Showing papers on "Biorthogonal system published in 2003"

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem

Abstract: We consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V1(x), V2(y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (‘‘windows’’), of lengths equal to the degrees of the potentials V1 and V2, satisfy systems of ODE’s with polynomial coefficients as well as PDE’s (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.

Wavelets in Electromagnetics and Device Modeling

Abstract: Preface. Notations and Mathematical Preliminaries. Intuitive Introduction to Wavelets. Basic Orthogonal Wavelet Theory. Wavelets in Boundary Integral Equations. Sampling Biorthogonal Time Domain Method (SBTD). Canonical Multiwavelets. Wavelets in Scattering and Radiation. Wavelets in Rough Surface Scattering. Wavelets in Packaging, Interconnects, and EMC. Wavelets in Nonlinear Semiconductor Devices. Index.

A class of regular biorthogonal linear-phase filterbanks: theory, structure, and application in image coding

Abstract: This paper discusses a method of regularity imposition onto biorthogonal linear-phase M-band filterbanks using the lattice structure. A lifting structure is proposed for lattice matrix parameterization where regularity constraints can be imposed. The paper focuses on cases with analysis and synthesis filterbanks having up to two degrees of regularity. Necessary and sufficient conditions for regular filterbanks in terms of the filter impulse response, frequency response, scaling function, and wavelets are revisited and are derived in terms of the lattice matrices. This also leads to a constraint on the minimum filter length. Presented design examples are optimized for the purpose of image coding, i.e., the main objectives are coding gain and frequency selectivity. Simulation results from an image coding application also show that these transforms yield improvement in the perceptual quality in the reconstruction images. The approach has also been extended to the case of integer/rational lifting coefficients, which are desirable in many practical applications.

A Riemann-Hilbert problem for biorthogonal polynomials

Abstract: We characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a Riemann-Hilbert problem. Our Riemann-Hilbert problem is different from the ones that were proposed recently by Ercolani and McLaughlin, Kapaev, and Bertola et al. We believe that our formulation may be tractable to asymptotic analysis.

Locally Supported, Piecewise Polynomial Biorthogonal Wavelets on Nonuniform Meshes

Abstract: In this paper, biorthogonal wavelets are constructed on nonuniform meshes. Both primal and dual wavelets are locally supported, continuous piecewise polynomials. The wavelets generate Riesz bases for the Sobolev spaces (H s ) for (|s| < 3/2). The wavelets at the primal side span standard Lagrange finite element spaces.

Orthogonal vs. Biorthogonal Wavelets for Image Compression

Abstract: Effective image compression requires a non-expansive discrete wavelet transform (DWT) be employed; consequently, image border extension is a critical issue. Ideally, the image border extension method should not introduce distortion under compression. It has been shown in literature that symmetric extension performs better than periodic extension. However, the non-expansive, symmetric extension using fast Fourier transform and circular convolution DWT methods require symmetric filters. This precludes orthogonal wavelets for image compression since they cannot simultaneously possess the desirable properties of orthogonality and symmetry. Thus, biorthogonal wavelets have been the de facto standard for image compression applications. The viability of symmetric extension with biorthogonal wavelets is the primary reason cited for their superior performance. Recent matrix-based techniques for computing a non-expansive DWT have suggested the possibility of implementing symmetric extension with orthogonal wavelets. For the first time, this thesis analyzes and compares orthogonal and biorthogonal wavelets with symmetric extension. Our results indicate a significant performance improvement for orthogonal wavelets when they employ symmetric extension. Furthermore, our analysis also identifies that linear (or near-linear) phase filters are critical to compression performance—an issue that has not been recognized to date. We also demonstrate that biorthogonal and orthogonal wavelets generate similar compression performance when they have similar filter properties and both employ symmetric extension. The biorthogonal wavelets indicate a slight performance advantage for low frequency images ; however, this advantage is significantly smaller than recently published results and is explained in terms of wavelet properties not previously considered. Acknowledgments I express my sincere gratitude to my advisor Dr. Amy Bell for her technical and financial support which made this thesis possible. Her constant encouragement, suggestions and ideas have been invaluable to this work. I immensely appreciate the time she devoted reviewing my writing and vastly improving my technical writing skills. Her thoroughness, discipline and work ethic are laudable and worthy of emulation. I would like to thank Dr. Brian Woerner and Dr. Lynn Abbott for reviewing my work and agreeing to serve on my committee. I am also grateful to Dr. Karen Duca in VBI for her financial support and the opportunity to work on some interesting biomedical signal processing problems. I am thankful to my fellow DSPCL colleagues Kishore Kotteri and Krishnaraj Varma for their technical help and insightful suggestions that went a long way in shaping this thesis. My interactions with them greatly improved my technical knowledge and research skills. I am also …

