Showing papers on "Biorthogonal system published in 2002"
Book•
11 Dec 2002
TL;DR: A model problem and its Discretization, a gorithms: Initial data, Prediction, Fluxes and Evolution, and a discussion of results.
Abstract: 1 Model Problem and Its Discretization.- 1.1 Conservation Laws.- 1.2 Finite Volume Methods.- 2 Multiscale Setting.- 2.1 Hierarchy of Meshes.- 2.2 Motivation.- 2.3 Box Wavelet.- 2.3.1 Box Wavelet on a Cartesian Grid Hierarchy.- 2.3.2 Box Wavelet on an Arbitrary Nested Grid Hierarchy.- 2.4 Change of Stable Completion.- 2.5 Box Wavelet with Higher Vanishing Moments.- 2.5.1 Definition and Construction.- 2.5.2 A Univariate Example.- 2.5.3 A Remark on Compression Rates.- 2.6 Multiscale Transformation.- 3 Locally Refined Spaces.- 3.1 Adaptive Grid and Significant Details.- 3.2 Grading.- 3.3 Local Multiscale Transformation.- 3.4 Grading Parameter.- 3.5 Locally Uniform Grids.- 3.6 Algorithms: Encoding, Thresholding, Grading, Decoding.- 3.7 Conservation Property.- 3.8 Application to Curvilinear Grids.- 4 Adaptive Finite Volume Scheme.- 4.1 Construction.- 4.1.1 Strategies for Local Flux Evaluation.- 4.1.2 Strategies for Prediction of Details.- 4.2 A gorithms: Initial data, Prediction, Fluxes and Evolution.- 5 Error Analysis.- 5.1 Perturbation Error.- 5.2 Stability of Approximation.- 5.3 Reliability of Prediction.- 6 Data Structures and Memory Management.- 6.1 Algorithmic Requirements and Design Criteria.- 6.2 Hashing.- 6.3 Data Structures.- 7 Numerical Experiments.- 7.1 Parameter Studies.- 7.1.1 Test Configurations.- 7.1.2 Discretization.- 7.1.3 Computational Complexity and Stability.- 7.1.4 Hash Parameters.- 7.2 Real World Application.- 7.2.1 Configurations.- 7.2.2 Discretization.- 7.2.3 Discussion of Results.- A Plots of Numerical Experiments.- B The Context of Biorthogonal Wavelets.- B.1 General Setting.- B.1.1 Multiscale Basis.- B.1.2 Stable Completion.- B.1.3 Multiscale Transformation.- B.2 Biorthogonal Wavelets of the Box Function.- B.2.1 Haar Wavelets.- B.2.2 Biorthogonal Wavelets on the Real Line.- References.- List of Figures.- List of Tables.- Notation.
166 citations
TL;DR: In this article, the authors show that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters.
Abstract: The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V1, V2 in two different variables, these kernels may be expressed in terms of finite dimensional “windows” spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V1, V2. The vectors formed by such subsequences satisfy “dual pairs” of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V1 or V2 and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V1 and V2. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters.
140 citations
TL;DR: In this article, the authors give a formula for compactly supported biorthogonal multi-wavelets, which makes construction of compactly supportable biorghogonal uniwavelets easy like in the construction of BOWs.
116 citations
01 Jan 2002
TL;DR: A major advantage of fuzzy wavenets techniques in comparison to most neurofuzzy methods is that the rules are validated online during learning by using a simple algorithm based on the fast wavelet decomposition algorithm.
