Showing papers on "Biorthogonal system published in 2000"

Wavelet families of increasing order in arbitrary dimensions

Abstract: We build discrete-time compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The associated scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, in-place calculation, and integer-to-integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. While we concentrate on the discrete-time case, some discussion of convergence and stability issues together with examples is given.

The binDCT: fast multiplierless approximation of the DCT

Abstract: This paper presents a family of fast biorthogonal block transforms called binDCT that can be implemented using only shift and add operations. The transform is based on a VLSI-friendly lattice structure that robustly enforces both linear phase and perfect reconstruction properties. The lattice coefficients are parameterized as a series of dyadic lifting steps providing fast, efficient, in place computation of the transform coefficients as well as the ability to map integers to integers. The new 8/spl times/8 transforms all approximate the popular 8/spl times/8 DCT closely, attaining a coding gain range of 8.77-8.82 dB, despite requiring as low as 14 shifts and 31 additions per eight input samples. Application of the binDCT in both lossy and lossless image coding yields very competitive results compared to the performance of the original floating-point DCT.

Linear-phase perfect reconstruction filter bank: lattice structure, design, and application in image coding

Abstract: A lattice structure for an M-channel linear-phase perfect reconstruction filter bank (LPPRFB) based on the singular value decomposition (SVD) is introduced. The lattice can be proven to use a minimal number of delay elements and to completely span a large class of LPPRFBs: all analysis and synthesis filters have the same FIR length, sharing the same center of symmetry. The lattice also structurally enforces both linear-phase and perfect reconstruction properties, is capable of providing fast and efficient implementation, and avoids the costly matrix inversion problem in the optimization process. From a block transform perspective, the new lattice can be viewed as representing a family of generalized lapped biorthogonal transform (GLBT) with an arbitrary number of channels M and arbitrarily large overlap. The relaxation of the orthogonal constraint allows the GLBT to have significantly different analysis and synthesis basis functions, which can then be tailored appropriately to fit a particular application. Several design examples are presented along with a high-performance GLBT-based progressive image coder to demonstrate the potential of the new transforms.

A general approach for analysis and application of discrete multiwavelet transforms

Abstract: This paper proposes a general paradigm for the analysis and application of discrete multiwavelet transforms, particularly to image compression. First, we establish the concept of an equivalent scalar (wavelet) filter bank system in which we present an equivalent and sufficient representation of a multiwavelet system of multiplicity r in terms of a set of r equivalent scalar filter banks. This relationship motivates a new measure called the good multifilter properties (GMPs), which define the desirable filter characteristics of the equivalent scalar filters. We then relate the notion of GMPs directly to the matrix filters as necessary eigenvector properties for the refinement masks of a given multiwavelet system. Second, we propose a generalized, efficient, and nonredundant framework for multiwavelet initialization by designing appropriate preanalysis and post-synthesis multirate filtering techniques. Finally, our simulations verified that both orthogonal and biorthogonal multiwavelets that possess GMPs and employ the proposed initialization technique can perform better than the popular scalar wavelets such as Daubechies'D8 wavelet and the D(9/7) wavelet, and some of these multiwavelets achieved this with lower computational complexity.

Spectral Transformation Chains and Some New Biorthogonal Rational Functions

Abstract: A discrete-time chain, associated with the generalized eigenvalue problem for two Jacobi matrices, is derived. Various discrete and continuous symmetries of this integrable equation are revealed. A class of its rational, elementary and elliptic functions solutions, appearing from a similarity reduction, are constructed. The latter lead to large families of biorthogonal rational functions based upon the very-well-poised balanced hypergeometric series of three types: the standard hypergeometric series 9 F 8, basic series 10ϕ9 and its elliptic analogue 10 E 9. For an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly.

A Lanczos-type method for multiple starting vectors

Abstract: Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of bior- thogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to mul- tiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left start- ing blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in deation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ look-ahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczos-type methods.

Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

Abstract: Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator onR is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions (CDP) is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev normsk¢k Hs.(0;1)/ for s 2 .i0:824926; 2:5/. In particular, the multiwavelets form Riesz bases for L2.(0; 1)/.

Apparatus, and associated method, for coding and decoding multi-dimensional biorthogonal codes

Abstract: Apparatus, and an associated method, by which to form biorthogonal codes utilized in a multi-dimensional modulation scheme. In one implementation, biorthogonal coding is provided in a cellular communication system in a manner that minimizes correlation of pairs of coordinates utilized in the communication system. Data that is to be communicated by a sending station is first modulated by a binary phase shift keying modulation. These first-modulated values are used by a mapper that maps the values to selected dimension values. And, the selected dimension values into which the first-modulated values are mapped are used to select biorthogonal code values.

Apparatus and method for encoding/decoding transport format combination indicator in CDMA mobile communication system

Abstract: An apparatus and method for encoding/decoding a transport format combination indicator (TFCI) in a CDMA mobile communication system In the TFCI encoding apparatus, a one-bit generator generates a sequence having the same symbols A basis orthogonal sequence generator generates a plurality of basis orthogonal sequences A basis mask sequence generator generates a plurality of basis mask sequences An operation unit receives TFCI bits that are divided into a first information part representing biorthogonal sequence conversion, a second information part representing orthogonal sequence conversion, and a third information part representing mask sequence conversion and combines an orthogonal sequence selected from the basis orthogonal sequence based on the second information, a biorthogonal sequence obtained by combining the selected orthogonal sequence with the same symbols selected based on the first information part, and a mask sequence selected based on the biorthogonal sequence and the third information part, thereby generating a TFCI sequence

Rationalizing the coefficients of popular biorthogonal wavelet filters

Abstract: Many wavelet filters found in the literature have irrational coefficients and thus require infinite precision implementation. One of the most popular filter pairs is the "9/7" biorthogonal pair of Cohen, Daubechies and Feauveau (1992), which is adopted in the FBI finger-print compression standard. We present a technique to rationalize the coefficients of wavelet filters that will preserve biorthogonality and perfect reconstruction. Furthermore, most of the zeros at z=-1 will also be preserved. These zeros are important for achieving regularity. The rationalized coefficients filters have characteristics that are close to the original irrational coefficients filters. Three popular pairs of filters, which include the "9/7" pair, are considered.

Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments

Abstract: We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms.

The LiftLT: fast-lapped transforms via lifting steps

Abstract: This paper introduces a class of multiband linear phase-lapped biorthogonal transforms with fast, VLSI-friendly implementations via lifting steps called the LiftLT. The transform is based on a lattice structure that robustly enforces both linear phase and perfect reconstruction properties. The lattice coefficients are parameterized as a series of lifting steps, providing fast, efficient, in-place computation of the transform coefficients. The new transform is designed for applications in image and video coding. Compared to the popular 8/spl times/8 DCT, the 8/spl times/16 LiftLT only requires one more multiplication, 22 more additions, and six more shifting operations. However, image coding examples show that the LiftLT is far superior to the DCT in both objective and subjective coding performance. Thanks to properly designed overlapping basis functions, the LiftLT can completely eliminate annoying blocking artifacts. In fact, the novel LT's coding performance consistently surpasses that of the much more complex 9/7-tap biorthogonal wavelet with floating-point coefficients. More importantly, the transform's block-based nature facilitates one-pass sequential block coding, region-of-interest coding/decoding, and parallel processing.

A class of totally positive refinable functions

Abstract: Refinable functions play a main role in the construction of wavelets; here we present a large class of refinable functions which can be identified through the explicit expression of their masks. These refinable functions can be considered as a generalization of cardinal B-splines, from which they borrow a few nice properties, like symmetry and total positivity, that are of relevant interest in most filtering applications. Moreover, each of the refinable functions here considered generates a multiresolution analysis. The associated wavelet bases and the corresponding biorthogonal bases are constructed, too.

