation.

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Lifting and Applications

Select this Article		Denoising via adaptive lifting schemes Wade K. Trappe and Kuo Juey R. Liu Proc. SPIE 4119, 302 (2000); http://dx.doi.org/10.1117/12.408614 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The lifting scheme is an effective method that provides flexible solutions for designing new perfect reconstruction filter bands. However, most of the existing applications of lifting are based upon stationary assumptions. As such, existing schemes tend to fit the data with a single, pre- determined model. These methods do not exploit the full flexibility provided by lifting. By exploiting the temporal interpretation of lifting, we incorporate adaptive filtering with the lifting scheme to cope with signals whose characteristics vary with time. In this paper, we study the proposed adaptive lifting scheme and its ability to decorrelate subbands. The decorrelation behavior is related proposed adaptive lifting scheme and its ability to decorrelate subbands. The decorrelation behavior is related to the coherence between the subbands, and simulations indicate improved decorrelation when compared with deterministic lifting. Our adaptive filterbank may be used in a thresholding scheme that can yield improved noise reduction capabilities compared to conventional wavelet thresholding schemes. We present a condition under which the proposed adaptive lifting denoising scheme can outperform a similar wavelet thresholding. Simulations are presented that indicate there is an SNR value at which the performance of adaptive lifting denoising surpasses wavelet denoising.

Select this Article		Minimum memory implementations of the lifting scheme Christos G. Chrysafis and Antonio Ortega Proc. SPIE 4119, 313 (2000); http://dx.doi.org/10.1117/12.408615 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract All publications on the lifting scheme up to now consider non-casual systems, where the assumption is that the whole input signal is buffered. This is problematic if we want to use lifting gin a low memory scenario. In this paper we present an analysis for making a lifting implementation of a filter bank causal, while at the same time reducing the amount of delay needed for the whole system. The amount of memory needed for the lifting implementation of any filter bank can br shown to be always smaller than the corresponding convolution implementation. The amount of memory saving sis filter bank dependent, it ranges form no savings for the Haar transform to 40 percent for a 2-10 filter bank. The amount of savings depends on the number of lifting steps as well as the length of the lifting steps used. We will also elaborate on the use of boundary extensions on each lifting step instead of the whole signal. This leads to lower memory requirements as well as simpler implementations.

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Functional Analysis of Wavelets II

Select this Article		Modular frames for Hilbert C*-modules and symmetric approximation of frames Michael Frank and David R. Larson Proc. SPIE 4119, 325 (2000); http://dx.doi.org/10.1117/12.408617 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We give a comprehensive introduction to a general modular frame construction in Hilbert C⁺-modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C⁺-modules over unital C*-algebras that admit an orthonormal modular Riesz basis. Interrelations and applications to classical frame theory are indicated. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. In particular, the existence and uniqueness of the closest tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closest tight frames often contains a multiple of its symmetric orthogonalization.

Select this Article		Gabor frames and operator algebras Jean P. Gabardo, Deguang Han, and David R. Larson Proc. SPIE 4119, 337 (2000); http://dx.doi.org/10.1117/12.408618 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We discuss three applications of operator algebra techniques in Gabor analysis: the parametrizations of Gabor frames, the incompleteness property, and the unique Gabor dual problem for subspace Gabor frames.

Select this Article		Compactly supported wavelets and representations of the Cuntz relations II Palle E. T. Jorgensen Proc. SPIE 4119, 346 (2000); http://dx.doi.org/10.1117/12.408619 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We show that compactly supported wavelets in L² (R) of scale N may be effectively parameterized with a finite set of spin vectors in C^N, and conversely that every set of spin vectors corresponds to a wavelet. The characterization is given in terms of irreducible representation of orthogonality relations defined form multiresolution wavelet filters.

Select this Article		Biorthogonality and multiwavelets in Hilbert spaces Wai-Shing Tang Proc. SPIE 4119, 356 (2000); http://dx.doi.org/10.1117/12.408620 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, we study various relations between Bessel sequences, frames, Riesz bases and biorthogonality in Hilbert spaces. We also consider the problem of existence of oblique multi wavelets in the setting of (1) and show that oblique multi wavelets exist under a very natural assumption.

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Multiwavelets

Select this Article		Image denoising using a local Gaussian scale mixture model in the wavelet domain Vasily Strela, Javier Portilla, and Eero P. Simoncelli Proc. SPIE 4119, 363 (2000); http://dx.doi.org/10.1117/12.408621 \| Cited 8 times Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The statistics of photographic images, when decomposed in a multiscale wavelet basis, exhibit striking non-Gaussian behaviors. The joint densities of clusters of wavelet coefficients are well-described as a Gaussian scale mixture: a jointly Gaussian vector multiplied by a hidden scaling variable. We develop a maximum likelihood solution for estimating the hidden variable from an observation of the cluster of coefficients contaminated by additive Gaussian noise. The estimated hidden variable is then used to estimate the original noise-free coefficients. We demonstrate the power of this model through numerical simulations of image denoising.

Select this Article		Multiwavelet characterization of function spaces adapted to the Navier-Stokes equations Joseph D. Lakey, S. Obeidat, and M. Cristina Pereyra Proc. SPIE 4119, 372 (2000); http://dx.doi.org/10.1117/12.408623 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We use wavelets based ona modification of the Geronimo- Hardin-Massopust construction to define localized extension/restriction operators form half-spaces to their full spaces/boundaries respectively. These operations are continuous in Sobolev and Morrey space norms. We also prove estimates for multiresolution projections of pointwise products of functions in these spaces. These are two of the key steps in extending results of Federbush and of Cannone and Meyer concerning solutions of Navier-Stokes with initial data in Sobolev and Morrey spaces to the case of half spaces and, ultimately, to more general domains with boundary.

Select this Article		Algorithm for the construction of symmetric and antisymmetric M-band wavelets Qiyu Sun Proc. SPIE 4119, 384 (2000); http://dx.doi.org/10.1117/12.408624 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, we give an algorithm to construct semi- orthogonal symmetric and anti-symmetric M-band wavelets. As an application, some semi-orthogonal symmetric and anti- symmetric or anti-symmetric M-band wavelets from a multiresolution, then that multiresolution has a symmetric scaling function.

Select this Article		Necessary conditions for the existence of multivariate multiscaling functions Carlos A. Cabrelli, Christopher E. Heil, and Ursula M. Molter Proc. SPIE 4119, 395 (2000); http://dx.doi.org/10.1117/12.408625 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper we outline the main ideas behind the recent proof of the authors that if a multivariate, multi-function refinement equation with an arbitrary dilation matrix has a continuous, compactly supported solution which has independent lattice translates, then the joint spectral radius of certain matrices restricted to an appropriate subspace is strictly less than one.

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Theoretical Issues

Select this Article		Classification of transforms Leon Cohen, Alan R. Lindsey, and Bruce W. Suter Proc. SPIE 4119, 407 (2000); http://dx.doi.org/10.1117/12.408626 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Lindsey and Suter have shown that for many transforms the transform of the convolution of two functions have the same functional form. We explain the origin of this result and derive the condition on the transformation kernel for when this should be the case. In addition we consider the general transform of the inverse and direct scale transform and obtain condition on the kernel so that the transform gives similar functional forms.

Select this Article		Perturbation stability of various coherent Riesz families Werner Kozek, Goetz E. Pfander, and Georg Zimmermann Proc. SPIE 4119, 411 (2000); http://dx.doi.org/10.1117/12.408627 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We compare three types of coherent Riesz families with respect to their perturbation stability under convolution with elements of a family of typical channel functions. This problem is of key relevance in the design of modulation signal sets for digital communication over time-invariant channels. Upper and lower bounds on the orthogonal perturbation are formulate din terms of spectral spread and temporal support of the prototype, and by the approximate design of worst case convolution kernels. Among the considered bases, the Weyl-Heisenberg structure which generates Gabor systems turns out to be optimal.

Select this Article		Lower bounds for finite wavelet and Gabor systems Ole Christensen and Alexander M. Lindner Proc. SPIE 4119, 420 (2000); http://dx.doi.org/10.1117/12.408628 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract For a function ΦϵL²(R) and parameters a>1, b>0, the corresponding wavelet family is the set of functions (function in paper). We show that for a dense set of functions (function in paper), every finite subfamily of the functions (function in paper) will be linearly independent. Under certain conditions, the result is also true for finite subsets (function in paper) of the irregular wavelet system (formula in paper), where (formula in paper). We estimate the corresponding lower Riesz bound, i.e., we find a positive number A such that (formula in paper) for all finite sequences (function in paper). We discuss cases from wavelet theory where it is crucial to have such estimates. We consider the same question for a Gabor family (formula in paper). A conjecture by Heil, Ramanathan and Topiwala states that when (formula in paper) and (formula in paper) consists of distinct points, then (formula in paper) will automatically be linearly independent. It is known that the conjecture holds in some special cases, e.g., under the assumption that g has support in a half line. We estimate the lower Riesz bound in that case.

