Showing papers on "Biorthogonal system published in 1994"
152 citations
TL;DR: In this article, the spectral radius of a linear operator acting on a wavelet basis is determined by using the wavelet's spectral radius as a measure of the Besov regularity of a compactly supported refinement equation.
Abstract: The Besov regularity of a compactly supported refinement equation solution is determined by the spectral radius of a linear operator acting on $\ell ^p (\mathbb{Z})$. The proof of this is obtained by using a wavelet basis. Exact criteria for Holder and Sobolev regularity follow immediately. Continuity, differentiability, and integrability can also be characterized.The results are applied to examples from the theory of orthonormal and biorthogonal wavelets and subdivision schemes for curve design.
151 citations
TL;DR: In this paper, the singular value decomposition (SVDC) is used to decompose complex spatiotemporal patterns into a few coherent modes that are often easier to interpret.
Abstract: The investigation of fluctuation phenomena in plasmas often necessitates the analysis of spatiotemporal signals. It is shown how such signals can be analyzed using the biorthogonal decomposition, which splits them into orthogonal spatial and temporal modes. The method, also referred to as the singular value decomposition, allows complex spatiotemporal patterns to be decomposed into a few coherent modes that are often easier to interpret. This is illustrated with two applications to fluctuating soft x‐ray and magnetic signals, as measured in a tokamak. Emphasis is given to the physical interpretation of the biorthogonal components and their link with known physical models is discussed. It is shown how new insight can be gained in the interpretation of spatiotemporal plasma dynamics.
142 citations
TL;DR: The authors discuss the solution of /spl gamma/(i) that is optimally close to an arbitrary desired function d(i) when the synthesis window h( i) and sampling pattern are given.
Abstract: Like the DFT/spl minus/a discrete version of the Fourier transform/spl minus/the recently developed discrete Gabor transform (DGT) provides a feasible vehicle to implement the very useful Gabor expansion. In general, the choice of the biorthogonal window function /spl gamma/(i) is not unique. The authors discuss the solution of /spl gamma/(i) that is optimally close to an arbitrary desired function d(i) when the synthesis window h(i) and sampling pattern are given. For d(i)=h(i), the resulting /spl gamma//sub opt/(i) directly leads to so-called orthogonal-like DGT. Combining the DGT, they believe that the result presented in this paper is rather significant for digital signal processing. >
42 citations
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TL;DR: In this article, the authors use generating functions to express orthogonality relations in the form of $q$-beta integrals, which are then used as a weight function for a new set of orthogonal or biorthogonal relations.
Abstract: We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal
37 citations
TL;DR: Several theorems are proved and the bridge between the continuous and discrete adaptive wavelet transform (AWT) is constructed explicitly, which is the implementation by means of wavelet chips and neurochips, in which each node is a daughter wavelet similar to a radial basis function using dyadic affine scaling.
Abstract: We prove several theorems and construct explicitly the bridge between the continuous and discrete adaptive wavelet transform (AWT). The computational efficiency of the AWT is a result of its compact support closely matching linearly the signal's time-frequency characteristics, and is also a result of a larger redundancy factor of the superposition-mother s(x) (super-mother), created adaptively by a linear superposition of other admissible mother wavelets. The super-mother always forms a complete basis, but is usually associated with a higher redundancy number than its constituent complete orthonormal (CON) bases. The robustness of super-mother suffers less noise contamination (since noise is everywhere, and a redundant sampling by bandpassings can suppress the noise and enhance the signal). Since the continuous super-mother has been created off-line by AWT (using least-mean-squares neural nets), we wish to accomplish fast AWT on line. Thus, we formulate AWT in discrete high-pass (H) and low-pass (L) filter bank coefficients via the quadrature mirror filter (QMF), a digital subband lossless coding. A linear combination of two special cases of the complete biorthogonal normalized (Cbi-ON) QMF [L(z),H(z),L+(z),H+(z)], called α-bank and β-bank, becomes a hybrid aα + bβ-bank (for any real positive constants a and b) that is still admissible, meaning Cbi-ON and lossless. Finally, the power of AWT is the implementation by means of wavelet chips and neurochips, in which each node is a daughter wavelet similar to a radial basis function using dyadic affine scaling.
37 citations
TL;DR: It is shown that one of the best substitutions for the Gaussian function in the Fourier domain is a squared sinusoid function that can form a biorthogonal windowfunction in the time domain.