Design of BFDM/OQAM systems based on biorthogonal modulated filter banks

Abstract: A new system of multicarrier modulation is described which takes advantage of the biorthogonality property to generalize the OFDM/OQAM technique into a biorthogonal frequency division multiplex scheme named BFDM/OQAM. A discrete-time analysis of this scheme leads to a discrete model of modulated transmultiplexer. Using a polyphase decomposition of this transmultiplexer, and assuming a distortion-free channel, we determine the mathematical conditions on the prototype filter allowing an exact cancellation of intersymbol and interchannel interference. Then, computationally efficient realizations of the modulators and demodulators are given. Lastly, some design examples are provided illustrating respective advantages of BFDM/OQAM and OFDM/OQAM.

Discrete Multiresolution Analysis Using Hermite Interpolation: Biorthogonal Multiwavelets

Abstract: We generalize Harten's interpolatory multiresolution representation to include Hermite interpolation. Compact Hermite interpolation with optimal order accuracy is used in both the decomposition and reconstruction algorithm. The resulting multiple basis functions (biorthogonal multiwavelets) are symmetric or skew-symmetric, compact, and analytic. Harten's approach has several advantages: the multiresolution scheme is inherently discrete, nonperiodic boundary conditions are easy to implement, and the representation can be extended to nonuniform grids in bounded domains. We demonstrate the compression features of the new multiple basis functions by application to several examples.

Theory of rate-distortion-optimal, constrained filterbanks-application to IIR and FIR biorthogonal designs

Abstract: We design filterbanks that are best matched to input signal statistics in M-channel subband coders, using a rate-distortion criterion. Previous research has shown that unconstrained-length, paraunitary filterbanks optimized under various energy compaction criteria are principal-component filterbanks that satisfy two fundamental properties: total decorrelation and spectral majorization. In this paper, we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints. Our results apply to a broad class of rate-distortion criteria, including the conventional coding gain criterion as a special case. A consequence of these properties is that optimal perfect-reconstruction (PR) filterbanks take the form of the cascade of principal-component filterbanks and a bank of pre- and post-conditioning filters. The proof uses variational techniques and is applicable to a variety of constrained design problems. In the second part of this paper, we apply the theory above to practical filterbank design problems. We give analytical expressions for optimal IIR biorthogonal filterbanks; our analysis validates a conjecture by several researchers. We then derive the asymptotic limit of optimal FIR biorthogonal filterbanks as filter length tends to infinity. The performance loss due to FIR constraints is quantified theoretically and experimentally. The optimal filters are quite different from traditional filters. Finally, a sensitivity analysis is presented.

Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems

Abstract: Summary. For analytic functions the remainder term of quadrature rules can be represented as a contour integral with a complex kernel function. From this representation different remainder term estimates involving the kernel are obtained. It is studied in detail how polynomial biorthogonal systems can be applied to derive sharp bounds for the kernel function. It is shown that these bounds are practical to use and can easily be computed. Finally, various numerical examples are presented.

Rapid design of biorthogonal wavelet transforms

Abstract: A rapid design methodology for biorthogonal wavelet transform cores has been developed based on a generic, scaleable architecture for wavelet filters. The architecture offers efficient hardware utilisation by combining the linear phase property of biorthogonal filters with decimation in a MAC-based implementation. The design has been captured in VHDL and parameterised in terms of wavelet type, data word length and coefficient word length. The control circuit is embedded within the cores and allows them to be cascaded without any interface glue logic for any desired level of decomposition. The design time to produce silicon layout of a biorthogonal wavelet system is typically less than a day. The silicon cores produced are comparable in area and performance to hand-crafted designs. The designs are portable across a range of foundries and are also applicable to FPGA and PLD implementations.