Abstract: The combination of wavelet theory and neural networks has lead to the development of wavelet networks. Wavelet networks are feed-forward neural networks using wavelets as activation function. They have been used in classification and identification problems with some success. Their strength lies on catching essential features in “frequency-rich” signals. In wavelet networks, both the position and the dilation of the wavelets are optimized besides the weights. Wavenet is another term to describe wavelet networks. Originally, wavenets did refer to neural networks using dyadic wavelets. In wavenets, the position and dilation of the wavelets are fixed and the weights are optimized by the network. We propose to adopt this terminology. The theory of wavenets has been generalized by the author to biorthogonal wavelets. This extension to biorthogonal wavelets has lead to the development of fuzzy wavenets. A serious difficulty with most neurofuzzy methods is that they do often furnish rules without a transparent interpretation. A solution to this problem is furnished by multiresolution techniques. The most appropriate membership functions are chosen from a dictionary of membership functions forming a multiresolution. The dictionary contains a number of membership functions that have the property to be symmetric, everywhere positive and with a single maxima. This family includes among others splines and some radial functions. The main advantage of using a dictionary of membership functions is that each term, such as “small”, “large” is well defined beforehand and is not modified during learning. The multiresolution properties of the membership functions in the dictionary function permit to fuse or split membership functions quite easily so as to express the rules under a linguistically understandable and intuitive form for the human expert. Different techniques, generally referred by the term “fuzzy-wavelet”, have been developed for data on a regular grid. Fuzzy wavenets extend these techniques to online learning. A major advantage of fuzzy wavenets techniques in comparison to most neurofuzzy methods is that the rules are validated online during learning by using a simple algorithm based on the fast wavelet decomposition algorithm. Significant applications of wavelet networks and fuzzy wavenets are discussed to illustrate the potential of thcse methods.
70 citations
TL;DR: In this paper, wavelet characterizations for anisotropic Besov spaces with respect to Lp-spaces with 0
69 citations
TL;DR: A general way for the construction of quinc unx interpolatory refinement masks associated with the quincunx lattice in R2 attain the optimal approximation order and smoothness order.
Abstract: We analyze the approximation and smoothness properties of quincunx fundamental refinable functions. In particular, we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in R2. Their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order. In addition, these examples are minimally supported with symmetry. For two special families of such quincunx interpolatory masks, we prove that their symbols are nonnegative. Finally, a general way of constructing quincunx biorthogonal wavelets is presented. Several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed.
45 citations
TL;DR: In this paper, a symmetric C2 orthogonal multiwavelet with multiplicity 4 and vanishing moments of order 4 was constructed, and its scaling function vector was supported on [−1, 1] and W1.75833.
43 citations
TL;DR: In this paper, a general class of M-channel linear phase perfect reconstruction filter banks (FBs) with rational coefficients is introduced. But the lattice coefficients are parameterized as a series of zero-order lifting steps, providing fast, efficient, in-place computation of the subband coefficients.
Abstract: This paper introduces a general class of M-channel linear phase perfect reconstruction filter banks (FBs) with rational coefficients. A subset of the presented solutions has dyadic coefficients, leading to multiplierless implementations suitable for low-power mobile computing. All of these FBs are constructed from a lattice structure that is VLSI-friendly, employs the minimum number of delay elements, and robustly enforces both linear phase and perfect reconstruction property. The lattice coefficients are parameterized as a series of zero-order lifting steps, providing fast, efficient, in-place computation of the subband coefficients. Despite the tight rational or integer constraint, image coding experiments show that these novel FBs are very competitive with current popular transforms such as the 8/spl times/8 discrete cosine transform and the wavelet transform with 9/7-tap biorthogonal irrational-coefficient filters.
42 citations
TL;DR: This analysis reveals clear evidence of the absence of low-dimensional chaos, at least for the time scales that can be analyzed with this database, and suggests that the spatiotemporal irregularities observed in the solar cycle are due to the superposition of regular structures with a stochastic background.
Abstract: We present a biorthogonal decomposition of the temporal and latitudinal distribution of sunspots recorded since 1874. We show that the butterfly diagrams can be interpreted as the result of approximately constant amplitudes and phases of two oscillations with periods close to 22 years. Our analysis reveals clear evidence of the absence of low-dimensional chaos, at least for the time scales that can be analyzed with this database. This result suggests that the spatiotemporal irregularities observed in the solar cycle are due to the superposition of regular structures with a stochastic background.
