Showing papers on "Biorthogonal system published in 1992"

Biorthogonal bases of compactly supported wavelets

Abstract: Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient condition for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.

A class of linear-phase regular biorthogonal wavelets

Abstract: A class of biorthogonal systems leading to linear-phase wavelets is presented. A notable feature of this structure is that the wavelets are derived from a filter bank where the lowpass analysis filter is constrained to be a halfband filter. The authors derive finite impulse response (FIR) biorthogonal solutions from a pair of Lagrange halfband filters. They also consider infinite impulse response (IIR) biorthogonal solutions based on a pair of zero-phase halfband filters derived from Butterworth halfband filters. >

Biorthogonal valence bond descriptions of electronic structure

Abstract: A valence bond approach based on nonorthogonal orbitals in a biorthogonal representation is presented. While the scheme suffers from the lack of a variational bound on the energy functional, it is shown that with a suitable optimization of the orbitals reliable molecular wavefunctions can be obtained. A review of the background theory is given emphasizing the similarity of this approach to the familiar spin-free unitary group formulation of quantum chemistry. The details of the computer implementation are discussed and the method is illustrated with model calculations on HF, H2O and F2O2.

Design of multidimensional non-separable regular filter banks and wavelets

Abstract: The design of multidimensional nonseparable wavelets based on iterated filter banks is investigated. To obtain regularity of the wavelet, a maximum number of zeros is put at aliasing frequencies in the lowpass filter. Two approaches are pursued. A direct method designs nonseparable perfect reconstruction filter banks based on cascade structures and with prescribed zeros both analytically (small cases) and numerically (larger cases). A second, indirect method maps biorthogonal one-dimensional banks with high regularity into multidimensional banks using the McClellan transformation. A number of properties relevant to perfect reconstruction and zero locations are shown in this case. Design examples are given in all cases, and the testing of regularity is discussed, together with a fast algorithm to compute iterated filters. >

Moment problem for a density matrix and a biorthogonal set of operator bases.

Abstract: The practical aspect of the moment problem for a density matrix for single-mode radiation is solved; namely, two simple explicit expressions for the density matrix in terms of its moments are presented. A biorthogonal set of operators is established, so that two bases are available for expressing an arbitrary operator.

Image coding using multiresolution Markov random fields

Abstract: The purpose of this paper is to propose a new scheme for image coding from the point of view of inverse problems. The goal is to find an approximation of the image that preserves edges for a given bit rate. In order to achieve better visual quality and to save computation time, the image is first decomposed using biorthogonal wavelets. We assume that wavelet coefficient sub-images can be modeled by Markov random fields (MRF) with line process. The sub- images are then approximated so their entropy decrease and edges are preserved. Thus, the visual quality of the reconstructed image is controlled. We also look at the MRF model and a monoresolution image approximation method, along with a short overview of wavelet-based multiresolution analysis. Finally, we describe the multiresolution coding scheme and give some results.

Block biorthogonal channel coding using wavelets

Abstract: The authors introduce block biorthogonal coding using wavelets, a channel coding method that uses the unique orthogonality properties of wavelet coefficient matrices (WCM) to efficiently encode information bits. The benefit of this algorithm is its diversity-based coding gains in fading and burst noise channels. The authors compare wavelet-Hadamard codes, which are obtained from wavelet-Hadamard matrices, a class of wavelet coefficient matrices, with traditional Hadamard codes and find them to be equivalent to additive white Gaussian noise and superior in fading and burst noise channels. >

Explicit orthogonalization of some biorthogonal bases for SU(n)⊇SO(n) and Sp(4)⊇U(2)

Abstract: Mutually related explicit algebraic‐polynomial expressions of the orthogonalization coefficients are proposed for the biorthogonal [polynomial (Bargmann–Moshinsky), stretched, antistretched, quasistretched, and their dual] bases of the two parametric (mixed tensor and covariant) irreducible representations (irreps) of SU(n) restricted to SO(n), as well as for the projected (Smirnov–Tolstoy) and dual bases of the five‐dimensional quasispin. The orthogonalization coefficients of the essentially simplified Gram–Schmidt process are expressed, up to explicitly given elementary factors, in terms of the numerator and denominator polynomials, represented as compositions of the generalized hypergeometric coefficients <3F2(...)‖μ≳ and <1F0(...)‖ν≳ and equivalent in the diagonal (denominator) and boundary cases to the Aλ(cabde) functions of Biedenharn and Louck. The distribution of zeros and the symmetry properties of the introduced polynomials are crucial for the conjectured solutions.

Performance analysis of trellis coded biorthogonal sequences in a variable rate system

Abstract: For digital communication systems with a given binary modulation scheme and variable information rate, a coding technique is described which employs a particular structure of trellis code. It consists of a convolutional encoder followed by a mapper which selects biorthogonal codewords. Based upon the state diagram of the system, performance is evaluated and then compared to that of more conventional solutions. Beside having good performance, this approach is very modular, i.e. its structure easily adapts to different information rates. Furthermore, it allows the time synchronization system to work at lower signal-to-noise ratios.

A new method for non-orthogonal decomposition

Abstract: A new algorithm for nonorthogonal decomposition is proposed and applied to Gabor decomposition of image. The algorithm is iterative and its advantages are discussed. Also a modified version of the algorithm is considered which increases the rate of convergence. Image simulations show that this method gives much lower reconstruction error that the method using biorthogonal functions, at a cost of a greater amount of computer time. >

Wavelets: multiresolution signal representation with applications to image, speech, radar, and other signal coding

Abstract: The problem of designing wavelets which are most appropriate for applications to multiresolution coding of image, speech, radar and other signals is addressed. The effects of regularity and zero moments on the design of wavelets and filter banks used to realize these wavelet decompositions are discussed, and insights pointed out. The use of vector quantization with wavelet transforms will be discussed. It is observed how wavelet decompositions are a compromise between optimality and complexity, where the optimality is determined from the minimization of bit rate and distortion, using rate distortion theory. The problem of designing wavelets yielding linear phase filtering, important for applications such as television coding and radar, is discussed and a number of approaches to solutions are described. These include the use of biorthogonal rather than orthogonal bases for wavelets which are realizable by general perfect reconstruction filter banks in which the analysis and synthesis filters are not time-reversed versions of each other. Methods for designing linear phase filters are briefly discussed and referenced. In the discussion on applications to radar signals, the relation of wavelet theory to a special signal called a chirplet is noted. Some connections of wavelets to splines and cardinal series are noted. Finally, wavelets which almost meet the uncertainty principle bound with equality are described.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.