Showing papers on "Biorthogonal system published in 2006"

Multiscale Poisson data smoothing

Abstract: Summary. The paper introduces a framework for non-linear multiscale decompositions of Poisson data that have piecewise smooth intensity curves. The key concept is conditioning on the sum of the observations that are involved in the computation of a given multiscale coefficient. Within this framework, most classical wavelet thresholding schemes for data with additive homoscedastic noise can be used. Any family of wavelet transforms (orthogonal, biorthogonal or second generation) can be incorporated in this framework. Our second contribution is to propose a Bayesian shrinkage approach with an original prior for coefficients of this decomposition. As such, the method combines the advantages of the Haar-Fisz transform with wavelet smoothing and (Bayesian) multiscale likelihood models, with additional benefits, such as extendability towards arbitrary wavelet families. Simulations show an important reduction in average squared error of the output, compared with the present techniques of Anscombe or Fisz variance stabilization or multiscale likelihood modelling.

Perfect reconstruction conditions and design of oversampled DFT-modulated transmultiplexers

Abstract: This paper presents a theoretical analysis of oversampled complex modulated transmultiplexers. The perfect reconstruction (PR) conditions are established in the polyphase domain for a pair of biorthogonal prototype filters. A decomposition theorem is proposed that allows it to split the initial system of PR equations, that can be huge, into small independent subsystems of equations. In the orthogonal case, it is shown that these subsystems can be solved thanks to an appropriate angular parametrization. This parametrization is efficiently exploited afterwards, using the compact representation we recently introduced for critically decimated modulated filter banks. Two design criteria, the out-of-band energy minimization and the time-frequency localization maximization, are examined. It is shown, with various design examples, that this approach allows the design of oversampled modulated transmultiplexers, or filter banks with a thousand carriers, or subbands, for rational oversampling ratios corresponding to low redundancies. Some simulation results, obtained for a transmission over a flat fading channel, also show that, compared to the conventional OFDM, these designs may reduce the mean square error.

Wavelets on manifolds: an optimized construction

Abstract: A key ingredient of the construction of biorthogonal wavelet bases for Sobolev spaces on manifolds, which is based on topological isomorphisms is the Hestenes extension operator. Here we firstly investigate whether this particular extension operator can be replaced by another extension operator. Our main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required by Dahmen and Schneider. In the second part of this paper, we investigate and optimize the Hestenes extension operator. The results of the optimization process allow us to implement the construction of biorthogonal wavelets from Dahmen and Schneider. As an example, we illustrate a wavelet basis on the 2-sphere.

Fast Poisson Noise Removal by Biorthogonal Haar Domain Hypothesis Testing

Abstract: Methods based on hypothesis tests (HTs) in the Haar domain are widely used to denoise Poisson count data. Facing large datasets or real-time applications, Haar-based denoisers have to use the decimated transform to meet limited-memory or computation-time constraints. Unfortunately, for regular underlying intensities, decimation yields discontinuous estimates and strong "staircase" artifacts. In this paper, we propose to combine the HT framework with the decimated biorthogonal Haar (Bi-Haar) transform instead of the classical Haar. The Bi-Haar filter bank is normalized such that the p-values of Bi-Haar coefficients (pBH) provide good approximation to those of Haar (pH) for high-intensity settings or large scales; for low-intensity settings and small scales, we show that pBH are essentially upper-bounded by pH. Thus, we may apply the Haar-based HTs to Bi-Haar coefficients to control a prefixed false positive rate. By doing so, we benefit from the regular Bi-Haar filter bank to gain a smooth estimate while always maintaining a low computational complexity. A Fisher-approximation-based threshold imple- menting the HTs is also established. The efficiency of this method is illustrated on an example of hyperspectral-source-flux estimation.