40 citations
TL;DR: In this article, the expression of the large N Harish Chandra-Itzykson-Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy.
33 citations
Book•
01 Jan 2002
TL;DR: This paper presents a meta-analysis of the Biorthogonal Decomposition of Sobolev Spaces in the context of Turbulent Flows, which aims to clarify the role of the Hodge Decompositions in the design of wavelets.
Abstract: 1 Wavelet Bases.- 1.1 Wavelet Bases in L2(?).- 1.1.1 General Setting.- 1.1.2 Characterization of Sobolev-Spaces.- 1.1.3 Riesz Basis Property in L2(?).- 1.1.4 Norm Equivalences.- 1.1.5 General Setting Continued.- 1.1.6 Further Wavelet Features.- 1.1.7 A Program for Constructing Wavelets.- 1.2 Wavelets on the Real Line.- 1.2.1 Orthonormal Wavelets.- 1.2.2 Biorthogonal B-Spline Wavelets.- 1.2.3 Interpolatory Wavelets.- 1.3 Wavelets on the Interval.- 1.3.1 Boundary Scaling Functions.- 1.3.2 Biorthogonal Scaling Functions.- 1.3.3 Biorthogonalization.- 1.3.4 Refinement Matrices.- 1.3.5 Biorthogonal Wavelets on (0, 1).- 1.3.6 Quantitative Aspects of the Biorthogonalization.- 1.3.7 Boundary Conditions.- 1.3.8 Other Bases.- 1.4 Tensor Product Wavelets.- 1.5 Wavelets on General Domains.- 1.5.1 Domain Decomposition and Parametric Mappings.- 1.5.2 Multiresolution and Wavelets on the Subdomains.- 1.5.3 Multiresolution on the Global Domain ?.- 1.5.4 Wavelets on the Global Domain.- 1.5.5 Univariate Matched Wavelets and Other Functions.- 1.5.6 Bivariate Matched Wavelets.- 1.5.7 Trivariate Matched Wavelets.- 1.5.8 Characterization of Sobolev Spaces.- 1.6 Vector Wavelets.- 2 Wavelet Bases for H(div) and H(curl).- 2.1 Differentiation and Integration.- 2.1.1 Differentiation and Integration on the Real Line.- 2.1.2 Differentiation and Integration on (0, 1).- 2.1.3 Assumptions for General Domains.- 2.1.4 Norm Equivalences.- 2.2 The Spaces H(div) and H (curl).- 2.2.1 Stream Function Spaces.- 2.2.2 Flux Spaces.- 2.2.3 Hodge Decompositions.- 2.3 Wavelet Systems for H (curl).- 2.3.1 Wavelets in H0(curl ?).- 2.3.2 Curl-Free Wavelet Bases.- 2.4 Wavelet Bases for H(div).- 2.4.1 Wavelet Bases in H(div ?).- 2.4.2 Divergence-Free Wavelet Bases.- 2.5 Helmholtz and Hodge Decompositions.- 2.5.1 A Biorthogonal Helmholtz Decomposition.- 2.5.2 Interrelations and Hodge Decompositions.- 2.6 General Domains.- 2.6.1 Tensor Product Domains.- 2.6.2 Parametric Mappings.- 2.6.3 Fictitious Domain Method.- 2.7 Examples.- 3 Applications.- 3.1 Robust and Optimal Preconditioning.- 3.1.1 Wavelet-Galerkin Discretizations.- 3.1.2 The Lame Equations for Almost Incompressible Material.- 3.1.3 The Maxwell Equations.- 3.1.4 Preconditioning in H(div ?).- 3.2 Analysis and Simulation of Turbulent Flows.- 3.2.1 Numerical Simulation of Turbulence.- 3.2.2 Divergence-Free Wavelet Analysis of Turbulence.- 3.2.3 Proper Orthogonal Decomposition (POD).- 3.2.4 Numerical Implementation and Validation.- 3.2.5 Numerical Results I: Data Analysis.- 3.2.6 Numerical Results II: Complexity of Turbulent Flows.- 3.3 Hardening of an Elastoplastic Rod.- 3.3.1 The Physical Problem.- 3.3.2 Numerical Treatment.- 3.3.3 Stress Correction and Wavelet Bases.- 3.3.4 Numerical Results I: Variable Order Discretizations.- 3.3.5 Numerical Results II: Plastic Indicators.- References.- List of Figures.- List of Tables.- List of Symbols.
