Showing papers on "Biorthogonal system published in 1995"

Spherical wavelets: efficiently representing functions on the sphere

Abstract: Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets. CR

A new class of two-channel biorthogonal filter banks and wavelet bases

Abstract: Proposes a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters. >

Wavelets and Multiscale Signal Processing

Abstract: Multi-resolution analysis: The continuous point of view The discrete point of view The multivariate case. Wavelets and conjugate quadrature filters: The general case The finite case Wavelets with compact support action of the FWT on oscillating signals. The regularity of scaling functions and wavelets: Regularity and oscillation The sub-division algorithms Spectral estimates of the regularity Estimates of the Lp-Sobolev exponent Applications. Biorthogonal wavelet bases: General principles of sub-band coding Unconditional biorthogonal wavelet bases Dual filters and biorthogonal Riesz bases Examples and applications. Stochastic processes: Linear approximation Linear approximation of images Approximation and compression of real images Piecewise stationary processes Non-linear approximation. Appendices: Quasi-analytic wavelet bases Multivariate constructions Multiscale unconditioned bases Notation.

Solution of two problems on wavelets

Abstract: We solve two problems on wavelets. The first is the nonexistence of a regular wavelet that generates a wavelet basis for the Hardy space ℍ2(ℝ). The second is the existence, given any regular wavelet basis for\(\mathbb{H}^2 (\mathbb{R})\), of aMulti-Resolution Analysis generating the wavelet. Moreover, we construct a regular scaling function for this Multi-Resolution Analysis. The needed regularity conditions are very mild and our proofs apply to both the orthonormal and biorthogonal situations. Extensions to more general cases in dimension 1 and higher are given. In particular, we show in dimension larger than 2 that a regular wavelet basis for\(\mathbb{L}^2 (\mathbb{R}^n )\) arises from a Multi-Resolution Analysis that is regular modulo the action of a unitary operator, which is whenn = 2 a Calderon-Zygmund operator of convolution type.

Role of anticausal inverses in multirate filter-banks .II. The FIR case, factorizations, and biorthogonal lapped transforms

Abstract: For pt. I see ibid., vol.43, no.5, p.1090, 1990. In part I we studied the system-theoretic properties of discrete time transfer matrices in the context of inversion, and classified them according to the types of inverses they had. In particular, we outlined the role of causal FIR matrices with anticausal FIR inverses (abbreviated cafacafi) in the characterization of FIR perfect reconstruction (PR) filter banks. Essentially all FIR PR filter banks can be characterized by causal FIR polyphase matrices having anticausal FIR inverses. In this paper, we introduce the most general degree-one cafacafi building block, and consider the problem of factorizing cafacafi systems into these building blocks. Factorizability conditions are developed. A special class of cafacafi systems called the biorthogonal lapped transform (BOLT) is developed, and shown to be factorizable. This is a generalization of the well-known lapped orthogonal transform (LOT). Examples of unfactorizable cafacafi systems are also demonstrated. Finally it is shown that any causal FIR matrix with FIR inverse can be written as a product of a factorizable cafacafi system and a unimodular matrix. >

The generalized Gabor transform

Abstract: The generalized Gabor transform (for image representation) is discussed. For a given function f(t), t/spl isin/R, the generalized Gabor transform finds a set of coefficients a/sub mr/ such that f(t)=/spl Sigma//sub m=-/spl infin///sup /spl infin///spl Sigma//sub r=-/spl infin///sup /spl infin///spl alpha//sub mr/g(t-mT)exp(i2/spl pi/rt/T'). The original Gabor transform proposed by D. Gabor (1946) is the special case of T=T'. The computation of the generalized Gabor transform with biorthogonal functions is discussed. The optimal biorthogonal functions are discussed. A relation between a window function and its optimal biorthogonal function is presented based on the Zak (1967) transform when T/T' is rational. The finite discrete generalized Gabor transform is also derived. Methods of computation for the biorthogonal function are discussed. The relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized Gabor transform can be extended to the finite discrete case. Efficient algorithms for the optimal biorthogonal function and generalized Gabor transform for the finite discrete case are proposed. >