Three-dimensional biorthogonal multiresolution time-domain method and its application to electromagnetic scattering problems

Abstract: A three-dimensional (3-D) multiresolution time-domain (MRTD) analysis is presented based on a biorthogonal-wavelet expansion, with application to electromagnetic-scattering problems. We employ the Cohen-Daubechies-Feauveau (CDF) biorthogonal wavelet basis, characterized by the maximum number of vanishing moments for a given support. We utilize wavelets and scaling functions of compact support, yielding update equations involving a small number of proximate field components. A detailed analysis is presented on algorithm implementation, with example numerical results compared to data computed via the conventional finite-difference time-domain (FDTD) method. It is demonstrated that for 3-D scattering problems the CDF-based MRTD often provides significant computational savings (in computer memory and run time) relative to FDTD, while retaining numerical accuracy.

Biorthogonal systems in Banach spaces

Abstract: We give biorthogonal system characterizations of Banach spaces that fail the Dunford-Pettis property, contain an isomorphic copy of $c_0$, 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.

New analytical probability of error expressions for classes of orthogonal signals in Rayleigh fading

Abstract: New exact expressions for the symbol-error rate and bit-error rate of coherent 3-ary and 4-ary orthogonal and transorthogonal signaling in slowly fading Rayleigh channels are derived. New exact error probability expressions for coherent 6-ary and 8-ary biorthogonal signaling in slow Rayleigh fading are also presented. The use of these exact expressions as accurate approximations for the error rates of M-ary orthogonal, biorthogonal, and transorthogonal signaling with arbitrary M is illustrated.

UWB multiple access performance using time hopped pulse position modulation with biorthogonal signaling

Abstract: This paper extends previous UMB multiple access (MA) performance characterization by combining pseudorandomly coded, time hopped pulse position modulation (TH-PPM) with 4-ary biorthogonal communication signaling. Communication performance for the hybrid TH-BPPM technique is first validated for a single user operating over an AWGN channel. Single user results are then extended and MA performance characterized for both synchronous and asynchronous networks containing up to 15 users. In both cases, the proposed 4-ary TH-BPPM technique with Gold coding achieves nearly identical performance as previously demonstrated MA methods employing binary TH-PPM. A key benefit afforded by the 4-ary TH-BPPM technique, given fixed average power and symbol length constraints, is a doubling of the effective data rate while achieving identical bit error rate performance.

Quasi-Biorthogonal Frame Multiresolution Analyses and Wavelets

Abstract: We introduce the concepts of quasi-biorthogonal frame multiresolution analyses and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal multiresolution analyses and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal frame multiresolution analyses to admit quasi-biorthogonal wavelet frames are given, and a non-trivial example of quasi-biorthogonal frame multiresolution analyses admitting quasi-biorthogonal frame wavelets is constructed. Finally, we characterize the pair of quasi-biorthogonal frame wavelets that is associated with quasi-biorthogonal frame multiresolution analyses.

A study of biorthogonal wavelets in digital watermarking

Abstract: A study of a family of biorthogonal wavelet filters for use in digital watermarking is presented. The filters are explicitly parametrized by two free parameters and can be used to provide diversity in watermarking. Diversity can be used to improve the security of the watermarking system from hostile attacks. Each filter has at least two vanishing moments, which is important for ensuring some degree of smoothness in the resulting wavelet function. Along with robustness and security, other factors, which impact upon the successful implementation of the filters in a watermarking application are analysed. The relationship between the strength of the inserted watermark and wavelet energy is also discussed.

Frame analysis for biorthogonal cosine-modulated filterbanks

Abstract: This paper addresses the efficient computation of frame bounds for cosine-modulated filterbanks. We derive explicit expressions for the eigenvalues of the frame operator that can be easily computed from the prototype's polyphase components. The number of channels and the downsampling factor may be even or odd, and the oversampling factor is supposed to be an integer. The analysis of low-delay, biorthogonal filterbanks; shows that prototypes solely designed to minimize the stopband energy may lead to wide open frames and, thus, to an undesirable numerical behavior. Because the computational cost of determining the frame bounds with the proposed method is very low, we can directly use the bounds during prototype optimization and obtain prototypes with minimum stopband energy under the condition of fixed frame bounds. Various design examples are presented.