Abstract: We discuss the semicontinuous short-time Fourier transform (STFT) and the semicontinual wavelet transform (WT) with Fourier-domain processing, which is suitable for optical implementation. We also systematically analyze the selection of the window functions, especially those based on the biorthogonality and the orthogonality constraints for perfect signal reconstruction. We show that one of the best substitutions for the Gaussian function in the Fourier domain is a squared sinusoid function that can form a biorthogonal window function in the time domain. The merit of a biorthogonal window is that it could simplify the inverse STFT and the inverse WT. A couple of optical architectures based on Fourier-domain processing for the STFT and the WT, by which real-time signal processing can be realized, are proposed.
34 citations
TL;DR: In this paper, the authors report the use of extended coupled cluster functional of Arponen, Bishop, and co-workers to implement a stationary biorthogonal response approach to calculate nonlinear molecular properties like hyperpolarizability, etc.
Abstract: In this paper, we report the use of extended coupled cluster functional of Arponen, Bishop, and co‐workers to implement a stationary biorthogonal response approach. The objective of this is to calculate nonlinear molecular properties like hyperpolarizability, etc. in a more convenient way.
32 citations
TL;DR: It is observed that the wavelets best suited for achieving the highest possible compression with this wavelet-based algorithm are Daubechies wavelets, while Coiflets lead to a faster decomposition algorithm at slightly lesser compression.
28 citations
31 Oct 1994
TL;DR: In this article, the optimal synthesis filter bank is the biorthogonal system followed by a scalar post filter, and the statistical conditions under which the BORG solution is still the best are discussed.
Abstract: In maximally decimated filter banks, the perfect reconstruction or biorthogonal solution is not necessarily the best choice when subband quantizers are present. Under suitable statistical assumptions, expressions for the best synthesis bank can be derived in terms of the analysis bank and other statistical quantities. We explore this topic for subband coders and the special case of transform coders. We highlight the statistical conditions under which the biorthogonal solution is still the best. We derive expressions for the Wiener filter matrix in terms of the joint statistics of appropriate signals. Special cases where the optimal synthesis filter bank is the biorthogonal system followed by a scalar post filter are also considered. >
25 citations
01 Feb 1994
TL;DR: In this paper, a pair of polynomial sets which are biorthogonal on the unit circle with respect to a complex weight function is considered, and it is shown that the bior-thogonality of these sets implies a q-beta integral, which in turn leads to pair of rational functions with qualitative properties reminiscent of the Szego theory for orthogonal polynomials.
Abstract: In this paper we first consider a pair of polynomial sets which are biorthogonal on the unit circle with respect to a complex weight function. We then show how the biorthogonality of this pair of polynomial sets implies a q-beta integral which in turn leads to a pair of biorthogonal rational functions. Finally we show that the asymptotics for these pairs of rational functions exhibit qualitative properties reminiscent of the Szego theory for orthogonal polynomials.
TL;DR: In this paper, experimental data from one-point measurements obtained in a transitional flow on a rotating flat disk are presented and analyzed by using biorthogonal decomposition techniques, and the analysis is performed at various Reynolds numbers from slightly above the onset of the first instability to the transition to turbulence.
Abstract: Experimental data from one‐point measurements obtained in a transitional flow on a rotating flat disk are presented and analyzed by using biorthogonal decomposition techniques. The analysis is performed at various Reynolds numbers from slightly above the onset of the first instability to the transition to turbulence. As Reynolds number increases, biorthogonal spectra become broader and the entropy characterizing the distribution of energy among the various biorthogonal modes increases. Details of this increase are studied by analyzing local entropy maxima corresponding to eigenvalue degeneracies. At these values of the Reynolds number, internal bifurcations, responsible for a lack of smoothness in the dependence of the flow with Reynolds number, are shown to occur.
11 Oct 1994
TL;DR: In this article, the problem of oblique projections in discrete shift-invariant spaces of 12 was studied and error bounds on the approximation were given for biorthogonal wavelets.
Abstract: We study the general problem of oblique projections in discrete shift-invariant spaces of 12 and we give error bounds on the approximation. We define the concept of discrete multiresolutions and wavelet spaces and show that the oblique projections on certain subclasses of discrete multiresolutions and their associated wavelet spaces can be obtained using perfect reconstruction filter banks. Therefore we obtain a discrete analog of the Cohen-Daubechies-Feauveau results on biorthogonal wavelets.
01 Jan 1994
TL;DR: In this paper, a Galerkin approach was used for the solution of an elliptic boundary value problem in one dimension with polynomial coefficients, where biorthogonal wavelets adapted to a differential operator with constant coefficients were used to set up the system of linear equations.