Theory and factorization for a class of structurally regular biorthogonal filter banks

Abstract: Regularity is a fundamental and desirable property of wavelets and perfect reconstruction filter banks (PRFBs). Among others, it dictates the smoothness of the wavelet basis and the rate of decay of the wavelet coefficients. This paper considers how regularity of a desired degree can be structurally imposed onto biorthogonal filter banks (BOFBs) so that they can be designed with exact regularity and fast convergence via unconstrained optimization. The considered design space is a useful class of M-channel causal finite-impulse response (FIR) BOFBs (having anticausal FIR inverses) that are characterized by the dyadic-based structure W(z)=I-UV/sup /spl dagger//+z/sup -1/UV/sup /spl dagger// for which U and V are M/spl times//spl gamma/ parameter matrices satisfying V/sup /spl dagger//U=I/sub /spl gamma//, 1/spl les//spl gamma//spl les/M, for any M/spl ges/2. Structural conditions for regularity are derived, where the Householder transform is found convenient. As a special case, a class of regular linear-phase BOFBs is considered by further imposing linear phase (LP) on the dyadic-based structure. In this way, an alternative and simplified parameterization of the biorthogonal linear-phase filter banks (GLBTs) is obtained, and the general theory of structural regularity is shown to simplify significantly. Regular BOFBs are designed according to the proposed theory and are evaluated using a transform-based image codec. They are found to provide better objective performance and improved perceptual quality of the decompressed images. Specifically, the blocking artifacts are reduced, and texture details are better preserved. For fingerprint images, the proposed biorthogonal transform codec outperforms the FBI scheme by 1-1.6 dB in PSNR.

Existence and construction of compactly supported biorthogonal multiple vector-valued wavelets

Abstract: In this paper, we introduce biorthogonal multiple vector-valued wavelets which are wavelets for vector fields. We proved that, like in the scalar and multiwavelet case, the existence of a pair of biorthogonal multiple vector-valued scaling functions guarantees the existence of a pair of biorthogonal multiple vector-valued wavelet functions. Finally, we investigate the construction of a class of compactly supported biorthogonal multiple vector-valued wavelets.

Estimation of spatial derivatives and identification of continuous spatio-temporal dynamical systems

Abstract: A new approach for the estimation of spatial derivatives and the identification of a class of continuous spatio-temporal dynamical systems from experimental data is presented in this study. The proposed identification approach is a combination of implicit Adams integration and an orthogonal forward regression algorithm (OFR), in which the operators are expanded using polynomials as basis functions. The noisy experimental data are de-noised by using biorthogonal spline wavelet filters and the spatial derivatives are estimated using a multiresolution analysis method. Finally, a bootstrap method is applied to refine the identified parameters from the OFR algorithm. The resulting identified models of the spatio-temporal evolution form a system of partial differential equations. Examples are provided to demonstrate the efficiency of the proposed method.

Simd lapped transform-based digital media encoding/decoding

Abstract: A block transform-based digital media codec achieves faster performance by re-mapping components of the digital media data into vectors or parallel units on which many operations of the transforms can be performed on a parallel or single-instruction, multiple data (SIMD) basis. In the case of a one-dimensional lapped biorthogonal transform, the digital media data components are re-mapped into vectors on which butterfly stages of both overlap pre-/post-filter and block transform portions of the lapped transform can be performed on a SIMD basis. In the case of a two-dimensional lapped biorthogonal transform, the digital media data components are re-mapped into vectors on which a Hadamard operator of both overlap pre-/post-filter and block transform can be performed on a SIMD basis.

Linear independence of pseudo-splines

Abstract: In this paper, we show that the shifts of a pseudo-spline are lin- early independent. This is stronger than the (more obvious) statement that the shifts of a pseudo-spline form a Riesz system. In fact, the linear independence of a compactly supported (reflnable) function and its shifts has been studied in several areas of approximation and wavelet theory (see e.g. (4), (9), (10), (13), (14), (16) and (19)). Furthermore, the linear independence of the shifts of a pseudo-spline is a necessary and su-cient condition for the existence of a compactly supported function whose shifts form a biorthogonal dual system of the shifts of the pseudo-spline.

The PDEs of Biorthogonal Polynomials Arising in the Two-Matrix Model

Abstract: The two-matrix model can be solved by introducing biorthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of biorthogonal polynomials (called windows) satisfy polynomial ODEs as well as deformation equations (PDEs) and finite difference equations (ΔE) which are all Frobenius compatible and define discrete and continuous isomonodromic deformations for the irregular ODE, as shown in previous works of ours. In the one matrix model an explicit and concise expression for the coefficients of these systems is known and it allows to relate the partition function with the isomonodromic tau-function of the overdetermined system. Here, we provide the generalization of those expressions to the case of biorthogonal polynomials, which enables us to compute the determinant of the fundamental solution of the overdetermined system of ODE + PDEs + ΔE.