Analysis of the CDF biorthogonal MRTD method with application to PEC targets

Abstract: We consider the biorthogonal Cohen-Daubechies- Feauveau (CDF) wavelet family in the context of a biorthogonal multiresolution time-domain (bi-MRTD) analysis. A disadvantage of previous bi-MRTD analyses is an inability to handle abrupt changes in material properties, particularly for a perfect electric conductor (PEC). A multiregion method is proposed to address PEC targets. The proposed method is based on the fact that the CDF bi-MRTD may be viewed as a linear combination of several conventional finite-difference time-domain (FDTD) solutions. The implementation of the connecting surface is also simplified. Several numerical results are presented, with comparison to analytic and FDTD results.

Fractional biorthogonal partners in channel equalization and signal interpolation

Abstract: The concept of biorthogonal partners has been introduced recently by the authors. The work presented here is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers, hence, the name fractional biorthogonal partners. The conditions for the existence of stable and of finite impulse response (FIR) fractional biorthogonal partners are derived. It is also shown that the FIR solutions (when they exist) are not unique. This property is further explored in one of the applications of fractional biorthogonal partners, namely, the fractionally spaced equalization in digital communications. The goal is to construct zero-forcing equalizers (ZFEs) that also combat the channel noise. The performance of these equalizers is assessed through computer simulations. Another application considered is the all-FIR interpolation technique with the minimum amount of oversampling required in the input signal. We also consider the extension of the least squares approximation problem to the setting of fractional biorthogonal partners.

Method of Generalized Moment Representations in the Theory of Rational Approximation (A Survey)

Abstract: We give a survey of the method of generalized moment representations introduced by Dzyadyk in 1981 and its applications to Pade approximations. In particular, some properties of biorthogonal polynomials are investigated and numerous important examples are given. We also consider applications of this method to joint Pade approximations, Pade–Chebyshev approximations, Hermite–Pade approximations, and two-point Pade approximations.

Method for compressing digital images

Abstract: The invention concerns a method for compressing data, in particular images, by transform, in which method this data is projected onto a base of localized orthogonal or biorthogonal functions, such as wavelets To quantize each of the localized functions with a quantization step that enables an overall set rate Rc to be satisfied, the method includes the following steps: a probability density model of coefficients in the form of a generalized Gaussian is associated with each subband, the parameters α and β of this density model are estimated, while minimizing the relative entropy, or Kullback-Leibler distance, between this model and the empirical distribution of coefficients of each subband, and from this model, for each subband, an optimum quantization step is determined such that the rate allocated is distributed in the various subbands and such that the total distortion is minimal As a preference, for each subband, the graphs of rate R and distortion D are deduced, from the parameters α and β, as a function of the quantization step and these graphs are tabulated to determine said optimum quantization step

Three-band biorthogonal interpolating complex wavelets with stopband suppression via lifting scheme

Abstract: Wavelet research has primarily focused on real-valued wavelet bases. However, the complex filterbanks provide much convenience for complex signal processing. For example, in radar and sonar signal processing, the complex signals from the I/Q receiver can be efficiently processed with complex filterbanks rather than real filterbanks. Specifically, the positive and negative Doppler frequencies imply different physical content in the moving target detector (MTD) and moving target identification (MTI); therefore, it is significant to design complex multiband filterbanks that can partition positive and negative frequencies into different subbands. We design two novel families of three-band biorthogonal interpolating complex filterbanks and wavelets by using the three-band lifting scheme. Unlike the traditional three-band filterbanks, the novel complex filterbank is composed of three channels, including the lowpass channel, the positive highpass channel whose passband distributes in the positive frequency region, and the negative highpass channel in the negative frequency region. Such a filterbank/wavelet naturally provides the ability to extract positive frequency components and negative frequency components from complex signals. Moreover, a novel set of design constraints are introduced to manipulate the stopband characteristic of highpass filters and are referred to as stopband suppression, which strengthens the traditional constraints of vanishing moments. Finally, a numerical method is given to further lower stopband sidelobes.

