Showing papers on "Biorthogonal system published in 2007"

Biorthogonal quantum systems

Abstract: Models of PT symmetric quantum mechanics provide examples of biorthogonal quantum systems. The latter incorporate all the structure of PT symmetric models, and allow for generalizations, especially in situations where the PT construction of the dual space fails. The formalism is illustrated by a few exact results for models of the form H=(p+ν)2+∑k>0μkexp(ikx). In some nontrivial cases, equivalent Hermitian theories are obtained and shown to be very simple: They are just free (chiral) particles. Field theory extensions are briefly considered.

Biorthogonal bases with local support and approximation properties

Abstract: We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p - 1. Working with Gaus-Lobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are two-dimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.

Biorthogonal wavelet trees in the classification of embedded signal classes for intelligent sensors using machine learning applications

Abstract: The paper deals with a method of constructing orthonormal bases of coordinates which maximize, through redundant dictionaries (frames) of biorthogonal bases, a class separability index or distances among classes. The method proposes an algorithm which consists of biorthogonal expansions over two redundant dictionaries. Embedded classes are often present in multiclassification problems. It is shown how the biorthogonality of the expansion can really help to construct a coordinate system which characterizes the classes. The algorithm is created for training wavelet networks in order to provide an efficient coordinate system maximizing the Cross Entropy function between two complementary classes. Sine and cosine wavelet packets are basis functions of the network. Thanks to their packet structure, once selected the depth of the tree, an adaptive number of basis functions is automatically chosen. The algorithm is also able to carry out centering and dilation of the basis functions in an adaptive way. The algorithm works with a preliminary extracted feature through shrinkage technique in order to reduce the dimensionality of the problem. In particular, our attention is pointed out for time-frequency monitoring, detection and classification of transients in rail vehicle systems and the outlier problem. In the former case the goal is to distinguish transients as inrush current and no inrush current and a further distinction between the two complementary classes: dangerous inrush current and no dangerous inrush current. The proposed algorithm is used on line in order to recognize the dangerous transients in real time and thus shut-down the vehicle. The algorithm can also be used in a general application of the outlier detection. A similar structure is used in developed algorithms which are currently integrated in the inferential modeling platform of the unit responsible for Advanced Control and Simulation Solutions within ABB's (Asea Brown Boveri) industry division. It is shown how impressive and rapid performances are achieved with a limited number of wavelets and few iterations. Real applications using real measured data are included to illustrate and analyze the effectiveness of the proposed method.

General biorthogonal projected bases as applied to second-order Møller-Plesset perturbation theory.

Abstract: With low-order scaling correlated wave function theories in mind, we present second quantization formalism as well as biorthonormalization procedures for general—singular or nonsingular—bases Of particular interest are the so-called projected atomic orbital bases, which are obtained from a set of atom-centered functions and feature a separation of occupied and virtual spaces We demonstrate the formalism by deriving and implementing second-order Moller-Plesset perturbation theory in it, and discuss the convergence and preconditioning of the iterative amplitude equations in detail

√3-Subdivision-Based Biorthogonal Wavelets

Abstract: A new efficient biorthogonal wavelet analysis based on the radic3 subdivision is proposed in the paper by using the lifting scheme. Since the radic3 subdivision is of the slowest topological refinement among the traditional triangular subdivisions, the multiresolution analysis based on the radic3 subdivision is more balanced than the existing wavelet analyses on triangular meshes and accordingly offers more levels of detail for processing polygonal models. In order to optimize the multiresolution analysis, the new wavelets, no matter whether they are interior or on boundaries, are orthogonalized with the local scaling functions based on a discrete inner product with subdivision masks. Because the wavelet analysis and synthesis algorithms are actually composed of a series of local lifting operations, they can be performed in linear time. The experiments demonstrate the efficiency and stability of the wavelet analysis for both closed and open triangular meshes with radic3 subdivision connectivity. The radic3-subdivision-based biorthogonal wavelets can be used in many applications such as progressive transmission, shape approximation, and multiresolution editing and rendering of 3D geometric models.

Fast lapped image transforms using 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.times.8 DCT, the 8.times.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.