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Novel Wavelet Features and Computations

Select this Article		Pseudo power signatures for image classification: SVD and projector approaches Vidya Venkatachalam Proc. SPIE 4119, 425 (2000); http://dx.doi.org/10.1117/12.408629 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Segmentation and classification are important problems with applications in areas like textural analysis and pattern recognition. Th is paper describes a single-0stage approach to solve the image segmentation/classification problem down to the pixel level, using energy density functions based on the wavelet transform. The energy density functions obtained, called Pseudo Power Signatures, are essentially functions of the scale and orientation, and are obtained using separable approximations to the 2D wavelet transform. A significant advantage of these representations is that they are invariant to signal magnitude, and spatial location within the object of interest. Further, they lend themselves to fast and simple classification routines. We provide a complete formulation of the signature determination problem for 2D, and propose an effective, albeit simple, technique based on a tensor singular value analysis, to solve the problem, We present an efficient computational algorithm, and a simulation result reflecting the strengths and limitations of this approach. We next present a detailed analysis of a more sophisticate method based on orthogonal projections to obtain signatures which are better representations of the underlying data.

Select this Article		Beamlet pyramids: a new form of multiresolution analysis suited for extracting lines, curves, and objects from very noisy image data David L. Donoho and Xiaoming Huo Proc. SPIE 4119, 434 (2000); http://dx.doi.org/10.1117/12.408630 \| Cited 1 time Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We describe a multiscale pyramid of line segments and develop algorithms which exploit that pyramid to recover image features - lines, curves, and blobs - from very noisy data.

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Sparsity and Statistical Independence in Image Representations

Select this Article		Learning sparse overcomplete image representations Bruno A. Olshausen and K. J. Millman Proc. SPIE 4119, 445 (2000); http://dx.doi.org/10.1117/12.408632 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We describe a method for learning an over complete set of basis functions for the purpose of modeling data with sparse structure. Such data re characterized by the fact that they require a relatively small number of non-zero coefficients on the basis functions to describe each data point. The sparsity of the basis function coefficients is modeled with a mixture-of-Gaussians distribution. One Gaussian captures non-active coefficients with a large-variance distribution centered at zero, while one or more other Gaussians capture active coefficients with a large-variance distribution. We show that when the prior is in such a form, there exist efficient methods for learning the basis functions as well as the parameters of the prior. The performance of the algorithm is demonstrated on a number of test cases and also on natural images. The basis functions learned on natural images are similar to those obtained with other methods, but the sparse from of the coefficient distribution is much better described. Also, since the parameters of the prior are adapted to the data, no assumption about sparse structure in the images need be made a priori, rather it is learned from the data.

Select this Article		Learning classes of efficient codes Te-Won Lee and Michael S. Lewicki Proc. SPIE 4119, 453 (2000); http://dx.doi.org/10.1117/12.408633 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We are interested in leaning efficient codes to represent classes of different images. The image classes are modeled using an ICA mixture model that assumes that the data was generated by several mutually exclusive data classes whose components are a mixture of non-Gaussian sources. The parameters of the model can be adapted using an approximate expectation maximization approach to maximize the data likelihood. We demonstrate that this method can learn classes of efficient codes to represent images that contain a variety of different structures. The learned codes can be used for image compression, de-noising and classification tasks. Compared to standard image coding methods, the ICA mixture model gives better encoding results because the codes are adapted to the structure of the data.

Select this Article		FOCUSS-based dictionary learning algorithms Kenneth Kreutz-Delgado and Bhaskar D. Rao Proc. SPIE 4119, 459 (2000); http://dx.doi.org/10.1117/12.408634 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Algorithms for data-driven learning of domain-specific over complete dictionaries are developed to obtain maximum likelihood and maximum a posteriori dictionary estimates based on the use of Bayesian models with concave/Schur- concave negative log-priors. Such priors are appropriate for obtaining sparse representations of environmental signals within an appropriately chosen dictionary. The elements of the dictionary can be interpreted as 'concepts,' features or 'words' capable of succinct expression of events encountered in the environment. This is a generalization of vector quantization in that one is interested in a description involving a few dictionary entries, but not necessarily as succinct as one entry. To learn an environmentally-adapted dictionary capable of concise expression of signals generated by the environment, we develop algorithms that iterate between a representative set of sparse representations found by variants of FOCUSS, an affine scaling transformation (ACT)-like sparse signal representation algorithm recently developed at UCSD, and an update of the dictionary using these sparse representations.

Select this Article		Sparsity vs. statistical independence from a best-basis viewpoint Naoki Saito, Brons M. Larson, and Bertrand Benichou Proc. SPIE 4119, 474 (2000); http://dx.doi.org/10.1117/12.408635 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We examine the similarity and difference between sparsity and statistical independence in image representations in a very concrete setting: use the best basis algorithm to select the sparsest basis and the least statistically- dependent basis from basis dictionaries for a given dataset. In order to understand their relationship, we use synthetic stochastic processes as well as the image patches of natural scene. Our experiments and analysis so far suggest the following: 1) Both sparsity and statistical independence criteria selected similar bases for most of our examples with minor differences; 2) Sparsity is more computationally and conceptually feasible as a basis selection criterion than the statistical independence, particularly for dat compression; 3) The sparsity criterion can and should be adapted to individual realization rather than for the whole collection of the realizations to achieve the maximum performance; 4) The importance of orientation selectivity of the local Fourier and brushlet dictionaries was not clearly demonstrated due to the boundary effect caused by the folding and local periodization.

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Wavelet Design I

Select this Article		Wavelets and radial basis functions: a unifying perspective Michael A. Unser and Thierry Blu Proc. SPIE 4119, 487 (2000); http://dx.doi.org/10.1117/12.408636 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Wavelets and radial basis functions (RBF) are two rather distinct ways of representing signals in terms of shifted basis functions. An essential aspect of RBF, which makes the method applicable to non-uniform grids, is that the basis functions, unlike wavelets, are non-local-in addition, they do not involve any scaling at all. Despite these fundamental differences, we show that the two types of representation are closely connected. We use the linear splines as motivating example. These can be constructed by using translates of the one-side ramp function, or, more conventionally, by using the shifts of a linear B-spline. This latter function, which is the prototypical example of a scaling function, can be obtained by localizing the one-side ramp function using finite differences. We then generalize the concept and identify the whole class of self-similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases. Conversely, we prove that, for any compactly supported scaling function, there exist a one-sided central basis function that spans the same multiresolution subspaces. The central property is that the multiresolution bases are generated by simple translation without any dilation.

Select this Article		Rhomboidal local cosine transform Douglas W. Warner, Dennis M. Healy, Jr., and Daniel N. Rockmore Proc. SPIE 4119, 494 (2000); http://dx.doi.org/10.1117/12.408637 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In the work, we describe a method for constructing non- separable multidimensional folding operators and discuss preliminary obtained with a discrete rhomboidal local cosine transform. Our construction extends related work by Xia and Suter and Bernardini and Kovacevic by generalizing the definition of folding operators to include the use of non- abelian symmetry groups. A family of prototypical dihedral folding operators allows one to decompose L²(R²) into n subspaces supported on approximate equiangular sectors. We draw directly on the representation theory of finite groups, making use of the group algebra structure. The folding operators do not incorporate windows. Instead, the folding operators are constructed directly by using elements of the matrix group SO(2n).

Select this Article		Multiresolution analysis structures generated by spline multiscaling functions Ada Cammilleri and Eduardo P. Serrano Proc. SPIE 4119, 506 (2000); http://dx.doi.org/10.1117/12.408638 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this work we expose some interesting properties of multiresolution structures generated from multiscaling functions. Particularly, we explore relations between different families of spline multiscaling functions embedded in a common multiresolution analysis.

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Wavelet Design II

Select this Article		Balanced nonseparable orthogonal multiwavelets with two and three vanishing moments on the quincunx grid Ana M. C. Ruedin Proc. SPIE 4119, 519 (2000); http://dx.doi.org/10.1117/12.408639 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We analyze the properties of orthogonality, short support, polynomial approximation order and balancing in the context of nonseparable bidimensional multi wavelets with quincunx decimation, and obtain conditions on the filter coefficients of the multi scaling function. These conditions are exploited to find examples of multi wavelets. The definition of balanced multi wavelets is extended to the bidimensional case for any dilation matrix. Relations between balancing and polynomial approximation order are investigated, and new are given. We find that for the dilation matrices chosen there can be no order are investigated, and new results are given. We find that for the dilation matrices chosen there can be no order 2 balanced multi wavelets of accuracy 2. The procedure for calculating the multi wavelet transform is outlined, and we given results of applying some of the wavelet found for image compression.