TL;DR: A formulation of the multiresolution time-domain (MRTD) algorithm using scaling and one-level wavelet basis functions, for orthonormal Daubechies and biorthogonal Cohen-Daubechie-Feauveau (CDF) wavelet families, and the issue of the analytic calculation of the MRTD coefficients is addressed.
Abstract: We present a formulation of the multiresolution time-domain (MRTD) algorithm using scaling and one-level wavelet basis functions, for orthonormal Daubechies and biorthogonal Cohen-Daubechies-Feauveau (CDF) wavelet families. We address the issue of the analytic calculation of the MRTD coefficients. This allows us to point out the similarities and the differences between the MRTD schemes based on the aforementioned wavelet systems and to compare their performances in terms of dispersion error and computational efficiency. The remainder of the paper is dedicated to the implementation of the CDF-MRTD method for scattering problems. We discuss the approximations made in implementing material inhomogeneities and validate the method by numerical examples.
TL;DR: It is shown how the sparsity of the density matrix is preserved for localized insulating systems as well as for itinerant metallic systems, which makes the multiresolution approach very attractable for linear- or close to linear-scaling electronic-structure calculations.
Abstract: A multiresolution density-matrix wavelet approach to electronic-structure calculations is proposed. A separable multidimensional biorthogonal interpolating multiwavelet and scaling representation of the Hamiltonian operator is introduced in which individual operator elements can be calculated locally at any scale. Issues regarding this representation are discussed, such as its low complexity in higher dimensions and its direct relation to finite-difference schemes. The density matrix is calculated via polynomial expansions in terms of the Hamiltonian. The expansions are improvements of the McWeeny purification within a grand canonical ensemble and are shown to be up to 20% more efficient in the number of matrix-matrix multiplications. The efficiency of the multiresolution representation of the density matrix compared to a real-space representation is analyzed. Within the multiresolution wavelet representation it is shown how the sparsity of the density matrix is preserved for localized insulating systems as well as for itinerant metallic systems. This is not possible within a real-space representation of the density matrix which has a very slow decay of its elements in the metallic phase. This makes the multiresolution approach very attractable for linear- or close to linear-scaling electronic-structure calculations.
TL;DR: This paper develops conditions for the existence of MIMO biorthogonal partners and conditions under which FIR solutions are possible and exploits the nonuniqueness of the solution, which will lead to the design of flexible fractionally spaced MIMo zero-forcing equalizers.
Abstract: Multiple input multiple output (MIMO) biorthogonal partners arise in many different contexts, one of them being multiwavelet theory. They also play a central role in the theory of MIMO channel equalization, especially with fractionally spaced equalizers. In this paper, we first derive some theoretical properties of MIMO biorthogonal partners. We develop conditions for the existence of MIMO biorthogonal partners and conditions under which FIR solutions are possible. In the process of constructing FIR MIMO biorthogonal partners, we exploit the nonuniqueness of the solution. This will lead to the design of flexible fractionally spaced MIMO zero-forcing equalizers. The additional flexibility in design makes these equalizers more robust to channel noise. Finally, other situations where MIMO biorthogonal partners occur are also considered, such as prefiltering in multiwavelet theory and deriving the vector version of the least squares signal projection problem.
TL;DR: A new wavelet method for the fast estimation of continuous Karhunen-Loeve eigenfunctions is developed by projecting the ensemble functions onto orthogonal or biorthogonal interpolating function spaces and the covariance matrix may be sparsified by a multiresolution decomposition.