Comparative Analysis of Shift Variance and Cyclostationarity in Multirate Filter Banks

Abstract: Multirate filter banks introduce periodic time-varying phenomena into their subband signals. The nature of these effects depends on whether the signals are regarded as deterministic or as random signals. We analyze the behavior of deterministic and wide-sense stationary (WSS) random signals in multirate filter banks in a comparative manner. While aliasing in the decimation stage causes subband energy spectra of deterministic signals to become shift-variant, imaging in the interpolation stage causes WSS random signals to become WS cyclostationary (WSCS). We provide criteria to quantify both shift variance and cyclic nonstationarity. For shift variance, these criteria separately assess the shift dependence of energy and of energy spectra. Similarly for nonstationarity, they separately assess the nonstationary behavior of signal power and of power spectra. We show that, under aliasing cancellation and perfect reconstruction constraints of paraunitary and biorthogonal filter banks, these criteria evaluate the behavior of deterministic and WSS random signals in a consistent, dual way. We apply our criteria to paraunitary and biorthogonal filter banks as well as to orthogonal block transforms, and show that, for critical signals such as lines or edges in image data, the biorthogonal 9/7 filters perform best among these

Composite Wavelet Bases with Extended Stability and Cancellation Properties

Abstract: The efficient solution of operator equations using wavelets requires that they generate a Riesz basis for the underlying Sobolev space and that they have cancellation properties of a sufficiently high order. Suitable biorthogonal wavelets were constructed on reference domains as the $n$-cube. Via a domain decomposition approach, these bases have been used as building blocks to construct biorthogonal wavelets on general domains or manifolds, where, in order to end up with local wavelets, biorthogonality was realized with respect to a modified $L_2$-scalar product. The use of this modified scalar product restricts the application of these so-called composite wavelets to problems of orders strictly larger than $-1$. Moreover, those wavelets with supports that extend to more than one patch generally have no cancellation properties. In this paper, we construct local, composite wavelets that are close to being biorthogonal with respect to the standard $L_2$-scalar product. As a consequence, they generate Riesz bases for the Sobolev spaces $H^s$ for the full range of $s$ allowed by the continuous gluing of functions over the patch interfaces, the properties of the primal and dual approximation spaces on the reference domain, and, in the manifold case, by the regularity of the manifold. Moreover, all these wavelets have cancellation properties of the full order induced by the approximation properties of the dual spaces on the reference domain. We illustrate our findings by a concrete realization of wavelets on a perturbed sphere.

Sampling theorem, bandlimited integral kernels and inverse problems

Abstract: We describe a novel approach based on the sampling theorem for studying eigenvalue problems associated with bandlimited integral kernels of convolution type. Two sets of functions biorthogonal to the eigenfunctions, one over the infinite interval and the other over a finite interval, are constructed and several identities satisfied by them are derived. The sampling theorem-based approach to the eigenvalue problem is further extended to construct the singular functions associated with the integral operator. It is shown that for the special case of the sinc-kernel, the eigenfunctions, the two biorthogonal sets and the singular functions reduce to the angular prolate spheroidal functions (or Slepian functions). Two methods are discussed for treating the inverse problem associated with bandlimited kernels—one employing the eigenfunctions and the biorthogonal sets and the other employing the singular functions. Numerical examples are included to illustrate the computation of eigenfunctions, biorthogonal sets and the singular functions and their application to the estimation of inverse solution.

Efficient Design of Cosine Modulated Filter Banks Based on Gradient Information

Abstract: An efficient iterative algorithm is first proposed for the design of biorthogonal cosine modulated filter banks (CMFBs) with nearly perfect reconstruction. The design problem is formulated as a quadratic unconstrained least-squares minimization problem, in which the gradient vector of the objective function with respective to unknown parameters can be obtained analytically. The algorithm is then modified for the design of orthogonal CMFBs. Design examples and comparisons are included that show that the proposed design method leads to filter banks with improved performance.

Peakons and Cauchy Biorthogonal Polynomials

Abstract: The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.

Supersymmetric biorthogonal quantum systems

Abstract: We discuss supersymmetric biorthogonal systems, with emphasis given to the periodic solutions that occur at spectral singularities of PT symmetric models. For these periodic solutions, the dual functions are associated polynomials that obey inhomogeneous equations. We construct in detail some explicit examples for the supersymmetric pairs of potentials V±(z)=−U(z)2±z(d∕dz)U(z) where U(z)≡∑k>0υkzk. In particular, we consider the cases generated by U(z)=z and z∕(1−z). We also briefly consider the effects of magnetic vector potentials on the partition functions of these systems.