Wavelets and perfect reconstruction subband coding with causal stable IIR filters

Abstract: We show for the first time that it is possible to have a large family of perfect reconstruction subband coding filter banks, which are causal and IIR both at the analysis and at the synthesis ends. Complete parametrization of such classes of filters leading to design methods are given. Parametrizations in frequency domain terms as well as in terms of state space descriptions of filters are provided for better numerical implementation of design methods. The development is valid for two-band as well as for multiband subband coding schemes. The iterated filter banks are shown to give rise to a new family of biorthogonal version of continuous wavelets, the scaled and dilated version of which represents arbitrary square integrable signals. Dilation factors larger than two are implicit in the development resulting from the fact that we consider multiband subband coding schemes. >

Biorthogonal Smooth Local Trigonometric Bases

Abstract: In this paper we discuss smooth local trigonometric bases. We present two generalizations of the orthogonal basis of Malvar and Coifman-Meyer: biorthogonal and equal parity bases. These allow natural representations of constant and, sometimes, linear components. We study and compare their approximation properties and applicability in data compression. This is illustrated with numerical examples.

Computationally attractive real Gabor transforms

Abstract: We present a Gabor transform for real, discrete signals and present a computationally attractive method for computing the transform. For the critically sampled case, we derive a biorthogonal function which is very localized in the time domain. Therefore, truncation of this biorthogonal function allows us to compute approximate expansion coefficients with significantly reduced computational requirements. Further, truncation does not degrade the numerical stability of the transform. We present a tight upper bound on the reconstruction error incurred due to use of a truncated biorthogonal function and summarize computational savings. For example, the expense of transforming a length 2048 signal using length 16 blocks is reduced by a factor of 26 over similar FFT-based methods with at most 0.04% squared error in the reconstruction. >

Enhancement of multichannel data in plasma physics by biorthogonal decomposition

Abstract: The biorthogonal decomposition (also referred to as the singular value decomposition) has recently emerged as a powerful tool for analysing and processing multichannel data. It is shown how this method can be used to enhance the signal-to-noise ratio of such data in a self-consistent way. Three applications to multichannel diagnostics that are routinely used on plasma devices, are presented: a soft X-ray camera, an XUV spectrometer and a microwave reflectometer. In each case, a physical interpretation of the enhancement is given. Furthermore, a robust criterion is presented, allowing data to be processed without making restrictive assumptions about their noise properties.

General analysis of two-band QMF banks

Abstract: The two-channel perfect-reconstruction quadrature-mirror-filter banks (PR QMF banks) are analyzed in detail by assuming arbitrary analysis and synthesis filters. Solutions where the filters are FIR or IIR correspond to the fact that a certain function is monomial or nonmonomial, respectively. For the monomial case, the design problem is formulated as a nonlinear constrained optimization problem. The formulation is quite robust and is able to design various two-channel filter banks such as orthogonal and biorthogonal, arbitrary delay, linear-phase filter banks, to name a few. Same formulation is used for causal and stable PR IIR filter bank solutions. >

Complex, linear-phase filters for efficient image coding

Abstract: With the exception of the Haar basis, real-valued orthogonal wavelet filter banks with compact support lack symmetry and therefore do not possess linear phase This has led to the use of biorthogonal filters for coding of images and other multidimensional data There are, however, complex solutions permitting the construction of compactly supported, orthogonal linear phase QMF filter banks By explicitly seeking solutions in which the imaginary part of the filter coefficients is small enough to be approximated to zero, real symmetric filters can be obtained that achieve excellent compression performance >

On the realizability of biorthogonal, m-dimensional two-band filter banks

Abstract: We show an algebraic approach for the design of ladder structures for causal biorthogonal filter banks. The key ingredient of the approach is known in literature as Euclid's algorithm. Using this algorithm we derive some strong result on the design freedom for ladder structures. In particular we show that the dimensionality of the problem plays an important role. We end by with some conjectures concerning the extensions to multichannel and noncausal filter banks. >