Stabile biortogonale Spline-Waveletbasen auf dem Intervall

Abstract: Based on the family of biorthogonal pairs of scaling functions consisting of cardinal B-splines and compactly supported dual generators on the whole real line, as presented by Cohen, Daubechies and Feauveau, we are concerned with the cnstruction of a new biorthogonal multiresolution analysis on the interval, such that the corresponding wavelet bases realize any desired order of moment condition on the interval. In contrast to previous approaches we choose the well-established Schoenberg spline basis on the interval with equidistant knots for primal multiresolution which has already been used by Chui and Quak to construct semiorthogonal spline bases on the interval. After giving an overview of the concrete construction, we discuss the cavorable properties of the constructed basis functions. The subsequent construction of the associated wavelets relies on the known method of stable completions, which has been presented by Dahmen, Kunoth and Urban as a helpful tool in constructing wavelet bases on the interval. Due to the fact, that we use all inner scaling functions, we use a high number of inner wavelets, so that the number of constructed boundary wavelets is very low, compared to former approaches. This is also true for the related wavelet basis with homogeneous or complementary boundary conditions. In view of applications, there are two interesting questions. Firstly, we investigate the condition number of the corresponding wavelet transforms. Secondly, we treat the stiffness matrix of the Laplace operator concerning to our basis and show, that its condition number is much better than the condition number in other approaches.

Exactly solvable two-dimensional complex model with a real spectrum

Abstract: Using supersymmetric intertwining relations of the second order in derivatives, we construct a two-dimensional quantum model with a complex potential for which all energy levels and the corresponding wave functions are obtained analytically. This model does not admit separation of variables and can be considered a complexified version of the generalized two-dimensional Morse model with an additional sinh −2 term. We prove that the energy spectrum of the model is purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. We explicitly find the symmetry operator, describe the biorthogonal basis, and demonstrate the pseudo-Hermiticity of the Hamiltonian of the model. The obtained wave functions are simultaneously eigenfunctions of the symmetry operator.

Biorthogonal Eigenfunction System in the Triple-Deck Limit

Abstract: The solutions of receptivity problems for a periodic-in-time actuator placed on the wall in a two-dimensional boundary layer and for a two-dimensional hump are discussed within the scope of the biorthogonal eigenfunction expansion technique in the limit of high Reynolds number when the triple-deck scaling is imposed. It is shown that the solutions obtained with the help of the biorthogonal eigenfunction system are equivalent to the solutions derived within the scope of the triple-deck theory.

Convergence of Vector Subdivision Schemes and Construction of Biorthogonal Multiple Wavelets

Abstract: Let φ = (φ1, . . . , φr) be a refinable vector of compactly supported functions in L2(IR). It is shown in this paper that there exists a refinable vector φ of compactly supported functions in L2(IR) such that φ is dual to φ if and only if the shifts of φ1, . . . , φr are linearly independent. This result is established on the basis of a complete characterization of the convergence of vector subdivision schemes associated with exponentially decaying masks. As an application of the general theory, two interesting examples of biorthogonal double wavelets are constructed.

Non-Hermitian polarizers: a biorthogonal analysis.

Abstract: The non-Hermitian operators of the ideal nonorthogonal multilayer optical polarizers are spectrally analyzed in the framework of skew-angular biorthonormal vector bases. It is shown that these polarizers correspond to skew projectors and their operators are generated by skew projectors, exactly as the canonical ideal polarizers correspond to Hermitian projectors. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors—the generating projectors. It is shown that if these nonorthogonal polarizers are looked upon as variable devices, two kinds of degeneracy may occur for suitable values of the inner parameter of the device: The corresponding operators may become normal (more precisely, Hermitian) or, on the contrary, very pathological—defective and singular. In the first case their eigenvectors and biorthogonal conjugate eigenvectors collapse into a unique pair of eigenvectors; in the second case their eigenvectors (as well as their biorthogonal conjugates) collapse into a single vector.

A 2/spl times/2 antennas ultra-wideband system with biorthogonal pulse position modulation

Abstract: The Golden code is a 2/spl times/2 space-time code that achieves the best known performance with all constellations carved from /spl Zopf/[i]. In this letter, we present the construction of a new coding scheme for 2M-ary biorthogonal pulse position modulations (BPPM) with M/spl ges/4. The proposed code satisfies all of the construction constraints of the Golden code and it has the additional advantage of being totally real making it suitable for low cost carrier-less ultra-wideband terminals. Namely, this totally real construction achieves full rate and full diversity with the best known coding gain and without any shaping losses for 2M-BPPM with M/spl ges/4.