Select this Article		Generalized Haar wavelets and frames Goetz E. Pfander Proc. SPIE 4119, 528 (2000); http://dx.doi.org/10.1117/12.408641 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Generalized Haar wavelets were introduced in connection with the problem of detecting specific periodic components in noisy signals. We showed that the non-normalized continuous wavelet transform of a periodic function taken with respect to a generalized Haar wavelet is periodic in time as well as in scale, and that generalized Haar wavelets are the only bounded functions with this property.

Select this Article		Directional complex-wavelet processing Felix Fernandes, Rutger van Spaendonck, Mark J. Coates, and C. Sidney Burrus Proc. SPIE 4119, 536 (2000); http://dx.doi.org/10.1117/12.408642 \| Cited 3 times Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Poor directional selectivity, a major disadvantage of the separable 2D discrete wavelet transform (DWT), has previously been circumvented either by using highly redundant, nonseparable wavelet transforms or by using restrictive designs to obtain a pair of wavelet trees. In this paper, we demonstrate that superior directional selectivity may be obtained with no redundancy in any separable wavelet transform. We achieve this by projecting the wavelet coefficients to separate approximately the positive and negative frequencies. Subsequent decimation maintains non-redundancy. A novel reconstruction step guarantees perfect reconstruction within this critically- sampled framework. Although our transform generates complex- valued coefficients, it may be implemented with a fast algorithm that uses only real arithmetic. We also explain how redundancy may be judiciously introduced into our transform to benefit certain applications. To demonstrate the efficacy of our projection technique, we show that it achieves state-of-the-art performance in a seismic image- processing application.

Select this Article		Canonical angles, subspace partitioning, and hyrbrid wavelet packets Robert A. Hedges Proc. SPIE 4119, 547 (2000); http://dx.doi.org/10.1117/12.408643 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In previous work, hybrid wavelet packets were introduced as a generalization of wavelet packets in which the choice of quadrature mirror filter (QMF) is selected adaptively within the wavelet packet analysis. This was motivated by the observation that for certain classes of signals, the choice of appropriate QMF is not only signal dependent, but may be scale dependent as well. Best Basis selection was generalized to provide a means of optimizing the representation of a given signal. This method can be viewed as an adaptive partitioning of signal subspaces. In such a scheme it is important to determine a small number of QMFs which will provide some diversity in the partitioning of the subspaces. In the current work, canonical correlations and canonical angles are used to quantify the difference between subspaces spanned by two filter pairs. Preliminary results indicate the utility of this method as a data independent metric for filter comparison.

Select this Article		Unconditional Haar bases for Lebesgue spaces on spaces of homogeneous type Hugo Aimar and Osvaldo Gorosito Proc. SPIE 4119, 556 (2000); http://dx.doi.org/10.1117/12.408644 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We show that spaces of homogeneous type are adequate structures on which the unbalanced wavelet of Girabardi and Sweldens, can be constructed with an additional geometric control for the size of the nested partitions, given by the underlying quasi-distance. Moreover, we show that if a non- degeneracy condition is satisfied, we can still apply the Calderon-Zygmund theory in order to get the characterization of L^p spaces.

Select this Article		Lifting scheme of biorthogonal wavelet transform based on discrete interpolatory splines Amir Z. Averbuch, Alexander B. Pevnyi, and Valery A. Zheludev Proc. SPIE 4119, 564 (2000); http://dx.doi.org/10.1117/12.408645 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, we present a new family of biorthogonal wavelet transforms and a related library of biorthogonal periodic symmetric waveforms. For the construction we used the interpolatory discrete splices which enabled us to design a library of perfect reconstruction filter banks. These filter banks are related to Buttersworth filters. The construction is performed in a 'lifting' manner. The difference from the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast Fourier transform. Two ways to choose the control filters are suggested. The proposed scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. These filters have linear phase property and the basic waveforms are symmetric. In addition, these filters yield perfect frequency resolution.

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Wavelets and Medical Imaging

Select this Article		Tomographic reconstruction with nonlinear diagonal estimators Jerome Kalifa, Andrew F. Laine, and Peter D. Esser Proc. SPIE 4119, 576 (2000); http://dx.doi.org/10.1117/12.408646 \| Cited 1 time Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In tomographic reconstruction, the inversion of the Radon transform in the presence of noise is numerically unstable. Reconstruction estimators are studied where the regularization is performed by a thresholding in a wavelet or wavelet packet decomposition. These estimators are efficient and their optimality can be established when the decomposition provides a near-diagonalization of the inverse Radon transform operator and a compact representation of the object to be recovered. Several new estimators are investigated in different decomposition. First numerical results already exhibit a strong metrical and perceptual improvement over current reconstruction methods. These estimators are implemented with fast non-iterative algorithms, and are expected to outperform Filtered Back- Projection and iterative procedures for PET, SPECT and X-ray CT devices.

Select this Article		Reconstruction in emission tomography via a Bayesian multiscale statistical framework Eric D. Kolaczyk and Robert D. Nowak Proc. SPIE 4119, 587 (2000); http://dx.doi.org/10.1117/12.408647 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Recently the authors introduced a general Bayesian statistical method for modeling and analysis in linear inverse problems involving certain types of count data. Emission-based tomography is medical imaging is a particularly important and common examples of this type of proem. In this paper we provide an overview of the methodology and illustrate its application to problems in emission tomography through a series of simulated and real- data examples. The framework rests on the special manner in which a multiscale representation of recursive dyadic partitions interacts with the statistical likelihood of data with Poisson noise characteristics. In particular, the likelihood function permits a factorization, with respect to location-scale indexing, analogous to the manner in which, say, an arbitrary signal allows a wavelet transform. Recovery of an object from tomographic data is the posed as a problem involving the statistical estimation of a multiscale parameter vector. A type of statistical shrinkage estimation is used, induced by careful choice of a Bayesian prior probability structure for the parameters. Finally, the ill-posedness of the tomographic imaging problem is accounted for by embedding the above-described framework within a larger, but simpler statistical algorithm problem, via the so-called Expectation-Maximization approach. The resulting image reconstruction algorithm is iterative in nature, entailing the calculation of two closed-form algebraic expression at each iteration. Convergence of the algorithm to a unique solution, under appropriate choice of Bayesian prior, can be assured.

Select this Article		Wavelet-based image processing for digital mammography Wei Qian, Lihua Li, Xuejun Sun, and Robert A. Clark Proc. SPIE 4119, 596 (2000); http://dx.doi.org/10.1117/12.408648 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract This paper is to evaluate the importance of image preprocessing using multiresolution and multiorientation wavelet transforms on the performance of a previously reported computer assisted diagnostic (CAD) method for breast cancer screening, using digital mammography. An analysis of the influence of WTs on image feature extraction for mass detection is achieved by comparing the discriminate ability of features extracted with and without wavelet based image preprocessing using computed ROC. Three indexes are proposed to assess the segmentation of the mass area with comparison to ground truth. Dat was analyzed on region-of- interest database that included mass and normal regions from digitized mammograms with ground truth. The metrics for measurement of segmentation of the mass clearly demonstrates the importance of image preprocessing methods. Similarly, the relative improvement in performance is observed in feature extraction, where the Az values are increased. The improvement depends on the feature characteristics. The use of methodology in this paper result sin a significant improvement in feature extraction for the previously proposed CAD detection method. We are therefore exploring additional improvement in wavelet based image preprocessing methods, including adaptive methods, to achieve a further improvement in performance on larger image databases.

Select this Article		Spatiotemporal directional analysis of 4D echocardiography Elsa D. Angelini-Casadevall, Andrew F. Laine, Shin Takuma, and Shunichi Homma Proc. SPIE 4119, 605 (2000); http://dx.doi.org/10.1117/12.408649 \| Cited 1 time Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Speckle noise corrupts ultrasonic data by introducing sharp changes in an echocardiographic image intensity profile, while attenuation alters the intensity of equally significant cardiac structures. These properties introduce inhomogeneity in the spatial domain and suggests that measures based on phase information rather than intensity are more appropriate for denoising and cardiac border detection. The present analysis method relies on the expansion of temporal ultrasonic volume data on complex exponential wavelet-like basis functions called Brushlets. These basis functions decompose a signal into distinct patterns of oriented textures. Projected coefficients are associated with distinct 'brush strokes' of a particular size and orientation. 4D overcomplete brushlet analysis is applied to temporal echocardiographic values. We show that adding the time dimension in the analysis dramatically improves the quality and robustness of the method without adding complexity in the design of a segmentation tool. We have investigated mathematical and empirical methods for identifying the most 'efficient' brush stroke sizes and orientations for decomposition and reconstruction on both phantom and clinical data. In order to determine the 'best tiling' or equivalently, the 'best brushlet basis', we use an entorpy-based information cost metric function. Quantitative validation and clinical applications of this new spatio-temporal analysis tool are reported for balloon phantoms and clinical data sets.