Abstract: We will be concerned with the solution of an elliptic boundary, value problem in one dimension with polynomial coefficients. In a Galerkin approach, we employ biorthogonal wavelets adapted to a differential operator with constant coefficients, and use the refinement equations to set up the system of linear equations with exact entries (up to round-off). For the solution of the linear equation, we construct a biorthogonal two-grid method with intergrid operators stemming from wavelet-type operators adapted to the problem.
16 Sep 1994
TL;DR: Biorthogonal wavelets or filterbanks are shown to be superior in coding gain performance than orthogonal ones for logarithmic subband decompositions as discussed by the authors, which is shown for maximally regular biorhogonal and orthogonality filters, as well as filters designed to optimize the subband coding grain.
Abstract: Biorthogonal wavelets or filterbanks are shown to be superior in coding gain performance than orthogonal ones for logarithmic subband decompositions (limited to iterative decomposition of the downsampled output of the analysis low-pass filter). As a consequence, for logarithmic decompositions, the optimal filter is not an ideal filter. This is shown for maximally regular biorthogonal and orthogonal filters, as well as filters designed to optimize the subband coding grain.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: Two generalizations of the original construction of Malvar, Coifman, and Meyer are presented: biorthogonal and equal parity bases, which have the advantage that constant and linear components, respectively, can be represented efficiently.
Abstract: We discuss smooth local trigonometric bases and their applications to signal compression. In image compression, these bases can
reduce the blocking effect that occurs in the Joint Photographic Experts Group (JPEG). We present and compare two generalizations of the original construction of Malvar, Coifman, and Meyer: biorthogonal and equal parity bases. These have the advantage that constant and linear components, respectively, can be represented efficiently. We show how they reduce blocking effects and improve the SNR.
TL;DR: For a maximally decimated non-uniform filter bank, the perfect reconstruction property is equivalent to biorthogonality as mentioned in this paper, and a number of properties of PR filter banks are derived.
TL;DR: The details of a simple and efficient scheme for performing variational biorthogonal valence bond calculations are presented and a variational bound on the energy functional is obtained through the use of a complete configuration expansion in a well‐chosen subset of orbitals.
Abstract: The details of a simple and efficient scheme for performing variational biorthogonal valence bond calculations are presented. A variational bound on the energy functional is obtained through the use of a complete configuration expansion in a well‐chosen subset of orbitals. The resultant wave functions are clearly dominated by the covalent (spin‐coupled) structures, with a negligible contribution from ionic structures. The orbitals obtained compare favorably with overlap enhanced atomic orbitals obtained by other valence bond approaches. The method is illustrated by calculations on water and dioxygen difluoride. © 1994 by John Wiley & Sons, Inc.
TL;DR: In this paper, a parametric family of dual filters, FvN(ξ), of length 2N + 1 is constructed, corresponding to a fixed filter Fn(λ) of finite length 2n + 1.
Abstract: Several families of biorthogonal wavelet bases are constructed with various properties. In particular, for a given filter, Fn(ξ), of finite length 2n + 1, a parametric family of dual filters, FvN(ξ), of length 2N + 1 is constructed. The parametric nature of the dual filters makes it possible to design the optimum dual filter Fv0N(ξ), corresponding to a fixed filter Fn(ξ).
TL;DR: In this paper, the authors show how uniformly propagating waves, togehter with their velocity, shape, and amplitude, can be extracted from a spatio-temporal signal consisting of the superposition of various traveling waves.
Abstract: By using biorthogonal decompositions, we show how uniformly propagating waves, togehter with their velocity, shape, and amplitude, can be extracted from a spatiotemporal signal consisting of the superposition of various traveling waves. The interaction between the different waves manifests itself in space-time resonances in case of a discrete biorthogonal spectrum and in resonant wavepackets in case of a continuous biorthogonal spectrum. Resonances appear as invariant subspaces under the biorthogonal operator, which leads to closed sets of algebraic equations. The analysis is then extended to superpositions of dispersive waves for which the (Fourier) dispersion relation is no longer linear. We then show how a space-time bifurcation, namely a qualitative change in the spatiotemporal nature of the solution, occurs when the biorthogonal operator is a nonholomorphic function of a parameter. This takes place when two eigenvalues are degenerate in the biorthogonal spectrum and when the spatial and temporal eigenvectors rotate within each eigenspace. Such a scenario applied to the superposition of traveling waves leads to the generation of additional waves propagating at new velocities, which can be computed from the spatial and temporal eigenmodes involved in the process (namely the shape of the propagating waves slightly before the bifurcation). An eigenvalue degeneracy, however, does not necessarily lead to a bifurcation, a situation we refer to as being self-avoiding. We illustrate our theoretical predictions by giving examples of bifurcating and self-avoiding events in propagating phenomena.