A New Class of MC-CDMA Systems Using Cyclic-Shift M-ary Biorthogonal Keying

Abstract: In this paper, a new class of MC-CDMA systems is proposed, which uses cyclic-shift M-ary biorthogonal keying (CS-MBOK) symbol mapping in terms of the polyphase Chu sequence with perfect autocorrelation function. In addition to the basic CS-MBOK mode, we also propose three other transmission modes for more flexible data rate and performance tradeoff in different applications. A low-complexity structure for CS-MBOK code correlation and demapping based on FFT/IFFT is also devised. Compared to the conventional MC-CDMA system, the new system is featured of greatly improved bandwidth efficiency and BER performance, and much lower peak-to-average power ratio (PAPR) of the transmit signal, especially for larger FFT size. Simulation results are included to confirm the above arguments.

Copyright Protection of Image Learning Objects using Wavelet-based Watermarking and Fuzzy Logic

Abstract: An adaptive watermarking algorithm is presented which exploits a biorthogonal wavelets-based human visual system (HVS) and a Fuzzy Inference System (FIS) to protect the copyright of im- ages in learning object repositories. Specifically, the HVS relies on the linear-phase property of biorthogonal wavelet filters (symmetric wavelets) in order to efficiently extract the masking information, while taking into account the local characteristics of the image. The FIS is utilized to compute the optimum watermark weighting function that would enable the embedding of the maximum-energy and impercep- tible watermark. The experimental results achieved demonstrate that the proposed algorithm is robust against both, signal processing and geometric attacks.

Image Information Hiding Using Second Order Biorthogonal Wavelets

Abstract: Image watermarking techniques have been widely studied since the need for copyright protection. In this paper, we present a method for image watermarking using the biorthogonal wavelet transform. The method is based on decomposing an image using the discrete wavelet transform (DWT), and then embedding a watermark in the most significant coefficients of the transform. Biorthogonal wavelets have the property of perfect reconstruction and smoothness (vanishing points). The method proved its robustness against several attacks.

Multiplierless Design of Biorthogonal Dual-Tree Complex Wavelet Transform using Lifting Scheme

Abstract: In this work, we present the design, implementation and application of two families of biorthogonal dual-tree complex wavelet transform (CWT) filters using lifting scheme The first design is achieved using exhaustive search with coding gain, DC leakage and directional selectivity as the fundamental criteria The second set of filters are derived from the biorthogonal design procedure that was recently suggested by Selesnick Furthermore, this paper also introduces a new theorem that suggests that lifting implementation of filters that are time-reversals of each other is closely related Various applications are presented to validate the proposed design scheme, including performance in denoising as well as in the JPEG-2000 image coding standard

Method for noise suppression during reception of circular-polarization electromagnetic wave by biorthogonal antenna assembly

Abstract: FIELD: antenna engineering; radio communication systems. ^ SUBSTANCE: proposed method intended to eliminate mutual impact among dipoles and to suppress noise including that similar to useful signal spectrum within wide spectrum of angles includes reception of circular-polarization electromagnetic wave by means of two orthogonal antenna assemblies relatively turned through angle of pi/4, conversion of signals received, generation of four reference signals, equalization of nine reference-signal readings with respect to time, detection of four combinations of hypothetical noise-signal implementations, estimation of each of four hypotheses for reliability and selection of most reliable of them, generation of four compensating signals corresponding to chosen hypothesis, correction of noise signals, phase equalization of corrected signals in each pair and between pairs, addition of signals in each pair and between pairs to produce output signal of biorthogonal antenna assembly; in the process four signals compensating for mutual impact of biorthogonal assembly antennas are produced by weighed addition of converted signals, whereupon four signals compensating for mutual impact of biorthogonal assembly antennas are used to generate four reference signals. ^ EFFECT: enhanced signal-to-noise ratio in reception. ^ 1 cl, 4 dwg