Select this Article		Activation detection in fMRI data via multiscale singularity detection Ralf Mekle, Andrew F. Laine, Gerard M. Perera, and Robert DeLaPaz Proc. SPIE 4119, 615 (2000); http://dx.doi.org/10.1117/12.408651 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Detection of active areas in the brain by functional magnetic resonance imaging (fMRI) is a challenging problem in medical imaging. Moreover, determining the onset and end of activation signal at specific locations in 3-space can determined networks of temporal relationships required for brain mapping. We introduce a method for activation detection in fMRI data via wavelet analysis of singular features. We pose the problem of determining activated areas as singularity detection in the temporal domain. Overcomplete wavelet expansion at integer scales are used to trace wavelet modulus maxima across scales to construct maxima lines. Form these maxima lines, singularities in the signal are located corresponding to the onset and end of an activation signal. We present result for simulated phantom waveforms and clinical fMRI dat from human finger tapping experiments. Different levels of noise were added to two waveforms of phantom data. No assumptions about specific frequency and amplitude of an activation signal were made prior to analysis. Detection was reliable for modest levels of random noise, but less precise at higher levels. For clinical fMRI data, activation maps were comparable to those of existing standard techniques.

Select this Article		Optimizing wavelets for the analysis of fMRI data Manuela Feilner, Thierry Blu, and Michael A. Unser Proc. SPIE 4119, 626 (2000); http://dx.doi.org/10.1117/12.408652 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Ruttiman et al. Have proposed to use the wavelet transform for the detection and localization of activation patterns in functional magnetic resonance imaging (fMRI). Their main idea was to apply a statistical test in the wavelet domain to detect the coefficients that are significantly different form zero. Here, we improve the original method in the case of non-stationary Gaussian noise by replacing the original z-test by a t-test that takes into account the variability of each wavelet coefficient separately. The application of a threshold that is proportional to the residual noise level. After the reconstruction by an inverse wavelet transform, further improves the localization of the activation pattern in the spatial domain.

Select this Article		Decomposition of fMRI data into multiple components Francois G. Meyer Proc. SPIE 4119, 638 (2000); http://dx.doi.org/10.1117/12.408653 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The goal of this work is to provide a new representation of functional magnetic resonance imaging (fMRI) time series. Functional neuroimaging aims at quantifying and localizing neuronal activity using imaging techniques. Functional MRI can detect and quantify hemodynamic changes induced by brain activation and neuronal activity. The time course of the fMRI signal at a given voxel inside the brain is represented with a structural model where each component of the model belongs to a subspace spanned by a small number of basis functions. The basis functions in different subspaces have very distinct time-frequency characteristics. The large scale trend of the signal is represented with a combination of large scale wavelets. The response to the stimulus is expanded on a small set of basis functions. Because it is critical to adapt the basis functions to the type of stimulus, the evoked response to a random presentation is expanded into small scale wavelets or wavelet packets, while the response to a periodic stimulus is represented with cosine or sine functions. We illustrate the estimation of the components of the model with several experiments.

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Wavelets and Approximation Theory

Select this Article		Image data compression with three-directional splines M. Charina, Costanza Conti, and Kurt Jetter Proc. SPIE 4119, 650 (2000); http://dx.doi.org/10.1117/12.408654 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper we consider a new method for image data compression. It is based on three-directional spline functions of low degree, viz. Piecewise constant functions, and piecewise cubic C^-1-functions. In the first case, a Haar wavelet type decomposition can be derived, and combined with standard thresholding techniques. In the second case, due to the fact that a splice basis is given by convolution products, the wavelet decomposition and thresholding can be computed on one factor of the convolution product only. Performance of the proposed method is discussed in section 3 where the reconstructed pictures are compared with the ones produced by the analogous decomposition methods provided by the MATLAB wavelet toolbox.

Select this Article		Cone-beam x-ray tomography using continuous wavelet transformation Shiying Zhao and Ge Wang Proc. SPIE 4119, 660 (2000); http://dx.doi.org/10.1117/12.408655 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Recently, medical computed tomography (CT) began a transition from fan-beam to cone-beam geometry with the introduction of multi-row-detector systems. Therefore, cone- beam techniques become important for medical CT. Despite recent advances, the approximate reconstruction method of Feldkamp remains the most commonly employed cone-beam reconstruction algorithm because of its computational efficiency and clinical applicability. Unfortunately, the derivation of the Feldkamp cone-beam reconstruction formula is based on geometric tilted fan neuristic. In this paper, we given a wavelet derivation of the Feldkamp cone-beam reconstruction method. It is found that the Feldkamp algorithm is an outcome of a zero-order approximation to the longitudinal wavelet decomposition of the object function to be reconstructed. Since the approximation is explicitly given, error estimates can be derived analytically. Theoretically, it also arises the possibility of improvement of the Feldkamp cone-beam algorithm.

Select this Article		Connection between p-frames and p-Riesz bases in locally finite SIS of L^p(R) Akram Aldroubi, Qiyu Sun, and Wai-Shing Tang Proc. SPIE 4119, 668 (2000); http://dx.doi.org/10.1117/12.408656 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, we consider the p-frame property of the space V_p(\Phi) with the generator \Phi being compactly supported function. Moreover, for the one-dimensional case, we show that the p-frame and p-Riesz basis properties are essentially the same for the space V_p(\Phi).

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Multiscale Image Processing

Select this Article		Information-theoretic interpretation of Besov spaces Hyeokho Choi and Richard G. Baraniuk Proc. SPIE 4119, 675 (2000); http://dx.doi.org/10.1117/12.408657 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Besov spaces classify signals and images through the Besov norm, which is based on a deterministic smoothness measurement. Recently, we revealed the relationship between the Besov norm and the likelihood of an independent generalized Gaussian wavelet probabilistic model. In this paper, we extend this result by providing an information- theoretical interpretation of the Besov norm as the Shannon codelength for signal compression under this probabilistic mode. This perspective unites several seemingly disparate signal/image processing methods, including denoising by Besov norm regularization, complexity regularized denoising, minimum description length processing, and maximum smoothness interpolation. By extending the wavelet probabilistic model, we broaden the notion of smoothness space to more closely characterize real-world data. The locally Gaussian model leads directly to a powerful wavelet- domain. Wiener filtering algorithm for denoising.

Select this Article		Mixed memory model for image processing and modeling with complex Daubechies wavelets Diego Clonda, Jean-Marc Lina, and Bernard Goulard Proc. SPIE 4119, 686 (2000); http://dx.doi.org/10.1117/12.408659 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, we propose a statistical modeling of images based on a decomposition with complex-valued Daubechies wavelets. These wavelets possess interesting properties that can be turned into account in the modeling to obtain a better characterization of the images. This characterization is achieved by statistically modeling the wavelet coefficient distribution via hidden Markov tree model. The wavelet coefficients in an image are organized into three tree structures and this type of model has already been used successfully in this context by independently modeling each of these trees. We propose a further refinement by considering the joint modeling of the three trees with a so- called mixed memory hidden Markov tree model. The mode is base don a memory mixture, a general approach to obtain an approximation of the joint distribution in the presence of factorial Markov models. The utilization of such model s is quite general and can be applied to various signal- processing problems. To illustrate the interest of this model as well as the relevance of using complex Daubechies wavelets, we evaluate their performance for a classification and a denoising application.

Select this Article		Multiplicative multiscale image decompositions: analysis and modeling Justin K. Romberg, Rudolf H. Riedi, Hyeokho Choi, and Richard G. Baraniuk Proc. SPIE 4119, 698 (2000); http://dx.doi.org/10.1117/12.408660 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Multiscale processing, in particular using the wavelet transform, has emerged as an incredibly effective paradigm for signal processing and analysis. In this paper, we discuss a close relative of the Haar wavelet transform, the multiscale multiplicative decomposition. While the Haar transform captures the differences between signal approximations at different scales, the multiplicative decomposition captures their ratio. The multiplicative decomposition has many of the properties that have made wavelets so successful. Most notably, the multipliers are a sparse representative for smooth signals, they have a dependency structure similar to wavelet coefficients, and they can be calculated efficiently. The multiplicative decomposition is also a more natural signal representation than the wavelet transform for some problems. For example, it is extremely easy to incorporate positivity constraints into multiplier domain processing. In addition, there is a close relationship between the multiplicative decomposition and the Poisson process; a fact that has been exploited in the field of photon-limited imaging. In this paper, we will show that the multiplicative decomposition is also closely tied with the Kullback-Leibler distance between two signals. This allows us to derive an n-term KL approximation scheme using the multiplicative decomposition.