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

Abstract: Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson's biorthogonal 10-W-9 functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6j-symbols in terms of Sklyanin algebra representations.

Efficient direct 2D architecture for lifted biorthogonal DWT

Abstract: The paper presents a new algorithm for 2D nonseparable lifted biorthogonal wavelet transform. The algorithm is derived by factoring complementary pairs of wavelet transform filters written as (L/spl times/M) tap 2D filters. The results are efficient architectures for real time signal processing, which do not require transpose memory for 2D processing of data. The proposed architecture exploits the in-place implementation inherited from the algorithm and can take advantage of both vertical and horizontal parallelism in the direct implementation. Processing in the architecture is scheduled carefully by pipelining the lifted steps, which allows two or four times faster processing than the direct implementation. The architecture therefore allows lowering of the clock frequency by two/four. The proposed architecture operates at high speed, consumes low power and has reduced computational complexity as compared to already published filter and lifting-based biorthogonal wavelet architectures.

Investigation of boundary algorithms for multiresolution analysis

Abstract: An investigation on the multiresolution time-domain (MRTD) method utilizing different wavelet levels in one mesh is presented. Contrary to adaptive thresholding techniques, only a rigid addition of higher order wavelets in certain critical cells is considered. Their effect is discussed analytically and verified by simulations of plain and dielectrically filled cavities with Daubechies' and Battle-Lemarie orthogonal, as well as Cohen-Daubechies-Feauveau (CDF) biorthogonal wavelets, showing their insufficiency unless used as a full set of expansion. It is pointed out that improvements cannot be expected from these fixed mesh refinements. Furthermore, an advanced treatment concerning thin metallization layers in CDF algorithms is presented, leading to a reduction in cell number by a factor of three per space dimension compared to conventional finite difference time domain (FDTD), but limited to very special structures with infinitely thin irises. All MRTD results are compared to those of conventional FDTD approaches.

Fast signal transforms with lifting steps

Abstract: This invention introduces a class of multi-band linear phase lapped biorthogonal transforms with fast, VLSI-friendly implementations via lifting steps called the LiftLT. The transform is based on a lattice structure which 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 as well as the ability to map integers to integers. Our main motivation of the new transform is its application in image and video coding. Comparing to the popular 8×8 DCT, the 8×16 LiftLT only requires 1 more multiplication, 22 more additions, and 6 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 LiftLT's coding performance consistently surpasses that of the much more complex 9/7-tap biorthogonal wavelet with floating-point coefficients. More importantly, our transform's block-based nature facilitates one-pass sequential block coding, region-of-interest coding/decoding as well as parallel processing.

Filter evaluation for DWT-domain image watermarking

Abstract: Similar to other wavelet-based image processing algorithms, the choice of wavelet filters generally affects the performance of a wavelet-based watermarking system. Reported is an evaluation of a set of biorthogonal integer wavelets under a multiresolution-watermarking framework. Further investigation is conducted to justify the robustness performance against attacks.

Uncorrelatedness and orthogonality for vector-valued processes

Abstract: For a square integrable vector-valued process f on the Loeb product space, it is shown that vector orthogonality is almost equivalent to componentwise scalar orthogonality. Various characterizations of almost sure uncorrelatedness for f are presented. The process f is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for f involving the biorthogonal representation for the conditional expectation of f with respect to the usual product σ-algebra is presented.

Multiwavelets and Integer Transforms

Abstract: In many applications of image processing, the given data are integer-valued. It is therefore desirable to construct transformations that map data of this type to an integer (or rational) ring. Calderbank, Daubechies, Sweldens, and Yeo [1] devised two methods for modifying orthogonal and biorthogonal wavelets so that they map integers to integers. The first method involves appropriately scaling the transform so that data that has been transformed and truncated can be recovered via the inverse wavelet transform. In developing this method, the authors of [1] created a useful factorization of the 4-tap Daubechies orthogonal wavelet transform [2]. We have observed that this factorization can be extended to 4-tap multiwavelets of arbitrary size. In this paper we will discuss this generalization and illustrate the factorization on two multiwavelets. In particular, the well-known Donovan, Geronimo, Hardin, and Massopust (DGHM) [3] multiwavelet transform can be scaled so that it maps integers to integers. Since this transform is (anti)symmetric in addition to orthogonal, regular, and compactly supported, the ability to modify it so that it maps integers to integers should be useful in image processing applications.