Selection of Mother Wavelet for Image Compression on Basis of Image

Abstract: Recently discrete wavelet transform and wavelet packet has emerged as popular techniques for image compression. This paper compares compression performance of Daubechies, Biorthogonal, Coiflets and other wavelets along with results for different frequency images. Based on the result, we propose that proper selection of mother wavelet on the basis of nature of images and improve the quality and compression ratio remarkably

An Intelligent Ballistocardiographic Chair using a Novel SF-ART Neural Network and Biorthogonal Wavelets

Abstract: This paper presents a comparative analysis of novel supervised fuzzy adaptive resonance theory (SF-ART), multilayer perceptron (MLP) and Multi Layer Perceptrons (MLP) neural networks over Ballistocardiogram (BCG) signal recognition. To extract essential features of the BCG signal, we applied Biorthogonal wavelets. SF-ART performs classification on two levels. At first level, pre-classifier which is self-organized fuzzy ART tuned for fast learning classifies the input data roughly to arbitrary (M) classes. At the second level, post-classification level, a special array called Affine Look-up Table (ALT) with M elements stores the labels of corresponding input samples in the address equal to the index of fuzzy ART winner. However, in running (testing) mode, the content of an ALT cell with address equal to the index of fuzzy ART winner output will be read. The read value declares the final class that input data belongs to. In this paper, we used two well-known patterns (IRIS and Vowel data) and a medical application (Ballistocardiogram data) to evaluate and check SF-ART stability, reliability, learning speed and computational load. Initial tests with BCG from six subjects (both healthy and unhealthy people) indicate that the SF-ART is capable to perform with a high classification performance, high learning speed (elapsed time for learning around half second), and very low computational load compared to the well-known neural networks such as MLP which needs minutes to learn the training material. Moreover, to extract essential features of the BCG signal, we applied Biorthogonal wavelets. The applied wavelet transform requires no prior knowledge of the statistical distribution of data samples.

Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions

Abstract: This article presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotationinvariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from classical tensor product wavelet theory.

Transmitter, receiver and its method for FDMA

Abstract: The invention is based on biorthogonal multi-subband filterbanks. The transmitting device converts the sending sequence symbols to be transmitted into multi serial sequence symbols by using series/parallel conversion; the biorthogonal multi-subband filterbanks modulates each serial sequence symbol to the subcarrier corresponding to different subband; partitioning the multicarrier signals outputted from the biorthogonal multi-subband filterbanks; adding the 'cycle prefix' into each data block that is transmitted to the RF; the receiving device receives the signal block and removes 'cycle prefix', and then makes frequency-domain equalization for each data block; the biorthogonal multi-subband filterbanks restores the transmitting signal from the data block.

Biorthogonal multiple wavelets generated by vector refinement equation

Abstract: Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis. In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (Mx - \alpha ), x \in \mathbb{R}^s } ,$$ where the vector of functions ϕ = (ϕ 1, …, ϕ r)T is in $$(L_2 (\mathbb{R}^s ))^r ,a = :(a(\alpha ))_{\alpha \in \mathbb{Z}^s } $$ is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M −n = 0. Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.

Transmit/receive pulse-shaping design in BFDM systems over time-frequency dispersive AWGN channel

Abstract: In this paper, we propose an optimal biorthogonal pulses design for a biorthogonal frequency division multiplexing (BFDM) systems operating on a time-frequency dispersive additive white Gaussian noise (AWGN) channel. The proposed transmit and receive pulse shapes which are expressed as a linear combination of the most localized Hermite waveforms and are therefore well time-frequency localized. We propose a method for designing biorthogonal pulse shapes that maximize the signal-to-interference-and-noise ratio (SINR). We demonstrate that the SINR of the BFDM systems operating over dispersive channels with additive white Gaussian noise converges to the value of signal-to-noise ratio (SNR).

2D Dual-Tree Complex Biorthogonal M-Band Wavelet Transform

Abstract: Dual-tree wavelet transforms have recently gained popularity since they provide low-redundancy directional analyses of images. In our recent work, dyadic real dual-tree decompositions have been extended to the M-band case, so adding much flexibility to this analysis tool. In this work, we propose to further extend this framework on two fronts by considering (i) biorthogonal and (ii) complex M-band dual-tree decompositions. Denoising results are finally provided to demonstrate the validity of the proposed design rules.

Obtaining Patterns for Classification of Power Quality Disturbances Using Biorthogonal Wavelets, RMS value and Support Vector Machines

Abstract: This paper proposes three strategies in order to obtain patterns that allow identification of power quality disturbances. These strategies use Discrete Wavelet Transform (using biorthogonal Wavelet) and RMS value. Disturbances under survey are: low frequency disturbances (such as flicker and harmonics) and high frequency disturbances (such as transient and sags). Due to time-frequency localization properties, Discrete Wavelet Transform permits decomposition of signals in different energy levels, which are used to characterize disturbances that contain information in frequency domain. Four wavelet families were studied and Biorthogonal showed excellent performance. Also, RMS value is used to characterize those disturbances that show big changes in magnitude. The combination of both strategies produces excellent results. Patterns are automatically classified by support vector machines (SVM). Thus, Radial Base Function (RBF) was used as kernel, because RBF requires only two parameters ( s and C) . Cross validation technique and grid search were used in this work. SVM exhibit a good performance as classifier despite similitude between some disturbance patterns