Abstract: This paper develops a new wavelet method for the fast estimation of continuous Karhunen-Loeve eigenfunctions. The method of snapshots is modified by projecting the ensemble functions onto orthogonal or biorthogonal interpolating function spaces. Under well-behaved piecewise smooth polynomial ensemble functions, the size of the covariance matrix produced is greatly reduced, without sacrificing much accuracy. Moreover, the covariance matrix C/spl tilde/ may be easily decomposed such that C/spl tilde/ = A/sup T/ A, and thus, the more stable singular value decomposition (SVD) algorithm may be applied. An interpolating scheme that reduces the computation of projecting the ensemble functions onto the biorthogonal subspace to a single sample is also developed. Furthermore, by projecting the ensemble functions onto wavelet spaces, the covariance matrix may be sparsified by a multiresolution decomposition. Error bounds for the eigenvalues between the sparsified and nonsparsified covariance matrix are also derived.
TL;DR: In this article, the concept of projectable refinable functions was introduced and it was shown that many multivariate refinability functions are projectable, i.e., they essentially carry the tensor product structure of a function even though themselves may be nontensor product (nonseparable) refinability.
TL;DR: A biorthogonal Jacobi--Davidson method (biJD), which can be viewed as an explicitly biorThogonalized, restarted Lanczos method, that uses the approximate solution of a correction equation to expand its basis.
Abstract: We propose a biorthogonal Jacobi--Davidson method (biJD), which can be viewed as an explicitly biorthogonalized, restarted Lanczos method, that uses the approximate solution of a correction equation to expand its basis. Through an elegant formulation, the algorithm allows for all the functionalities and features of the Jacobi--Davidson method (JD), but it also includes some of the advantages of nonsymmetric Lanczos.
The main motivation for this work stems from a correction equation and a restarting scheme that are possible with biJD but not with JD. Specifically, a correction equation using the left approximate eigenvectors available in biJD yields cubic asymptotic convergence, as opposed to quadratic with the JD correction equation. In addition, a successful restarting scheme for symmetric JD depends on the Lanczos three-term recurrence and thus can only apply to biJD. Finally, methods that require a multiplication with the adjoint of the matrix need to be reconsidered on today's computers with memory hierarchies, as this multiplication can be performed with minimal additional cost.
We describe the algorithm, its features, and the possible functionalities. In addition, we develop an appropriate correction equation framework and analyze the effects of the new restarting scheme. Our numerical experiments confirm that biJD is a highly competitive method for a difficult problem.
TL;DR: Two types of vector wavelet transforms are described that are dual to each other but have similar properties and three pre(post)processing algorithms that do not degrade the approximation accuracy of the vector wavelets.
Abstract: We present new multiwavelet transforms of multiplicity 2 for manipulation of discrete-time signals. The transforms are implemented in two phases: (1) pre(post)-processing, which transforms the scalar signal into a vector signal (and back) and (2) wavelet transforms of the vector signal. Both phases are performed in a lifting manner. We use the cubic interpolatory Hermite splines as a predicting aggregate in the vector wavelet transform. We present new pre(post)-processing algorithms that do not degrade the approximation accuracy of the vector wavelet transforms. We describe two types of vector wavelet transforms that are dual to each other but have similar properties and three pre(post)processing algorithms. As a result, we get fast biorthogonal algorithms to transform discrete-time signals that are exact on sampled cubic polynomials. The bases for the transform are symmetric and have short support.
TL;DR: This paper proposes the use of wavelets for the identification of an unknown sparse system whose impulse response (IR) is rich in spectral content and uses biorthogonal wavelets which fulfil both of these two requirements to provide additional gain in performance.