Select this Article		Non-Euclidean pyramids Maria A. Munoz Barrutia, Thierry Blu, and Michael A. Unser Proc. SPIE 4119, 710 (2000); http://dx.doi.org/10.1117/12.408661 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We propose to design the reduction operator of an image pyramid so as to minimize the approximation error in the l_p sense where p can take non-integer values. The underlying image model is specified using arbitrary shift- invariant basis functions such as splines. The solution is determined by an iterative optimization algorithm, based on digital filtering. Its convergence is accelerated by the use of first and second derivatives. For p equals 1, our modified pyramid is robust to outliers; edges are preserved better than in the standard case where p equals 2. For 1 < p < 2, the pyramid decomposition combines the qualities of l₁ and l₂ approximations. The method is applied to edge detection and its improved performance over the standard formulation is determined.

Select this Article		Karhunen-Loeve multispectral and multiscale image restoration Jean-Luc Starck, Philippe Querre, and David L. Donoho Proc. SPIE 4119, 721 (2000); http://dx.doi.org/10.1117/12.408662 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We introduce in this paper the notion of WT-KLT and apply it to the problem of noise removal. Decorrelating first the data in the spatial domain using the WT and afterwards using the KLT in spectral domain allows us to derive a roust noise modeling in the WT-KLT space, and hence to filter the transformed data in an efficient way. Experiments are performed in order to derive (i) the best way to calculate the covariance matrix in the case of noisy data, (ii) the best method to correct the noisy WT-KLT coefficients. Finally we investigate if the curvelet transform could be an alternative to the wavelet transform for color image filtering.

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Multirate Filter Bank Applications

Select this Article		Iterated oversampled filter banks and wavelet frames Ivan W. Selesnick and Levent Sendur Proc. SPIE 4119, 733 (2000); http://dx.doi.org/10.1117/12.408663 \| Cited 1 time Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract This paper takes up the design of wavelet tight frames that are analogous to Daubechies orthonormal wavelets - that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. The oversampled dyadic DWT considered in this paper is based on a single scaling function and tow distinct wavelets. Having more wavelets than necessary gives a closer spacing between adjacent wavelets within the same scale. As a result, the transform is nearly shift-invariant, and can be used to improve denoising. Because the associated time- frequency lattice preserves the dyadic structure of the critically sampled DWT it can be used with tree-based denoising algorithms that exploit parent-child correlation.

Select this Article		Minimum-redundancy ISI-free FIR filterbank transceivers Yuan-Pei Lin and See-May Phoong Proc. SPIE 4119, 745 (2000); http://dx.doi.org/10.1117/12.408664 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The discrete multitone modulation system (DMT) has been demonstrated to be a very useful technique for high speed transmission over frequency selective channels such as the digital subscriber loops. The DMT system can be realized using a filterbank transceiver, the synthesis bank as the transmitter and the analysis bank as the receiver. With proper time domain equalization, the channel can usually be modeled as an FIR filter with order L. It is known that if a redundancy of length L is introduced, FIR filterbank transceivers with zero ISI property can be achieved. For example, in DFT based DMT system, redundancy is introduced by adding a cyclic prefix of length L. In this paper, we will derive the minimum length of redundancy required for FIR filterbank transceivers with ISI free property. For a given channel, we will show that the minimum length is directly related to the Smith form of an appropriately defined channel matrix.

Select this Article		Issues in nonuniform filter banks Soura Dasgupta and Ashis Pandharipande Proc. SPIE 4119, 756 (2000); http://dx.doi.org/10.1117/12.408665 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract This paper concerns biorthogonal filter banks. It is shown that a tree structured filter bank is biorthogonal if it is equivalent to a tree structured filter bank whose matching constituent levels on the analysis and synthesis sides are themselves biorthogonal pairs. We then show that a stronger statement can be made about dyadic Filter Banks in general: That a dyadic filter bank is biorthogonal if both the analysis and synthesis banks can be decomposed into dyadic trees. We further show that these decompositions are stability preserving. These results thus generalize earlier comparable results for orthogonal filter banks.

Select this Article		Nonuniform principal component filter banks: definitions, existence, and optimality Sony J. Akkarakaran and Palghat P. Vaidyanathan Proc. SPIE 4119, 768 (2000); http://dx.doi.org/10.1117/12.408666 \| Cited 1 time Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The optimality of principal component filter banks (PCFBs) for data compression has been observed in many works to varying extents. Recent work by the authors has made explicit the precise connection between the optimality of uniform orthonormal filter banks (FBs) and the principal component property: The PCFB is optimal whenever the minimization objective is a concave function of the subband variances of the FB. This gives a unified explanation of PCFB optimality for compression, denoising and progressive transmission. However not much is known for the case when the optimization is over a class of nonuniform Fbs. In this paper we first define the notion of a PCFB for a class of nonuniform orthonormal Fbs. We then show how it generalizes the uniform PCFBs by being optimal for a certain family of concave objectives. Lastly, we show that existence of nonuniform PCFBs could imply severe restrictions on the input power spectrum. For example, for the class of unconstrained orthonormal nonuniform Fbs with any given set of decimators that are not all equal, there is no PCFB if the input spectrum is strictly monotone.

Select this Article		Multiresolution analysis with decimated wavelet transform for image zooming Frederic Nicolier and Frederic Truchetet Proc. SPIE 4119, 780 (2000); http://dx.doi.org/10.1117/12.408667 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Blurred images are produced by interpolation process. A wavelet-based magnification method is proposed that both increases the resolution of an image and adds local high- frequency information, in order to provide digitally zoomed images with sharp edges. Wavelet transforms computed by the decimated Mallat's algorithm present pyramidal aspect. This pyramidal analysis combined with a prediction of high- frequency coefficients is used to produce a magnified image. The prediction is based on a zero-crossings representation and on the construction of a multiscale edge-signature database. Performances are evaluated for synthetic and noisy images. A significant improvement regarding some classical methods is observed.

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Advances in Multiple Description Coding

Select this Article		Optimal overcomplete subband expansions for multiple description coding Sanjeev Mehrotra Proc. SPIE 4119, 787 (2000); http://dx.doi.org/10.1117/12.408669 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We study the use of overcomplete subband expansions for multiple description coding. The system proposed uses an overcomplete subband expansion followed by independent coding of each of the subbands. The entire coding of each of the subbands is entirely contained within a single description. Using analysis in the polyphase domain, we study finding frequency response of optimal filterbanks and the rate allocation across the subbands to minimize the end- to-end reconstruction error subject to a transmission rate constraint when a distribution over channel states is known. A general analysis of the problem is given and results are shown for a 3 by 2 expansion of a first-order autoregressive process.

Select this Article		Setting priorities: a new SPIHT-compatible algorithm for image compression Diego Dugatkin and Michelle Effros Proc. SPIE 4119, 799 (2000); http://dx.doi.org/10.1117/12.408670 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We introduce a new algorithm for progressive or multiresolution image compression. The algorithm improves on the Set Partitioning in Hierarchical Trees (SPIHT) algorithm by replacing the SPIHT encoder. The new encoder optimizes the multiresolution code performance relative to a user- defined probability distribution over the code's rates or resolutions. The new algorithm's decoder is identical to the SPIHT decoder. The resulting code achieves the optimal expected performance across resolutions subject to the constraints imposed by the use of the SPIHT decoder and the distribution over resolutions set by the user. The encoder optimization yields performance improvements at the rates or resolutions of greatest importance at the expense of performance degradation at low priority rates or resolutions. The algorithm is fully compatible at the decoder with the original SPIHT algorithm. In particular, the decoder requires no knowledge of the priority function employed at the encoder. Experimental results on an image containing both text and photographic material yield up to 0.86 dB performance improvement over SPIHT at the resolution of highest priority.

Select this Article		Graphical statistical modeling of wavelet coefficients and its applications Igor V. Kozintsev, Shu Xiao, and Kannan Ramchandran Proc. SPIE 4119, 806 (2000); http://dx.doi.org/10.1117/12.408671 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Wavelet transform-based methods are currently used in a variety of image and video processing applications and are popular candidates for future image and video processing standards. Very little, however, has been done to develop efficient and simple stochastic models for wavelet image data. In this paper we review some existing modeling approaches for wavelet image data. Inspired by our recent estimation-quantization image coder, we introduce an efficient graphical stochastic model for wavelet image coefficients. Specifically, we propose to model wavelet image coefficients as Gaussian random variables with parameters determined by an underlying hidden Markov-type process. This stochastic model is defined using a factor graph framework. We test our model for denoising images corrupted by additive Gaussian noise. Our results are among the state-of-the-art in the field and they indicate the promise of the proposed model.

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Denoising

Select this Article		Shift-invariant Gibbs-free denoising algorithm based on wavelet transform footprints Pier L. Dragotti and Martin Vetterli Proc. SPIE 4119, 821 (2000); http://dx.doi.org/10.1117/12.408672 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In recent years wavelet have had an important impact on signal processing theory and practice. The effectiveness of wavelets is mainly due to their capability of representing piecewise smooth signals with few non-zero coefficients. Away from discontinuities, the inner product between a wavelet and a smooth function will be either zero or very small. At singular points, a finite number of wavelets concentrated around the discontinuity lead to non-zero inner products. This ability of wavelet transform to pack the main signal information in few large coefficients is behind the success of wavelet based denoising algorithms. Indeed, traditional approaches simply consist in thresholding the noisy wavelet coefficients, so the few large coefficients carrying the essential information are usually kept while small coefficients mainly containing, so the few large coefficients carrying the essential information are usually kept while small coefficients mainly containing noise are canceled. However, wavelet denoising suffers of two main drawbacks: it is not shift-invariant and it exhibits pseudo Gibbs phenomenon around discontinuities.

