Showing papers on "Biorthogonal system published in 2004"
TL;DR: A new construction of lifted biorthogonal wavelets on surfaces of arbitrary two-manifold topology for compression and multiresolution representation, and wavelet constructions for bilinear, bicubic, and B-spline subdivision are presented.
Abstract: We present a new construction of lifted biorthogonal wavelets on surfaces of arbitrary two-manifold topology for compression and multiresolution representation. Our method combines three approaches: subdivision surfaces of arbitrary topology, B-spline wavelets, and the lifting scheme for biorthogonal wavelet construction. The simple building blocks of our wavelet transform are local lifting operations performed on polygonal meshes with subdivision hierarchy. Starting with a coarse, irregular polyhedral base mesh, our transform creates a subdivision hierarchy of meshes converging to a smooth limit surface. At every subdivision level, geometric detail is expanded from wavelet coefficients and added to the surface. We present wavelet constructions for bilinear, bicubic, and biquintic B-spline subdivision. While the bilinear and bicubic constructions perform well in numerical experiments, the biquintic construction turns out to be unstable. For lossless compression, our transform is computed in integer arithmetic, mapping integer coordinates of control points to integer wavelet coefficients. Our approach provides a highly efficient and progressive representation for complex geometries of arbitrary topology.
73 citations
TL;DR: This paper suggests an alternative denoising procedure based on the Efromovich-Pinsker algorithm that is optimal over a wide class of noise distributions and implies an optimal method for recovering the derivative of a noisy signal.
Abstract: Multiwavelets are relative newcomers into the world of wavelets. Thus, it has not been a surprise that the used methods of denoising are modified universal thresholding procedures developed for uniwavelets. On the other hand, the specific of a multiwavelet discrete transform is that typical errors are not identically distributed and correlated, whereas the theory of the universal thresholding is based on the assumption of identically distributed and independent normal errors. Thus, we suggest an alternative denoising procedure based on the Efromovich-Pinsker algorithm. We show that this procedure is optimal over a wide class of noise distributions. Moreover, together with a new cristina class of biorthogonal multiwavelets, which is introduced in this paper, the procedure implies an optimal method for recovering the derivative of a noisy signal. A Monte Carlo study supports these conclusions.
73 citations
TL;DR: This work treats nonparametric stochastic regression using smooth design-adapted wavelets built by means of the lifting scheme, and uses weighted average interpolation to construct biorthogonal wavelets with a higher number of vanishing analyzing moments.
Abstract: We treat nonparametric stochastic regression using smooth design-adapted wavelets built by means of the lifting scheme. The proposed method automatically adapts to the nature of the regression problem, that is, to the irregularity of the design, to data on the interval, and to arbitrary sample sizes (which do not need to be a power of 2). As such, this method provides a uniform solution to the usual criticisms of first-generation wavelet estimators. More precisely, starting from the unbalanced Haar basis orthogonal with respect to the empirical design measure, we use weighted average interpolation to construct biorthogonal wavelets with a higher number of vanishing analyzing moments. We include a lifting step that improves the conditioning through constrained local semiorthogonalization. We propose a wavelet thresholding algorithm and show its numerical performance both on real data and in simulations including white, correlated, and heteroscedastic noise.
48 citations
TL;DR: In this article, the authors give biorthogonal system characterizations of Banach spaces that fail the Dunford-Pettis property, contain an isomorphic copy of c0, or fail the hereditary Dunford Pettis condition.
Abstract: We give biorthogonal system characterizations of Banach spaces that fail the Dunford-Pettis property, contain an isomorphic copy of c0, or fail the hereditary Dunford-Pettis property. We combine this with previous results to show that each infinite dimensional Banach space has one of three types of biorthogonal systems.
46 citations
26 Jun 2004
TL;DR: A coevolutionary genetic algorithm is developed that searches the space of biorthogonal wavelets and the lifting technique, which defines a wavelet as a sequence of digital filters, provides a compact representation and an efficient way of handling necessary constraints.
Abstract: Finding a good wavelet for a particular application and type of input data is a difficult problem. Traditional methods of wavelet design focus on abstract properties of the wavelet that can be optimized analytically but whose influence on its real-world performance are not entirely understood. In this paper, a coevolutionary genetic algorithm is developed that searches the space of biorthogonal wavelets. The lifting technique, which defines a wavelet as a sequence of digital filters, provides a compact representation and an efficient way of handling necessary constraints. The algorithm is applied to a signal compression task with good results.