Abstract: This paper proposes the use of wavelets for the identification of an unknown sparse system whose impulse response (IR) is rich in spectral content. The superior time localization property of wavelets allows for the identification and subsequent adaptation of only the nonzero IR regions, resulting in lower complexity and faster convergence speed. An added advantage of using wavelets is their ability to partially decorrelate the input, thereby further increasing convergence speed. Good time localization of nonzero IR regions requires high temporal resolution while good decorrelation of the input requires high spectral resolution. To this end we also propose the use of biorthogonal wavelets which fulfil both of these two requirements to provide additional gain in performance. The paper begins with the development of the wavelet-basis (WB) algorithm for sparse system identification. The WB algorithm uses the wavelet decomposition at a single scale to identify the nonzero IR regions and subsequently determines the wavelet coefficients of the unknown sparse system at other scale levels that require adaptation as well. A special implementation of the WB algorithm, the successive-selection wavelet-basis (SSWB), is then introduced to further improve performance when certain a priori knowledge of the sparse IR is available. The superior performance of the proposed methods is corroborated through simulations.
TL;DR: A stabilization of the two-step lifting construction of Sweldens for wavelet bases on irregular meshes on the interval yields biorthogonal bases in which the wavelets on both the primal and the dual side have a chosen number of vanishing moments and have local support.
Abstract: We propose a stabilization of the two-step lifting construction of Sweldens for wavelet bases on irregular meshes on the interval. The method yields biorthogonal bases in which the wavelets on both the primal and the dual side have a chosen number of vanishing moments and have local support. We combine it with a constrained local semiorthogonalization to obtain a stabilized variant.
Numerical results show that the wavelet bases are well-conditioned, having approximately the same condition numbers as wavelet bases obtained by global semiorthogonalization, while the average support is not much larger than in the unstabilized construction.
TL;DR: In this paper, the hyperbolic tensor product wavelet bases are used for approximating nonlinearly functions from anisotropic Besov spaces, which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.
Abstract: In L2(0, 1)2) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.
TL;DR: The ILT with 5-b integer coefficients are found to have similar coding gain and image coding performances as their real-valued counterparts, and multiplier-less lapped transforms with very low implementation complexity are obtained.
Abstract: This paper proposes new integer approximations of the lapped transforms, called the integer lapped transforms (ILT), and studies their applications to image coding. The ILT are derived from a set of orthogonal sinusoidal transforms having short integer coefficients, which can be implemented with simple integer arithmetic. By employing the same scaling constants in these integer sinusoidal transforms, integer versions of the lapped orthogonal transform (LOT), the lapped biorthogonal transform (LBT), and the hierarchical lapped biorthogonal transform (HLBT) are developed. The ILTs with 5-b integer coefficients are found to have similar coding gain (within 0.06 dB) and image coding performances as their real-valued counterparts. Furthermore, by representing these integer coefficients as sum of powers-of-two coefficients (SOPOT), multiplier-less lapped transforms with very low implementation complexity are obtained. In particular, the implementation of the eight-channel multiplier-less integer LOT (ILOT), LBT (ILBT), and HLBT (IHLBT) require 90 additions and 44 shifts, 98 additions and 59 shifts, and 70 additions and 38 shifts, respectively.
13 May 2002
TL;DR: The general lattice factorization of a polyphase matrix of a particular class of N-channel oversampled linear-phase perfect reconstruction filter banks with a decimation factor M smaller than N is provided.
Abstract: We investigate a special class of N-channel oversampled linear-phase perfect reconstruction filter banks with a decimation factor M smaller than N. We deal with systems in which all analysis and synthesis filters have the same FIR length and share the same center of symmetry. We provide the general lattice factorization of a polyphase matrix of a particular class of these oversampled filter banks. The lattice structure is based on the singular value decomposition for non-square matrices. 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 show that the present systems with the lattice structure cover a wide range of linear-phase perfect reconstruction filter banks. We also show several design examples.
17 Jun 2002
TL;DR: This work presents a biorthogonal wavelet construction based on Catmull-Clark-style subdivision volumes that provides the ability to reverse the subdivision process after a lattice has been re-shaped at a very fine level of detail, for example using an automatic fitting method.