Select this Article		Image denoising using orthonormal finite ridgelet transform Minh N. Do and Martin Vetterli Proc. SPIE 4119, 831 (2000); http://dx.doi.org/10.1117/12.408673 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract This paper addresses the image denoising problem using a newly proposed digital image transform: the finite rigdelet transform (FRIT). The transform is invertible, non-redundant and achieved via fast algorithms. Furthermore this transform can be designed to be orthonormal thus indicating its potential in many other image processing applications. We then propose various improvements on the initial design of the FRIT in order to make it to have better energy compaction and to reduce the border effect. Experimental results show that the new transform outperforms wavelets in denoising images with linear discontinuities.

Select this Article		Wavelet and multirate denoising for signal-dependent noise Bruno Aiazzi, Luciano Alparone, Stefano Baronti, and Andrea Garzelli Proc. SPIE 4119, 843 (2000); http://dx.doi.org/10.1117/12.408674 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, after reviewing a general model to deal with signal-dependent image noise, the well known local linear minimum mean squared error (LLMMSE) filter is derived for the most general case. Signal-dependent noise filtering is approached in a multiresolution framework either by LLMMSE processing ratios of combinations of lowpass images, which are tailored to the noise model in order to mitigate its signal-dependence, or by thresholding a normalized nonredundant wavelet transform designed to yield signal- independent noisy coefficients as well. Experimental results demonstrate that the Laplacian pyramid approach largely outperform LLMMSE filtering on a unique scale and is still superior to wavelet denoising by soft-thresholding.

Select this Article		Wavelet despiking of fractographs Jean-Marie Aubry and Naoki Saito Proc. SPIE 4119, 853 (2000); http://dx.doi.org/10.1117/12.408675 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Fractographs are elevation maps of the fracture zone of some broken material. The technique employed to create these maps often introduces noise composed of positive or negative 'spikes' that must be removed before further analysis. Since the roughness of these maps contains useful information, it must be preserved. Consequently, conventional denoising techniques cannot be employed. We use continuous and discrete wavelet transforms of these images, and the properties of wavelet coefficients related to pointwise Hoelder regularity, to detect and remove the spikes.

Select this Article		Wavelet denoising by MSFI and NSFI EnBing Lin Proc. SPIE 4119, 861 (2000); http://dx.doi.org/10.1117/12.408676 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Image denoising in wavelet transform domain using thresholding is one of the most effective techniques of image enhancement. Even though the thresholding on the wavelet transform domain generally works well, one can still visually note the remaining noise in the resulting image, especially in the non-edge areas. In this paper, we use multi-scaling function interpolation and nonseparable scaling function interpolation to smooth the noise image. Several simulation results will be presented and comparisons between MSFI and NSFI methods will be shown.

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Poster Session

Select this Article		Early recognition of Alzheimer's disease in EEG using recurrent neural network and wavelet transform Arthur A. Petrosian, Danil Prokhorov, and Randolph B. Schiffer Proc. SPIE 4119, 870 (2000); http://dx.doi.org/10.1117/12.408570 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The diagnosis of Alzheimer's disease (AD) at the present time remains dependent upon clinical symptomatology. Lifetime accuracy in the best clinics remains 86-89 percent, and mean diagnostic delay in the clinical course of the disease remains 3.6 years after symptomatic onset. Although EEG is an obvious quantitative parameter related to the illness, its limitation is the absence of an identified set of features that discriminates AD EEG abnormalities form those due to confounding conditions. As a consequence, no computerized method exists up to date that can reliably detect those abnormalities. The objective of this study is to develop a robust computerized method for early detection of AD in EEG. We explore the ability of specifically designed and trained recurrent neural networks (RNN), combined with wavelet preprocessing, to discriminate between EEGs of early onset AD patients and their age-matched control subjects. We have used a similar approach previously for predicting the onset of epileptic seizure in EEG. The RNNs are chosen because they can implement extremely nonlinear decision boundaries and possess memory of the state, which is crucial for the considered task. The result on eyes-closed resting EEG reveals particularly favorable network behavior when applied to wavelet-filtered subbands as opposed to original signal data.

Select this Article		Adaptive Kalman filtering deconvolution via dyadic wavelet transform Enqing Dong, Guizhong Liu, and Zhongping Zhang Proc. SPIE 4119, 878 (2000); http://dx.doi.org/10.1117/12.408571 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract A new approach of adaptive Kalman filtering deconvolution (AKFD) is developed basing on dyadic wavelet transform. The technique discards the assumption of stationarity for signals in predictive deconvolution, and overcomes improving resolution at the price of decreasing signal-to-noise (SNR) obviously. The technique can well compress the reflection waveforms, but the noises are not variable in substance. So it has a better ability of resistance noise. Suppression false reflections in dyadic wavelet transform domain is better than by applying AKFD in time domain. In addition the technique also has the characteristic of adaptive Kalman filter in every band for a signal respectively, it enhances the adaptation of Kalman filtering, so the resolution is obvious higher than that one in time domain. A great deal of numerical models and real seismic data indicate that the technique has obvious effect. At the same time, the technique also overcomes the drawback of increasing the low- frequency component of AKFD in time domain. A great deal of numerical models and real seismic data indicate that the technique has obvious effect. The approach not only suits for seismic data, but also can be used for reference to another similar signal processing.

Select this Article		Application of local discriminant bases discrimination algorithm for theater missile defense Mary L. Cassabaum, Harry A. Schmitt, Hai-Wen Chen, and Jack G. Riddle Proc. SPIE 4119, 886 (2000); http://dx.doi.org/10.1117/12.408572 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The local discriminant bases (LDB) method is a powerful algorithmic framework that was originally developed by Coifman and Saito in 1994 as a technique for analyzing object classification problems. LDB is a feature extraction algorithm which selects a best-basis from a library of orthogonal bases based on relative entropy or a similar metric. The localized nature of these orthogonal basis functions often results in features that are easier to interpret and more intuitive than those obtained form more conventional methods. An evaluation of the best-basis technique using LDB was conducted with IR sensor data. In particular, our data set consisted of the intensity fluctuations of subpixel targets collected don a focal plane array. This 1D dat set provides a useful benchmark against current feature estimation/extraction algorithms as well as preparation for the much more difficult 2D problem. Significantly, LDB is an automated procedure. This has a number of potential advantages, including the ability to: (1) easily handle an increased threat set; and (2) significantly improve the productivity of the feature estimation 'expert' by removing them from the mechanics of the classification process.

Select this Article		Multiscale edge representation applied to image fusion Paul Scheunders Proc. SPIE 4119, 894 (2000); http://dx.doi.org/10.1117/12.408573 \| Cited 1 time Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper the fusion of multimodal images into one greylevel image is aimed at. A multiresolution technique, based on the wavelet multiscale edge representation is applied. The fusion consists of retaining only the modulus maxima of the wavelet coefficients from the different bands and combining them. After reconstruction, a synthetic image is obtained that contains the edge information from all bands simultaneously. Noise reduction is applied by removing the noise-related modulus maxima. In several experiments on test images and multispectral satellite images, we demonstrate that the proposed technique outperforms mapping techniques, as PCA and SOM and other wavelet-based fusion techniques.

Select this Article		Robust lifting wavelet transform for subimage extraction Shigeru Takano and Koichi Niijima Proc. SPIE 4119, 902 (2000); http://dx.doi.org/10.1117/12.408574 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We present a novel method to extract subimages from a huge reference image by using integer-type lifting wavelet transforms. Our integer to integer lifting wavelet transform contains controllable free parameters in the lifting term, which is constructed based on an integer version of Haar wavelet transform. Such free parameters are trained following a vanishing criterion for low frequency components of query images. The trained parameters have characteristics of the query images. We apply a lifting wavelet transform with such parameters to a reference image and check whether they satisfy our vanishing criterion or not, to extract target subimages.

Select this Article		Image compression with adaptive Haar-Walsh tilings Maj Lindberg and Lars F. Villemoes Proc. SPIE 4119, 911 (2000); http://dx.doi.org/10.1117/12.408575 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We perform adaptive joint space and frequency tilings including all levels in the Haar-Walsh wavelet packet tree for 2D signals. The method gives surprisingly good results in terms of nonlinear approximation. The visual quality of the compressed images with this method is the same as the quality using twice the number of coefficients for wavelets and standard wavelet packets when Haar filters are used. When all levels are allowed the cost for description of the location of the winning coefficients is not negligible. A tiling information vector is introduced for description of the chosen basis and the original image can be easily and quickly reconstructed using this information. For image compression, this tilting information vector is compressed to only those nodes which correspond to kept coefficients, and this makes the adaptive scheme competitive.