Obtaining patterns for classification of power quality disturbances using biorthogonal wavelets, RMS value and support vector machines

Abstract: This paper proposes three strategies in order to obtain patterns that allow identification of power quality disturbances. These strategies use Discrete Wavelet Transform (using biorthogonal Wavelet) and RMS value. Disturbances under survey are: low frequency disturbances (such as flicker and harmonics) and high frequency disturbances (such as transient and sags). Due to time-frequency localization properties, Discrete Wavelet Transform permits decomposition of signals in different energy levels, which are used to characterize disturbances that contain information in frequency domain. Four wavelet families were studied and Biorthogonal showed excellent performance. Also, RMS value is used to characterize those disturbances that show big changes in magnitude. The combination of both strategies produces excellent results. Patterns are automatically classified by support vector machines (SVM). Thus, Radial Base Function (RBF) was used as kernel, because RBF requires only two parameters (sigma and C). Cross validation technique and grid search were used in this work. SVM exhibit a good performance as classifier despite similitude between some disturbance patterns.

A Combinatorial Representation with Schröder Paths of Biorthogonality of Laurent Biorthogonal Polynomials

Abstract: Combinatorial representation in terms of Schroder paths and other weighted plane paths are given of Laurent biorthogonal polynomials (LBPs) and a linear functional with which LBPs have orthogonality and biorthogonality. Particularly, it is clarified that quantities to which LBPs are mapped by the corresponding linear functional can be evaluated by enumerating certain kinds of Schroder paths, which imply orthogonality and biorthogonality of LBPs.

Biorthogonal exponential sequences with weight function exp(²+) on the real line and an orthogonal sequence of trigonometric functions

Abstract: Some orthogonal functions can be mapped onto other orthogonal functions by the Fourier transform. In this paper, by using the Fourier transform of Stieltjes-Wigert polynomials, we derive a sequence of exponential functions that are biorthogonal with respect to a complex weight function like exp(q1(ix + p 1 ) 2 + q 2 (ix + p 2 ) 2 ) on (-∞,∞). Then we restrict these introduced biorthogonal functions to a special case to obtain a sequence of trigonometric functions orthogonal with respect to the real weight function exp(-qx 2 ) on (-∞, ∞).

A New Bidirectionally Motion-Compensated Orthogonal Transform for Video Coding

Abstract: Motion-compensated lifted wavelets have received much interest for video compression. While they are biorthogonal, they may substantially deviate from orthonormality due to motion compensation, even if based on an orthogonal or near-orthogonal wavelet. A temporal transform for video sequences that maintains orthonormality while permitting flexible motion compensation would be very desirable. We have recently introduced such a transform for unidirectional motion compensation from one previous frame. In this paper, we extend this idea to bidirectional motion compensation. Orthonormality is maintained for arbitrary integer-pixel motion compensation by cascading a sequence of incremental orthogonal 3×3 transforms. The energy of three input pictures is accumulated in two temporal low-bands while the temporal high-band is zero if the input pictures are identical after motion compensation. Further, the motion-compensated orthogonal transforms can be cascaded to build a dyadic wavelet decomposition. The new bidirectionally motion-compensated orthogonal transform compares favorably with the lifted 5/3 wavelet in video coding experiments with integer-pixel motion compensation.

A Lattice Structure of Biorthogonal Linear-Phase Filter Banks with Higher-Order Feasible Building Blocks

Abstract: This paper proposes a lattice structure of biorthogonal linear-phase filter banks (BOLPFBs) using new building blocks which can obtain long filters with fewer number of building blocks than conventional ones. The structure is derived from a generalization of the building blocks of first-order LPFBs. Furthermore, the proposed building blocks are applicable for both even and odd number of channels. The resulting FBs have good performance in stopband attenuation and low implementation costs.

IEEE 802.11b Complementary Code Keying and Complementary Signals Derived from Biorthogonal Sequences

Abstract: Two classes of complementary signal sets are compared in terms of their complementary properties and their error probabilities for channels with thermal noise and multipath interference. One class consists of the high-rate (11 Mbps and 5.5 Mbps) signals employed in the IEEE 802.11b standard, and the other class includes full-rate (11 Mbps) complex signals derived from biorthogonal sequences and half-rate (5.5 Mbps) biorthogonal signals. We examine several types of complementary properties of each class of signals and give performance comparisons for the signals when employed on channels in which thermal noise is the only disturbance and channels with thermal noise and multipath interference. For standard IEEE 802.11b complementary-code-key (CCK) modulation, we find the performance is strongly dependent on the differential multipath delay. For systems that employ binary error-control coding, the signals that are based on biorthogonal modulation are superior to the two IEEE 802.11b CCK signal sets.