Abstract: We present a biorthogonal wavelet construction based on Catmull-Clark-style subdivision volumes. Our wavelet transform is the three-dimensional extension of a previously developed construction of subdivision-surface wavelets that was used for multiresolution modeling of large-scale isosurfaces. Subdivision surfaces provide a flexible modeling tool for surfaces of arbitrary topology and for functions defined thereon. Wavelet representations add the ability to compactly represent large-scale geometries at multiple levels of detail. Our wavelet construction based on subdivision volumes extends these concepts to trivariate geometries, such as time-varying surfaces, free-form deformations, and solid models with non-uniform material properties. The domains of the repre-sented trivariate functions are defined by lattices composed of arbitrary polyhedral cells. These are recursively subdivided based on stationary rules converging to piecewise smooth limit-geometries. Sharp features and boundaries, defined by specific polygons, edges, and vertices of a lattice are explicitly represented using modified subdivision rules. Our wavelet transform provides the ability to reverse the subdivision process after a lattice has been re-shaped at a very fine level of detail, for example using an automatic fitting method. During this coarsening process all geometric detail is compactly stored in form of wavelet coefficients from which it can be reconstructed without loss.
Posted Content•
TL;DR: A backward biorthogonalization approach is proposed, which modifies biorhogonal functions so as to generate orthogonal projections onto a reduced subspace, 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.
TL;DR: It is shown that the optimal non-overlapping basis functions in the minimal mean square error sense can be found by solving an eigenvalue problem without numerical search when the feasible overlapping basis functions are given.
TL;DR: It is shown that these wavelets of variable polynomial order up to four dimensions have a much sparser representation of discontinuous functions as compared to tensor product wavelets, which allows for a more compact and efficient representation.
07 Aug 2002
TL;DR: In this article, a new family of lapped biorthogonal transform is introduced, called the IntLBT, which is implemented by a series of dyadic lifting steps and provides very fast, efficient in-place computation of the transform coefficients.
Abstract: By scaling all discrete cosine transform (DCT) intermediate output coefficients of the lapped transform and employing the type-II and type-IV DCT based on lifting steps, a new family of lapped biorthogonal transform is introduced, called the IntLBT. When all the elements with a floating point of each lifting matrix in the IntLBT are approximated by binary fractions, the IntLBT is implemented by a series of dyadic lifting steps and provides very fast, efficient in-place computation of the transform coefficients, and all internal nodes have finite precision. When each lifting step in the IntLBT is implemented using the same nonlinear operations as those used in the well known integer-to-integer wavelet transform, the IntLBT maps integers to integers, so it can express lossless image information. As an application of the novel IntLBT to lossy image compression, simulation results demonstrate that the IntLBT has significantly less blocking artefacts, higher peak signal-to-noise ratio, and better visual quality than the DCT. More importantly, the IntLBT's coding performance is approximately the same as that of the much more complex Cohen-Daubechies-Feauveau (CDF) 9/7-tap biorthogonal wavelet with floating-point coefficients, and in some cases even surpasses that of the CDF 9/7-tap biorthogonal wavelet.
07 Nov 2002
TL;DR: A new system of multicarrier modulation is described, which takes advantage of the biorthogonality property, to generalize the oversampled OFDM technique into a bIORthogonal frequency division multiplex scheme named BFDM/QAM.
Abstract: A new system of multicarrier modulation is described, which takes advantage of the biorthogonality property, to generalize the oversampled OFDM technique into a biorthogonal frequency division multiplex scheme named BFDM/QAM. 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 filters, allowing an exact cancellation of intersymbol and interchannel interference, i.e. a perfect reconstruction. Then, computationally efficient realizations of the modulators and demodulators are given. Lastly, some design examples are provided.
TL;DR: In this paper, it was shown that a biorthogonal wavelet basis with weak decay conditions yields an unconditional basis in all spaces L p (R d ) with 1 < p < 1, provided that the wavelet set functions satisfy weak decay condition.
Abstract: We prove that a biorthogonal wavelet basis yields an unconditional basis in all spacesL p (R d ) with 1< p <1, provided the biorthogonal wavelet set functions satisfy weak decay conditions. The biorthogonal wavelet set is associated with an arbitrary dilation matrix in any dimension.