Select this Article		Tunable continuous wavelet and frame transforms James F. Scholl, Delmar L. Barker, Harry A. Schmitt, and John D. Langan Proc. SPIE 4119, 922 (2000); http://dx.doi.org/10.1117/12.408576 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Continuous wavelet transforms (CWT) and frames have always been useful for noise suppression, edge detection and medical signal processing. However, these transforms are generally shied away from since computational complexity prevents their widespread use. However, recently developed processor technology that uses analog rather than digital signal processing hardware may be the ideal means to implement and apply these algorithms. It is then appropriate to consider new types of frames and continuous wavelet systems. We propose two families of tunable continuous wavelet systems with widely varying frame bounds and scaling behavior, and illustrate examples of computations involving these systems.

Select this Article		Very low bit-rate video coding using a new kind of compact biorthogonal wavelet Weidong Hou, Yu L. Mo, Hong Liu, and Haiping Hu Proc. SPIE 4119, 934 (2000); http://dx.doi.org/10.1117/12.408577 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Video communication aiming at public switched telephone network (PSTN) applied with voice-band modem is attractive because of its low-cost facilities and the wide coverage of PSTN around the world, The key technique of video transmission over PSTN with voice-band modem is very low bit-rate video coding. Video coding based on discrete wavelet transform has become a hot research topic. But while in very low bit-rate video coding applications, the peak signal to noise ratio (PSNR) and the visual quality of image reconstructions are not very satisfactory by using the general orthogonal or biorthogonal wavelet which does not match well with human visual system characteristics. In this paper, a new kind of compact biorthogonal wavelet based on the modulation transfer function for human visual system model is used in very low bit-rate video coding scheme, in which a new improved Goh's 3D wavelet transform and motion compression technique are applied. The experimental results indicate that the new coding scheme using the constructed compact biorthogonal wavelet has a good performance in average PSNR, compression ratios and visual quality of image reconstruction when compared to the other motion-compensated 2D and 3D coding schemes based on the general biorthogonal wavelet transform.

Select this Article		Q-wave: a flexible and low-complexity quadtree wavelet encoder Marco Grangetto, Enrico Magli, and Gabriella Olmo Proc. SPIE 4119, 942 (2000); http://dx.doi.org/10.1117/12.408579 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, the novel Q-wave algorithm for image coding is prosed. Q-wave, which provides progressive transmission, is designed with the aim of limiting the computational complexity at the expenses of a slight quality degradation with respect to popular coders such as SPIHT. It is particularly suitable for low bit rate compression, and is then promising for applications such as intraframe video coding, Internet browsing, image transmission over band- limited channels. Moreover, Q-wave can be applied also to non standard wavelet decompositions without modifications.

Select this Article		Performance of integer wavelets for the lossy compression of smooth images Marco Grangetto, Enrico Magli, and Gabriella Olmo Proc. SPIE 4119, 950 (2000); http://dx.doi.org/10.1117/12.408580 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract It is well known that the image compression task can be effectively accomplished by means of the wavelet transform. A new method has ben recently propose for the computation of this transform, i.e. the lifting scheme (LS). Besides being computationally more efficient than the classical filter bank scheme, the LS also enables the computation of a wavelet transform which maps integers to integers, so allowing for the design of joint lossless and lossy compression schemes. The performance of this integer wavelet transform (IWT) has already been studied in the literature, and compared to that of the discrete wavelet transform (DWT) in the lossy case; it has been found that in most cases the DWT achieves slightly better performance with respect to the DWT. In this paper we show that this result does not hold in the case of lossy compression of smooth images, as the IWT has a much larger loss of performance, making it ineffective for the compression of such images. We first select measures of image smoothness; then we study the IWT lossy compression performance in the presence of different degrees of smoothness, on a set containing various kinds of images. Finally, we relate the IWT compression performance to the degree of image smoothness.

Select this Article		Wavelet-based algorithm for acoustic detection of moving ground and airborne targets Amir Z. Averbuch, Valery A. Zheludev, and Inna Kozlov Proc. SPIE 4119, 959 (2000); http://dx.doi.org/10.1117/12.408581 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We detect the presence of a vehicle or an air borne target form a certain class via the analysis of its acoustic signature against an existing database of recorded and processed acoustic signals. To achieve this detection with no false alarms we construct the acoustic signatures of certain targets to be found by the distribution of the energies among blocks which consists of wavelet packet coefficients. We developed an efficient procedure for adaptive selection of the characteristic blocks. We modified the CART algorithm in order to utilize it as a decision unit in our scheme. A wide series of field experiments manifested a remarkable efficiency of the algorithm. The detecting had been achieved practically with no false alarms even under severe conditions such a the acoustic recording of sought- after object was a superposition of the acoustics emitted from other targets that belong to other classes. The detection was even immune to severe noisy surroundings.

Select this Article		Efficient seismic volume compression using the lifting scheme Faouzi M. Khene and Samir H. Abdul-Jauwad Proc. SPIE 4119, 971 (2000); http://dx.doi.org/10.1117/12.408582 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract An advanced seismic compression technique is proposed to mange seismic data in a world of ever increasing data volumes in order to maintain productivity without compromising interpretation results. A separable 3D discrete wavelet transform using long biorthogonal filters is used. The computation efficiency of the DWT is improved by factoring the wavelet filters using the lifting scheme. In addition, the lifting scheme offers: 1) a dramatic reduction of the required auxiliary memory, 2) an efficient combination with parallel rendering algorithms to perform arbitrary surface and volume rendering for interactive visualization, and 3) an easy integration in the parallel I/O seismic data loading routines. The proposed technique is tested on a seismic volume from the Stratton field in South Texas. The resulting 3-level multiresolution decomposition yields 21 detail sub-volumes and a unique low-resolution sub-volume. The detail wavelet coefficients are quantized with an adaptive threshold uniform scalar quantizer. The scale-dependent thresholds are determined with the Stein unbiased risk estimate principle. As the approximation coefficients represents a smooth low-resolution version of the input data they are only quantized using a uniform scalar quantizer. Finally, a run-length plus a Huffman encoding are applied for binary coding of the quantized coefficients.

Select this Article		Attenuation and dispersion of a monopulse in layered and lossy media using an adaptive multiscale wavelet collocation method Yongzhong Wang, Ru-Shan Wu, and Wen-Bing Wang Proc. SPIE 4119, 983 (2000); http://dx.doi.org/10.1117/12.408583 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract An adaptive multiscale collocation method is applied to analyze the attenuation and dispersion of a monopulse in layered and lossy media. From numerical results, it is found that sufficient resolution can be provided in the solution's regions where large gradients and dramatic fluctuation occur. The new wavelet method can thus be regarded as an excellent algorithm for a wave equation's solution with large gradients and dramatic fluctuation.

Select this Article		Combining speckle reduction and data compression for synthetic aperture radar (SAR) images in wavelet domain Punya Thitimajshima Proc. SPIE 4119, 991 (2000); http://dx.doi.org/10.1117/12.408584 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The wavelet transform is widely used in both speckle reduction and dat compression of SAR image. Thus, it is very efficient to integrate these two procedures in a single process. IN this research, an input image is first subject to a logarithmic operation. The image is then transformed by using multi level wavelet decomposition. The variance of noise is estimated from the data to determine the threshold, which is used for soft-thresholding the wavelet coefficients. For each subband, the obtained wavelet coefficients are quantized and finally entropy encoded to produce the output bit stream of the image. The advantage of this method is that both speckle reduction and image compression are performed in wavelet domain. Experimental results on JERS-1/SAR images are also given.

Select this Article		Improved zerotree coding algorithm for wavelet image compression Jun Chen, Yunsong Li, and Chengke Wu Proc. SPIE 4119, 996 (2000); http://dx.doi.org/10.1117/12.408585 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract A listless minimum zerotree coding algorithm based on the fast lifting wavelet transform with lower memory requirement and higher compression performance is presented in this paper. Most state-of-the-art image compression techniques based on wavelet coefficients, such as EZW and SPIHT, exploit the dependency between the subbands in a wavelet transformed image. We propose a minimum zerotree of wavelet coefficients which exploits the dependency not only between the coarser and the finer subbands but also within the lowest frequency subband. And a ne listless significance map coding algorithm based on the minimum zerotree, using new flag maps and new scanning order different form Wen-Kuo Lin et al. LZC, is also proposed. A comparison reveals that the PSNR results of LMZC are higher than those of LZC, and the compression performance of LMZC outperforms that of SPIHT in terms of hard implementation.