38 citations
TL;DR: This paper presents a 2-dimensional biorthogonal DWT processor design based on the residue number system that is able to fit into a 1,000,000-gate FPGA device and be able to complete a first level 2-D DWT decomposition of a 32/spl times/32-pixel image in 205 /spl mu/s.
Abstract: Discrete wavelet transform has been incorporated as part of the JPEG2000 image compression standard and is used in many consumer imaging products. This paper presents a 2-dimensional biorthogonal DWT processor design based on the residue number system. The symmetric extension scheme is employed to reduce distortion at image boundaries. Hardware complexity reduction and utilization improvement are achieved by hardware sharing. Our implementation results show that the design is able to fit into a 1,000,000-gate FPGA device and is able to complete a first level 2-D DWT decomposition of a 32/spl times/32-pixel image in 205 /spl mu/s.
37 citations
TL;DR: In this article, the Sklyanin algebra admits realizations by difference operators acting on theta functions, and the authors prove this conjecture and also give natural biorthogonal and orthogonal bases for the representation space.
Abstract: The Sklyanin algebra admits realizations by difference operators acting on theta functions. Sklyanin found an invariant metric for the action and conjectured an explicit formula for the corresponding reproducing kernel. We prove this conjecture and also give natural biorthogonal and orthogonal bases for the representation space. Moreover, we discuss connections with elliptic hypergeometric series and integrals and with elliptic 6j-symbols.
37 citations
TL;DR: In this paper, the authors give a characterization of biorthogonal wavelets arising from MRA's of multiplicity D entirely in terms of the dimension function, which improves the previous characterization in [8] removing an unnecessary angle condition.
Abstract: We give a characterization of biorthogonal wavelets arising from MRA’s of multiplicity D entirely in terms of the dimension function. This improves the previous characterization in [8] removing an unnecessary angle condition. Besides we characterize Riesz wavelets arising from MRA’s, and present new proofs based on shift-invariant space theory, generalizing the 1-dimensional results appearing in [17].
32 citations
TL;DR: A lattice structure for a special class of N-channel oversampled linear-phase perfect reconstruction filterbanks with a decimation factor M smaller than N is investigated and able to provide fast implementation and allows us to determine the filter coefficients by solving an unconstrained optimization problem.
Abstract: We investigate a lattice structure for a special class of N-channel oversampled linear-phase perfect reconstruction filterbanks with a decimation factor M smaller than N. We deal with systems in which all analysis and synthesis filters have the same finite impulse response (FIR) length and share the same center of symmetry. We provide the minimal lattice factorization of a polyphase matrix of a particular class of these oversampled filterbanks (FBs). All filter coefficients are parameterized by rotation angles and positive values. The resulting lattice structure is able to provide fast implementation and allows us to determine the filter coefficients by solving an unconstrained optimization problem. We consider next the case where we are given the generalized lapped pseudo-biorthogonal transform (GLPBT) lattice structure with specific parameters, and we a priori know the correlation matrix of noise that is added in the transform domain. In this case, we provide an alternative lattice structure that suppress the noise. We show that the proposed systems with the lattice structure cover a wide range of linear-phase perfect reconstruction FBs. We also introduce a new cost function for oversampled FB design that can be obtained by generalizing the conventional coding gain. Finally, we exhibit several design examples and their properties.
28 citations
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TL;DR: In this article, the Sklyanin algebra admits realizations by difference operators acting on theta functions, and also gives natural biorthogonal and orthogonal bases for the representation space.
Abstract: The Sklyanin algebra admits realizations by difference operators acting on theta functions. Sklyanin found an invariant metric for the action and conjectured an explicit formula for the corresponding reproducing kernel. We prove this conjecture, and also give natural biorthogonal and orthogonal bases for the representation space. Moreover, we discuss connections with elliptic hypergeometric series and integrals and with elliptic 6j-symbols.
26 citations
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TL;DR: In this article, biorthogonal polynomials for a measure over the complex plane which consists in the exponential of a potential V(z,z*) and in a set of external sources at the numerator and at the denominator were constructed.