RF simplex spread spectrum receiver and method with symbol deinterleaving prior to bit estimating

Abstract: A method and system are provided for the reception of a biorthogonally-modulated RF signal in a simplex free-space system. The method is found to substantially diminish the coupling between the error probabilities of those bits associated with signal selection and those bits associated with signal polarity in prior art biorthogonal receivers. By decoupling the error probabilities, the polarity information error rate is reduced and consequently the overall error rate of the receiver system is improved. The receiver system preserves soft-decision reliability information for the signal selection bits in a manner so that the information is insensitive to uncompensated amplitude fluctuations in the received signal. A symbol deinterleaver in the receiver is located between the correlators and the bit estimators. The method and system are applicable to all known biorthogonal signal sets but requires a specific interleaving structure in the corresponding transmitter system.

RF simplex spread spectrum receiver and method

Abstract: A method and system are provided for the reception of a biorthogonally-modulated RF signal in a simplex free-space system. The method is found to substantially diminish the coupling between the error probabilities of those bits associated with signal selection and those bits associated with signal polarity in prior art biorthogonal receivers. By decoupling the error probabilities, the polarity information error rate is reduced and consequently the overall error rate of the receiver system is improved. The receiver system preserves soft-decision reliability information for the signal selection bits in a manner so that the information is insensitive to uncompensated amplitude fluctuations in the received signal. The method and system are applicable to all known biorthogonal signal sets but requires a specific interleaving structure in the corresponding transmitter system.

Performance analysis of CDMA systems by using biorthogonal codes

Abstract: A new CDMA system is proposed, which is composed of very low rate convolutional codes and biorthogonal codes. Convolutional encoders with rate 1/64 and constraint length 7 and 128 rows/spl times/64 chips biorthogonal codes generated from Walsh codes with 64 rows/spl times/64 chips are used for encoding and spreading. A Viterbi decoder is used for demodulating the newly designed CDMA system. The performance of proposed CDMA system using biorthogonal codes is better than that of the CDMA system using FEC and orthogonal modulation, and from the hardware complexity point of view, the amount of computation for implementing the proposed CDMA system is increased only a little.

Non-Gaussian Analysis and Hypergroups

Abstract: UDC 517.515 In recent years two approaches to the generalization of the Gaussian infinite-dimensional analysis (white noise analysis) to non-Gaussian measures have appeared: one of them is based on spectral theory for families of commuting self-adjoint operators [1, 2] and the other proceeds from biorthogonal expansions [3, 4]. In this note we show that in the one-dimensional model case the second approach can be extensively generalized if the characters of an Ll-hypergroup are used in its construction instead of exponential functions. For the properties of hypergroups applied below see [5-7] (here we use the term "Ll-hypergroup" instead of "hypercomplex system with locally compact base" according to [7]). 1. Consider a commutative Ll-hypergroup H I with locally compact basis Q, nonnegative structure measure, and multiplicative measure din(x) (x e Q); H I is assumed to be normal (Q 9 x ~-~ x* • Q is

Biorthogonal coupling coefficients of uq(n)

Abstract: The coupling (Wigner-Clebsch-Gordan) coefficients and their isofactors for the unitary quantum algebras uq(n) with the repeating irreducible representations in the coproduct decomposition are considered. Generalizing the U(n) case, the biorthogonal systems of the uq(n) isofactors with the dual multiplicity labels are constructed by means of the recoupling technique in terms of the isofactors with simpler multiplicity structure. A first construction, correlated with the inverted Littlewood-Richardson rules, gives the bilinear combinations of isofactors after applying the proportionality of the q-recoupling (Racah) coefficients to the boundary q-isofactors. An alternative recursive construction gives the nonorthogonal q-isofactors satisfying the most elementary boundary conditions and proportional to the uq(n-1) recoupling coefficients for some less restricted values of parameters. Some multiplicity-free and more general uq(n) recoupling coefficients are found, the blocks (bilinear combinations) of which (equal to the resubducing coefficients of the complementary chains of q-algebras) are proposed to use for the orthonormalization of some uq(n) biorthogonal isofactors, including the general uq(3) case.