Select this Article		Calculating rhythmicity of infant breathing using wavelets Katherine E. Macey, Wyatt H. Page, Ronald M. Harper, Paul M. Macey, and Rodney P. K. Ford Proc. SPIE 4119, 1002 (2000); http://dx.doi.org/10.1117/12.408586 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract Breathing signals are one set of physiological data that may provide information regarding the mechanisms that cause SIDS. Isolated breathing pauses have been implicated in fatal events. Other features of interest include slow amplitude modulation of the breathing signal, a phenomenon whose origin is unclear, and periodic breathing. The latter describes a repetitive series of apnea, and may be considered an extreme manifestation of amplitude modulation with successive cessations of breathing. Rhythmicity is defined to assess the impact of amplitude modulation on breathing signals and describes the extent to which frequency components remain constant for the duration of the signal. The wavelet transform was used to identify sections of constant frequency components within signals. Rhythmicity can be evaluated for all the frequency components in a signal, for individual frequencies. The rhythmicity of eight breathing epochs from sleeping infants at high and low risk for SIDS was calculated. Initial results show breathing from infants at high risk for SIDS exhibits greater rhythmicity of modulating frequencies than breathing from low risk infants.

Select this Article		Nonuniform spatially adaptive wavelet packets Philippe Carre and Christine Fernandez-Maloigne Proc. SPIE 4119, 1011 (2000); http://dx.doi.org/10.1117/12.408587 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, we propose a new decomposition scheme for spatially adaptive wavelet packets. Contrary to the double tree algorithm, our method is non-uniform and shift- invariant in the time and frequency domains, and is minimal for an information cost function. We prose some-restrictions to our algorithm to reduce the complexity and permitting us to provide some time-frequency partitions of the signal in agreement with its structure. This new 'totally' non-uniform transform, more adapted than Malvar, Packets or dyadic double-tree decomposition, allows the study of all possible time-frequency partitions with the only restriction that the blocks are rectangular. It permits one to obtain a satisfying Time-Frequency representation, and is applied for the study of EEG signals.

Select this Article		Exponential-quadratic optimal signal reconstruction in noisy filter banks Haris Vikalo, Alper T. Erdogan, Babak Hassibi, and Thomas Kailath Proc. SPIE 4119, 1020 (2000); http://dx.doi.org/10.1117/12.408589 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We consider the design of synthesis filters in noisy filter bank system, using an exponential-quadratic criterion. We assume that the analysis filters have been design to achieve good coding of the input signal. Then we design the synthesis filters to minimize reconstruction error according to the adopted criterion. When the synthesis filters are restricted to be FIR, the design can be cast as a constraint analytic centering problem. To this end, we first employ standard state-space techniques to obtain a set of H(infinity) optimal FIR synthesis filters. Among these, we select the so-called risk-sensitive synthesis filters by additionally minimizing exponential-quadratic cost function. We provide numerical examples to illustrate the procedure.

Select this Article		Perspective invariant function using dyadic wavelet transform Mahmoud I. Khalil and Mohamed M. Bayoumi Proc. SPIE 4119, 1029 (2000); http://dx.doi.org/10.1117/12.408590 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract A perspective invariant function has been derived. It is based on analyzing the object boundary using the dyadic wavelet transform. Five dyadic wavelet transfer levels are used to define the invariant function. The selection of the dyadic levels is discussed. Experimental results test the recognition power of the proposed invariant function. In addition, the stability of the invariant function is examined.

Select this Article		Evaluation of a multiscale enhancement protocol for digital mammography Ralf Mekle, Andrew F. Laine, Suzanne Smith, Cory Singer, Tova Koenigsberg, and Marc Brown Proc. SPIE 4119, 1038 (2000); http://dx.doi.org/10.1117/12.408591 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We have carried out a receiver operating characteristics (ROC) study for the enhancement of mammographic features in digitized mammograms. The study evaluated the benefits of multi-scale enhancement methods in terms of diagnostic performance of radiologists. The enhancement protocol relied on multi-scale expansions and non-linear enhancement functions. Dyadic spline wavelet functions were used together with a sigmoidal non-linear enhancement function. We designed a computer interface ona softcopy display and performed an ROC study with three radiologists, who specialized in mammography. Clinical cases were obtained from a national mammography database of digitized radiographs prepared by the University of South Florida and Harvard Medical School. Our study focused on dense mammograms, i.e. mammograms of density 3 and 4 on the American College of Radiology breast density rating, which are the most difficult cases in screening, were selected. To compare the performance of radiologists with an without using multi-scale enhancement, two groups of 30 cases each were diagnosed. Each group contained 15 cases of cancerous and 15 cases of normal mammograms. Conventional ROC analysis was applied, and the resulting ROC curves indicated improved diagnostic performance when radiologists used multi-scale non-linear enhancement.

Select this Article		Automatic detection of epileptic events in scalp EEG Susana I. Isaacson, Carlos E. D'Attellis, and Ricardo O. Sirne Proc. SPIE 4119, 1050 (2000); http://dx.doi.org/10.1117/12.408592 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In this work we present a technique base don wavelet analysis for EEGs processing of epileptic patients explored with scalp electrodes. Our aim is to provide a contribution to the automatic treatment of EEGs in the following areas of clinical applications: detection of transients, data reduction in long-term records and tracking of crisis propagation. We show the results obtained with the prosed method applied to several EEGs records.

Select this Article		Discrete wavelet approach to multifractality Susana I. Isaacson, Susana C. Gabbanelli, and Jorge R. Busch Proc. SPIE 4119, 1058 (2000); http://dx.doi.org/10.1117/12.408593 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract The use of wavelet techniques for the multifractal analysis generalizes the box counting approach, and in addition provides information on eventual deviations of multifractal behavior. By the introduction of a wavelet partition function W_q and its corresponding free energy (beta) (q), the discrepancies between (beta) (q) and the multifractal free energy r(q) are shown to be indicative of these deviations. We study with Daubechies wavelets (D₄) some 1D examples previously treated with Haar wavelets, and we apply the same ideas to some 2D Monte Carlo configurations, that simulate a solution under the action of an attractive potential. In this last case, we study the influence in the multifractal spectra and partition functions of four physical parameters: the intensity of the pairwise potential, the temperature, the range of the model potential, and the concentration of the solution. The wavelet partition function W_q carries more information about the cluster statistics than the multifractal partition function Z_q, and the location of its peaks contributes to the determination of characteristic sales of the measure. In our experiences, the information provided by Daubechies wavelet sis slightly more accurate than the one obtained by Haar wavelets.

Select this Article		Differential affine motion estimation for medical image registration Senthil Periaswamy, John B. Weaver, Dennis M. Healy, Jr., Daniel N. Rockmore, Peter J. Kostelec, and Hany Farid Proc. SPIE 4119, 1066 (2000); http://dx.doi.org/10.1117/12.408594 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We have applied techniques from differential motion estimation in the context of automatic registration of medical images. This method uses optical-flow and Fourier technique for local/global registration. A six parameter affine model is used to estimate shear, rotation, scale and translation. We show the efficacy of this method with images of similar and different contrasts.

Select this Article		Multiscale nonrigid data registration using adaptive basis functions Thomas Dinsenbacher, Gustavo K. Rohde, Douglas P. Hardin, Akram Aldroubi, and Benoit M. Dawant Proc. SPIE 4119, 1076 (2000); http://dx.doi.org/10.1117/12.408595 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract When performing registrations, it is often crucial to maintain certain structure of the template data T - the data being deformed into the subject data S - as well as to keep the deformation field smooth. Current approaches to registration often impose smoothness through heuristic means, but building it into the model has proven to be more difficult due mainly to computational constraints.

Select this Article		Gerchberg-Papoulis algorithm and the finite Zak transform Andrzej K. Brodzik and Richard Tolimieri Proc. SPIE 4119, 1084 (2000); http://dx.doi.org/10.1117/12.408597 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract We propose a new, time-frequency formulation of the Gerchberg-Papoulis algorithm for extrapolation of band- limited signals. The new formulation is obtained by translating the constituent operations of the Gerchberg- Papoulis procedure, the truncation and the Fourier transform, into the language of the finite Zak transform, a time-frequency tool intimately related to the Fourier transform. We will show that the use of the Zak transform results in a significant reduction of the computational complexity of the Gerchberg-Papoulis procedure and in an increased flexibility of the algorithm.

Select this Article		Superrium of perturbation for sampling in shift-invariant subspaces Wen Chen, Shuichi Itoh, and Junji Shiki Proc. SPIE 4119, 1094 (2000); http://dx.doi.org/10.1117/12.408599 Online Publication Date: May 20, 2003 Full Text: \| Download PDF
	+ -	Show Abstract In the more general framework 'shift invariant subspace', the paper obtains a different estimate of sampling in function subspace to our former work, by using the Frame Theory. The derived formula is easy to be calculated and the estimate is relaxed in some shift invariant subspaces.

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