Abstract: We construct biorthogonal polynomials for a measure over the complex plane which consists in the exponential of a potential V(z,z*) and in a set of external sources at the numerator and at the denominator. We use the pseudonorm of these polynomials to calculate the resolvent integral for correlation functions of traces of powers of complex matrices (under certain conditions).
TL;DR: A new lifting pattern is developed, namely, the progressive lifting pattern, which allows us to pairwise generate M-band interpolating filterbanks and wavelets by the order from lowpass to highpass filters.
Abstract: Recently, the lifting scheme was generalized to the multidimensional and multiband cases and was used to design M-band interpolating scaling filters and their duals. Based on this idea, we develop a new lifting pattern, namely, the progressive lifting pattern. This pattern allows us to pairwise generate M-band interpolating filterbanks and wavelets by the order from lowpass to highpass filters. A complete lifting procedure is divided into M - 1 simple steps, in each step, a pair of filters (the l'th filter and its dual) are generated. In this way, an M -band biorthogonal interpolating filterbank/wavelet is determined by M(M - 1) lifting filters. The first 2(M 1) lifting filters completely characterize the two scaling filters as well as the vanishing moments of bandpass and highpass filters; the residual (M - 1) (M - 2) lifting filters are used to pairwise optimize the bandpass and highpass filters in terms of the criterion of stopband energy minimization. The obtained M-band biorthogonal interpolating filterbanks and wavelets provide excellent frequency characteristics, in particular, low stopband sidelobes. Furthermore, the pattern is also utilized to design signal-adapted interpolating filterbanks and their rational coefficient counterparts in terms of subband coding gain. The obtained filterbanks achieve large subband coding gains. The rational coefficient filterbanks preserve the biorthogonality and allow wavelet transforms from integers to integers and a unifying lossy/lossless coding framework at the cost of a slight degradation in subband coding gain.
17 May 2004
TL;DR: This work investigates the implementation of two lifting coefficient sets, rational and irrational, for the biorthogonal 9/7 wavelet and finds that the best hardware and PSNR performance is obtained using the rational coefficient set quantized with gain compensation and lumped scaling.
Abstract: The lifting structure has been shown to be computationally efficient for implementing filter banks. The hardware implementation of a filter bank requires that the lifting coefficients be quantized. The quantization method determines compression performance, hardware size, hardware speed and energy. We investigate the implementation of two lifting coefficient sets, rational and irrational, for the biorthogonal 9/7 wavelet. Six different approaches are used to find optimal quantized lifting coefficients from these sets. We find that the best hardware and PSNR performance is obtained using the rational coefficient set quantized with gain compensation and lumped scaling.
TL;DR: In this article, a backward biorthogonalization approach is proposed to generate orthogonal projections onto a reduced subspace, which is relevant to problems amenable to be represented by a general linear model.
Abstract: A backward biorthogonalization approach is proposed, which
modifies biorthogonal functions so as to generate orthogonal
projections onto a reduced subspace. The technique is relevant to
problems amenable to be represented by a general linear model. In
particular, problems of data compression, noise reduction, and
sparse representations may be tackled by the proposed approach.
03 Mar 2004
TL;DR: In this article, two criteria for dictionary redundancy elimination are discussed, one of them operates by disregarding linearly dependent atoms, whilst the other selects linearly independent atoms, implemented by the modified Gram-Schmidt orthogonalization with pivoting technique, and is suitable for handling the effect of quasi-linear dependence, most likely to be present in a redundant dictionary.
Abstract: Two criteria for dictionary redundancy elimination are discussed. One of them operates by disregarding linearly dependent atoms, whilst the other selects linearly independent atoms. The latter is implemented by the modified Gram–Schmidt orthogonalisation with pivoting technique, and is suitable for handling the effect of ‘quasi-linear dependence’, most likely to be present in a redundant dictionary. The corresponding reciprocal waveforms are easily obtained within the workings of the selection process. Such waveforms are biorthogonal to the selected atoms and allow computation of the respective coefficients of the linear combination approximating an arbitrary signal at best in a minimum distance sense.
TL;DR: In this paper, the Villemoes machine is used to compute the Sobolev smoothness of a refinable function, which involves the computation of the spectral radius of a special matrix which has at least quadratic time complexity with respect to the refinement mask size.