Wavelet-based reconstruction of irregularly-sampled images: application to stereo imaging

Abstract: We are concerned with the reconstruction of a regularly-sampled image based on irregularly-spaced samples thereof. We propose a new iterative method based on a wavelet representation of the image. For this representation we use a biorthogonal spline wavelet basis implemented on an oversampled grid. We apply the developed algorithm to disparity-compensated stereoscopic image interpolation. Under disparity compensation, the resulting sampling grids are irregular and require the irregular/regular interpolation. We show the experimental results on real-world images and we compare our results with other methods proposed in the literature.

Wavelet-based MC-CDMA cellular systems

Abstract: This paper presents work carried out on a multi-carrier code division multiple access (MC-CDMA) for a down-link cellular radio systems. The MC-CDMA system carries out energy spreading in all available sub-channels or in the frequency domain. It also performs multi-carrier modulation using a set of orthogonal (or biorthogonal) filters. The optimisation of these filters is done based on their time-frequency characteristics and depends also on the application. Here the biorthogonal filters are implemented with M-band wavelet-based system in a form of cosines modulated filters to configure a perfect reconstruction system. The simulation results show that this WB-MC-CDMA system suffers lower interchannel interference, thus making its response more robust against multipath fading and narrow band interference or jamming signal.

An algebraic construction of k-balanced multiwavelets via the lifting scheme

Abstract: Multiwavelets have been revealed to be a successful generalization within the context of wavelet theory Recently Lebrun and Vetterli have introduced the concept of “balanced” multiwavelets, which present properties that are usually absent in the case of classical multiwavelets and do not need the prefiltering step In this work we present an algebraic construction of biorthogonal multiwavelets by means of the well-known “lifting scheme” The flexibility of this tool allows us to exploit the degrees of freedom left after satisfying the perfect reconstruction condition in order to obtain finite k-balanced multifilters with custom-designed properties which give rise to new balanced multiwavelet bases All the problems we deal with are stated in the framework of banded block recursive matrices, since simplified algebraic conditions can be derived from this recursive approach

Design of biorthogonal M-channel cosine-modulated FIR/IIR filter banks

Abstract: We investigate on the problem of designing biorthogonal filter banks obtained by the cosine-modulation of an analysis and a synthesis prototype. The design procedure is developed in a general framework including both finite impulse response (FIR) and causal infinite impulse response (IIR) prototypes. Numerical examples are presented to show the effectiveness of the method.

Grid refinement in multiscale dynamic optimization

Abstract: In the present work we explore an adaptive discretization scheme for dynamic optimization problems applied to input and state estimation. The proposed method is embedded into a solution methodology where the dynamic optimization problem is approximated by a hierarchy of successively refined finite dimensional problems. Information on the solution of the coarser approximations is used to construct a fully adaptive, problem dependent discretization where the finite dimensional spaces are spanned by biorthogonal wavelets arising from B-splines. We demonstrate exemplarily that the proposed strategy is capable to identify accurate discretization meshes which are more economical than uniform meshes with respect to the ratio of approximation quality vs. number of used trial functions.

Perturbative method for generalized spectral decompositions

Abstract: Imposing analytic properties to states and observables we construct a perturbative method to obtain a generalized biorthogonal system of eigenvalues and eigenvectors for quantum unstable systems. A decay process can be described using this generalized spectral decomposition, and the final generalized state is obtained.

Optimizing the recognition rates of unconstrained handwritten numerals using biorthogonal spline wavelets

Abstract: In this paper an approach for off-line recognition of unconstrained handwritten numerals is presented. This approach uses the Cohen-Daubechies-Feauveau (CDF) family of biorthogonal spline wavelets as a feature extractor for absorbing local variations in handwritten characters and a multilayer cluster neural network as classifier. Experiments with the bases CDF 2/2, CDF 2/4, CDF 3/3 and CDF 3/7 were performed using the handwritten numeral database of Concordia University of Canada. The results show that CDF biorthogonal wavelets yield a performance improvement of 2.4% in numeral recognition, compared to the results obtained with the Haar wavelets.