19 Apr 1994
TL;DR: An algebraic approach to the design of different kinds of discrete wavelet transforms (orthogonal and biorthogonal single-/multiwavelet transforms, multiwavelet-like transforms) is taken and modifications of this multiwavelets-like transform are given with respect to practical applications.
Abstract: An algebraic approach to the design of different kinds of discrete wavelet transforms (orthogonal and biorthogonal single-/multiwavelet transforms, multiwavelet-like transforms) is taken. The different transforms are analysed with respect to computational efforts, approximation properties and symmetry. The design of the orthogonal and biorthogonal single-/multiwavelets requires the solution of a system of linear and nonlinear equations. Only the biorthogonal case enables symmetric coefficients. The basis matrix of the multiwavelet-like transform is easy to compute, orthogonal and ultimately symmetric. Modifications of this multiwavelet-like transform are given with respect to practical applications. >
01 Jan 1994
TL;DR: In this article, the first application of the bi-orthogonal dilated electron propagator to the treatment of molecular resonances is presented for the energy and the width of the 2IIg shape resonance in e−N2 scattering.
Abstract: Preliminary results from the first application of the bi-orthogonal dilated electron propagator to the treatment of molecular resonances are presented for the energy and the width of the 2IIg shape resonance in e–N2 scattering. The corresponding resonant Feynman–Dyson amplitudes (FDAS) are plotted to get a quantitative affirmation of the topology of the lowest unoccupied molecular orbital (LUMO) for N2. It is shown that a plot of the resonant FDAS offer new insights into the role of correlation in the formation and decay of molecular shape resonances. © 1994 John Wiley & Sons, Inc.
31 Oct 1994
TL;DR: In this paper, a biorthogonal-like sequence (BLS) theory and its application to the generalized Gabor (1946) expansion (equivalently, the generalized short-time Fourier transform/filterbank summation) are presented.
Abstract: A biorthogonal-like sequence (BLS) theory and its application to the generalized Gabor (1946) expansion (equivalently, the generalized short-time Fourier transform/filterbank summation) are presented. A pair of BLSs are defined to be two sequences satisfying a biorthogonal-like condition (BLC). We show that two collections in a Hilbert space generated by a pair of BLSs in the joint time-frequency domain are complete, and either can be used as an analysis filter, and the other as a synthesis filter, for a generalized Gabor expansion of discrete-time signals. An efficient algorithm for computation of BLSs is addressed in detail, and examples are presented to illustrate our results. >
TL;DR: This paper improves the traditional biorthogonal-function-based method by providing an algorithm of complexity O( L log 2 L ) for computing the bIORthorgonal sequence for both the critical and oversampled cases, and an algorithm for computingThe Gabor coefficients from the biorstogonal function with the efficiency of the 2D-FFT.
13 Nov 1994
TL;DR: The selection is considered of scaling functions for optimal signal representation by multidimensional biorthogonal wavelet frames by aiming at the minimization of the mean-square approximation error at each level of the decomposition.
Abstract: The selection is considered of scaling functions for optimal signal representation by multidimensional biorthogonal wavelet frames. Criterion for optimality is the minimization of the mean-square approximation error at each level of the decomposition. Conditions are given under which the approximation error of the decomposition approaches zero as the level increases. Optimal and suboptimal families of filters for the realization of the scaling functions are explicitly defined. >
TL;DR: It is shown that, in cases where the blur filter can be considered as a low-pass filter of a biorthogonal multiresolution analysis, it is possible to obtain an efficient family of regularization operators from the convolution operator alone.
Abstract: One well-known method to solve ill-posed problems in image restoration is to use regularization techniques. A new scheme is proposed for digital image restoration based on a regularization method and on the biorthogonal wavelet transform. We show that, in cases where the blur filter can be considered as a low-pass filter of a biorthogonal multiresolution analysis, it is possible to obtain an efficient family of regularization operators from the convolution operator alone.
TL;DR: In this paper, the use of a biorthogonal basis for continuous wavelet transformations is explored, thus relaxing the so-called admissibility condition on the analyzing wavelet.
Abstract: The use of a biorthogonal basis for continuous wavelet transformations is explored, thus relaxing the so‐called admissibility condition on the analyzing wavelet. As an application, the eigenvalues and corresponding radial eigenfunctions of the Hamiltonian of relativistic hydrogen‐like atoms are determined.
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TL;DR: In this article, the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegő and for their four parameter generalization to biorthogonal rational functions were studied.
Abstract: We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegő and for their four parameter generalization to ${}_4\phi_3$ biorthogonal rational functions on the unit circle.