TL;DR: In this paper, an approach for a biorthogonal basis constructed by the lifting scheme is presented, which allows the choice of the smoothness at least of the primal wavelet in a very natural way.
Abstract: Several applications require to retrieve a certain pattern from a signal where the actual scaling of the pattern is not known before. We consider applications like evaluation of mass spectrograms, detection of component wearout by observing the current of an engine, detection of pollutions of rotor spinning machines, decomposition of (audio) signals into time-frequency atoms.
We try to solve this problem with discrete wavelet transforms where the wavelet function is constructed to match the given pattern. An approach for a biorthogonal basis constructed by the lifting scheme is presented. This method is rather simple and fast and allows the choice of the smoothness at least of the primal wavelet in a very natural way. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this article, the authors extend the lifting scheme of Sweldens to the case of biorthogonal wavelet functions and show how to construct a sequence of pairs of pairs from one pair of wavelet or an orthonormal scaling function.
Abstract: Let @f be an orthonormal scaling function with approximation degree p - 1, and let B"nbe the B-spline of order n. Then, spline type scaling functions defined by @?"n = @? * B"n (n = 1, 2, ...) possess higher approximation order, p + n - 1, and compact support. The corresponding biorthogonal wavelet functions are also constructed. This technique is extended to the case of biorthogonal scaling function system. As an application of the method supplied in this paper, one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or an orthonormal scaling function. In particular, if both the method and the lifting scheme of Sweldens (see [1]) are applied, then all pairs of biorthogonal spline type scaling functions shown in references [2] and [3] can be constructed from the Haar scaling function.
TL;DR: A rigorous analysis of the multi-resolution time domain (MRTD) is provided, employing positive sampling functions and their biorthogonal dual technique, which demonstrates superiority over the FDTD in terms of memory and speed.
Abstract: The multi-resolution time domain (MRTD) technique for electromagnetic field equations was proposed by Krumpholz, Katehi et al., using Battle-Lemarie wavelets. The basis principle behind the MRTD is the wavelet-Galerkin time domain (WGTD) approach. Despite its effectiveness in space discretization, the complexity ofthe MRTD makes it unpopular. Recently, the WGTD was significantly simplified by Cheong et al. based on the approximate sampling property ofthe shifted versions ofthe Daubechies compactly supported wavelets. In this paper, we provide a rigorous analysis ofthe MRTD, employing positive sampling functions and their biorthogonal dual. We call our approach as the sampling biorthogonal time-domain (SBTD) technique. The introduced sampling and dual functions are both originated from Daubechies scaling functions of order 2 (referred as to D2), and form a biorthonormal system. This biorthonormal system has exact interpolation properties and demonstrates superiority over the FDTD in terms ofmemory and speed. Numerical examples and comparisons with the traditional FDTD results are provided.
TL;DR: In this paper, a very neat extension of Tian and Wells' family of biorthogonal Coifman wavelets by recovering other "missing'' members of the BWC wavelet system is presented.
Abstract: Wavelet systems with a maximum number of balanced vanishing moments are known to be extremely useful in a variety of applications such as image and video compression. Tian and Wells recently created a family of such wavelet systems, called the biorthogonal Coifman wavelets, which have proved valuable in both mathematics and applications. The purpose of this work is to establish along with direct proofs a very neat extension of Tian and Wells' family of biorthogonal Coifman wavelets by recovering other "missing'' members of the biorthogonal Coifman wavelet systems.
TL;DR: This paper uses biorthogonal spline wavelets to develop numerical schemes for multidimensional advection-reaction equations that have the ability to carry out adaptive compression while preserving the total mass.
23 May 2004
TL;DR: This architecture performs 1-D DBWT decomposition of an N/sub 0/-sample input signal with K decomposition levels in N/ sub 0//2 clock cycles, at least twice as fast as other known DBWT architectures.
Abstract: Biorthogonal wavelets offer improved coding gain and an efficient treatment of signal boundaries. In this paper, we propose a high-speed/high-throughput architecture for 1-D Discrete Biorthogonal Wavelet Transform (DBWT). This architecture performs 1-D DBWT decomposition of an N/sub 0/-sample input signal with K decomposition levels in N/sub 0//2 clock cycles. Therefore, it is at least twice as fast as other known DBWT architectures. The architecture offers efficient hardware utilisation for VLSI implementation by combining the linear phase property of biorthogonal filters with decimation.