Numerical stability of biorthogonal wavelet transforms

Abstract: For orthogonal wavelets, the discrete wavelet and wave packet transforms and their inverses are orthogonal operators with perfect numerical stability. For biorthogonal wavelets, numerical instabilities can occur. We derive bounds for the 2-norm and average 2-norm of these transforms, including efficient numerical estimates if the numberL of decomposition levels is small, as well as growth estimates forL → ∞. These estimates allow easy determination of numerical stability directly from the wavelet coefficients. Examples show that many biorthogonal wavelets are in fact numerically well behaved.

Oblique and biorthogonal multiwavelet bases with fast-filtering algorithms

Abstract: We construct oblique multi-wavelets bases which encompass the orthogonal multi-wavelets and the biorthogonal uni-wavelets of Cohen, Deaubechies and Feauveau. These oblique multi- wavelets preserve the advantages of orthogonal and biorthogonal wavelets and enhance the flexibility of wavelet theory to accommodate a wider variety of wavelet shapes and properties. Moreover, oblique multi-wavelets can be implemented with fast vector-filter-bank algorithms. We use the theory to derive a new construction of biorthogonal uni-wavelets.

McClellan transformation and the construction of biorthogonal wavelet bases of L2R2

Abstract: Bidimensional wavelet bases are constructed by means of McClellan's transformation applied to a pair of one-dimensional biorthogonal wavelet filters. It is shown that under some conditions on the transfer function F(ω1,ω2) associated to the McClellan transformation and on the dilation matrix D, it is possible to construct symmetric compactly supported biorthogonal wavelet bases of L2(R2). Finally, the construction method is illustrated by means of numerical examples.

Wavelets as signaling waveforms over partial-response channels

Abstract: Wavelet transform theory, in the context of multiresolution analysis, has brought out useful properties of Hilbert spaces generated by bases consisting of translates of a single function In particular, biorthogonal wavelet decomposition schemes provide a simple decomposition and reconstruction arrangement by using scaling functions and duals whose translates form biorthogonal bases Based on this concept, schemes are developed for data transmission over a partial response channel where one waveform is used to modulate the transmit data and another, its dual, is used to demodulate the data The schemes do not require a precoder and are not restricted to binary data In one of the schemes suited for bandlimited channels, it is possible to construct several different waveform pairs with a moderate sacrifice in bandwidth

Two-dimensional dyadic subband tree modeling with embedded quantizers

Abstract: We represent the effects of scalar quantization by a nonlinear gain-plus-additive-noise model for the PDF-optimized quantizer in a 2-D dyadic subband tree structure. Based on the equivalent models and the polyphase decomposition approach, we compute the complete mean square error (MSE) using cyclostationary concepts. Then, the optimal filter coefficients satisfying paraunitary or biorthogonal conditions, bit allocations and compensation vectors are obtained such that the above MSE is minimized. This design procedure is then used in compressing the Lena image.

Configuration interaction studies using biorthogonal approach to VB basis

Abstract: In the present article, we have attempted a systematic procedure for use of biorthogonal techniques to the configuration interaction studies in molecules using nonorthogonal valence bond (VB) orbitals. The procedure developed is integral-driven and a program based on this has been developed. Test runs of the program have been carried out in case of full and truncated configuration spaces. 29 refs., 3 tabs.

A complement to a derivation of discrete Gabor expansions

Abstract: In the previous discrete Gabor expansion (DGE) presented by Wexler and Raz (1990), the ratio of the signal length L to the number of frequency channels N is restricted to be an integer. If L is a power of 2, then the oversampling rate is limited to 1, 2, 4, 8, etc. We derive a complementary condition to the derivation of the discrete Gabor expansion of Wexler et. al. and give a general pointwise biorthogonal relationship that relaxes the constraint on N. Consequently, the resulting Gabor expansion applies for integer as well as rational oversampling rates. >