TL;DR: An optimum bit loading scheme is found that distributes the bit rate transmitted across the various subchannels, where the precise subch channels are assigned to each user, and an optimum transceiver is found.
27 Oct 2004
TL;DR: The use of adaptive biorthogonal wavelet packet bases in a probabilistic approach to texture analysis is discussed, thus combining the advantages of bior fourth wavelets (FIR, linear phase) with those of a coherent texture model.
Abstract: We discuss the use of adaptive biorthogonal wavelet packet bases in a probabilistic approach to texture analysis, thus combining the advantages of biorthogonal wavelets (FIR, linear phase) with those of a coherent texture model. The computation of the probability uses both the primal and dual coefficients of the adapted biorthogonal wavelet packet basis. The computation of the biorthogonal wavelet packet coefficients is done using a lifting scheme, which is very efficient. The model is applied to the classification of mosaics of Brodatz textures, the results showing improvement over the performance of the corresponding orthogonal wavelets.
TL;DR: In this paper, a strengthened Cauchy-Schwarz inequality for one-dimensional biorthogonal wavelets is proved for a class of Hilbert spaces defined in terms of weighted Fourier transforms, which contain as relevant examples the standard Sobolev spaces H ( s ) as well as their homogeneous version.
01 Jan 2004
TL;DR: This work surveys the most relevant results regarding wavelets and presents different biorthogonal wavelet constructions based on highly efficient and simple-to-use lifting operations that are suitable for lossy and lossless compression of large-scale data sets.
Abstract: Multiresolution data representations provide an indispensable tool for the compression, progressive transmission, and visualization of scientific data. Wavelet transforms based on B-spline scaling functions are frequently used to obtain continuous surface- and volume representations at multiple levels of resolution. Starting with a fine-resolution data set, wavelet decomposition provides a sequence of coarser approximations based on B-spline scaling functions and a set of wavelet coefficients containing the geometric differences with respect to the finer levels. The inverse transform can be used to reconstruct the finer levels of resolution from these wavelet coefficients within linear computation time. Biorthogonal wavelet transforms facilitate local computation of decomposition and reconstruction. We survey the most relevant results regarding wavelets and present different biorthogonal wavelet constructions based on highly efficient and simple-to-use lifting operations. Our approaches are suitable for lossy and lossless compression of large-scale data sets.
20 Sep 2004
TL;DR: This work considers parametrisations of biorthogonal filters, which can be used to generate a large keyspace and yield filters with very different quality, and shows their properties regarding compression and encryption.
Abstract: Selective encryption is used to encrypt parts of a bitstream, in our case images which are compressed by a wavelet based method. One approach is to keep the filter secret which is used for the transformation. Parameterised wavelet filters can be used to generate a large keyspace, however, in the case of orthogonal filters obtained by a variant of Pollen's factorisation it turns out that different parameters yield filters with very different quality and in particular worse quality as compared to the standard biorthogonal filters usually used for compression. To eventually overcome these limitations, we consider parametrisations of biorthogonal filters in this work. We discuss methods to create such filters, and show their properties regarding compression and encryption.
TL;DR: This note presents a wavelets‐Galerkin scheme for the numerical solution of a Stokes problem by using the scaling function of a symmetric biorthogonal spline wavelets that can be modified to generate the divergence‐free wavelets.
Abstract: This note presents a wavelets-Galerkin scheme for the numerical solution of a Stokes problem by using the scaling function of a symmetric biorthogonal spline wavelets that can be modified to generate the divergence-free wavelets. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 193–198, 2004
TL;DR: In this article, the phase space of N damped linear oscillators is endowed with a bilinear map under which the evolution operator is symmetric, which allows properties familiar from conservative systems to be recovered, e.g., eigenvectors are 'orthogonal' under the map and obey sum rules, initial value problems are readily solved and perturbation theory applies to the complex eigenvalues.
Abstract: The phase space of N damped linear oscillators is endowed with a bilinear map under which the evolution operator is symmetric. This analogue of self-adjointness allows properties familiar from conservative systems to be recovered, e.g., eigenvectors are 'orthogonal' under the bilinear map and obey sum rules, initial-value problems are readily solved and perturbation theory applies to the complex eigenvalues. These concepts are conveniently represented in a biorthogonal basis.