Biorthogonal polynomials for 2-matrix models with semiclassical potentials

Abstract: We consider the biorthogonal polynomials associated to the two-matrix model where the eigenvalue distribution has potentials V_1,V_2 with arbitrary rational derivative and whose supports are constrained on an arbitrary union of intervals (hard-edges). We show that these polynomials satisfy certain recurrence relations with a number of terms d_i depending on the number of hard-edges and on the degree of the rational functions V_i'. Using these relations we derive Christoffel-Darboux identities satisfied by the biorthogonal polynomials: this enables us to give explicit formulae for the differential equation satisfied by d_i+1 consecutive polynomials, We also define certain integral transforms of the polynomials and use them to formulate a Riemann-Hilbert problem for (d_i+1) x (d_i+1) matrices constructed out of the polynomials and these transforms. Moreover we prove that the Christoffel-Darboux pairing can be interpreted as a pairing between two dual Riemann-Hilbert problems.

Array Pattern Synthesis Based on Weighted Biorthogonal Modes

Abstract: A novel approach to arbitrary array pattern synthesis utilizing weighted biorthogonal modes is presented in this paper. The algorithm, involving arrays with arbitrary geometry and incorporating mutual coupling effect, can accurately determine element excitations required to yield desired field pattern. The mutual orthogonality of modes is used to ensure the agreement between the desired and computed patterns. The effectiveness of the method is demonstrated with several numerical examples.

Diagrammatic tools for generating biorthogonal multiresolutions

Abstract: Elsewhere we have introduced a construction to produce biorthogonal multiresolutions from given subdivisions. This construction was formulated in matrix terms, which is appropriate for curves and tensor-product surfaces. For mesh surfaces of non-tensor connectivity, however, matrix notation is inconvenient. This work presents the construction for regular meshes using diagrams (stencils, masks) and interactions between diagrams to replace matrices and matrix multiplication. Regular triangular meshes with butterfly subdivision and a variant of Loop subdivision due to Litke, et al. are used as examples.

Compressing the Laplacian Pyramid

Abstract: The Laplacian pyramid (LP) is one of the earliest examples of multiscale representation of visual data. It is well known that an LP is overcomplete or redundant by construction, and has lower compression efficiency compared to critical representations such as wavelets and subband coding. In this paper, we propose to improve the rate-distortion (R-D) performance of the LP through critical representation. We consider an LP with biorthogonal decimation and interpolation filters, and show that the detail signals lie in lower-dimensional subspaces. This allows them to be represented using fewer coefficients than the original spatial representations. We derive orthogonal bases for these subspaces and represent the detail signals in terms of their projections onto these bases. Simulation results suggest that higher compression ratios can be achieved with the critical representation than with the standard LP with usual or dual frame based reconstructions.

Integrals of rational symmetric functions, two matrix models and biorthogonal polynomials

Abstract: We give a new method for the evaluation of a class of integrals of rational symmetric functions in N pairs of variables {x_a, y_a}_{a=1,... N} arising in coupled matrix models, valid for a broad class of two-variable measures. The result is expressed as the determinant of a matrix whose entries consist of the associated biorthogonal polynomials, their Hilbert transforms, evaluated at the zeros and poles of the integrand, and bilinear expressions in these. The method is elementary and direct, using only standard determinantal identities, partial fraction expansions and the property of biorthogonality. The corresponding result for one-matrix models and integrals of rational symmetric functions in N variables {x_a}_{a=1,... N} is also rederived in a simple way using this method

A parallel GNFS algorithm with the biorthogonal block lanczos method for integer factorization

Abstract: Currently, RSA is a very popular, widely used and secure public key cryptosystem, but the security of the RSA cryptosystem is based on the difficulty of factoring large integers. The General Number Field Sieve (GNFS) algorithm is the best known method for factoring large integers over 110 digits. Our previous work on the parallel GNFS algorithm, which integrated the Montgomery’s block Lanczos algorithm to solve the large and sparse linear systems over GF(2), has one major disadvantage, namely the input has to be symmetric (we have to symmetrize the input for nonsymmetric case and this will shrink the rank). In this paper, we successfully implement the parallel General Number Field Sieve (GNFS) algorithm and integrate with a new algorithm called the biorthogonal block Lanczos algorithm for solving large and sparse linear systems over GF(2). This new algorithm is based on the biothorgonal technique, can find more solutions or dependencies than Montgomery’s block Lanczos method with less iterations. The detailed experimental results on a SUN cluster will be presented as well.

Biorthogonal Recombination Nonuniform Cosine-Modulated Filter Banks and their Multiplier-Less Realizations

Abstract: This paper studies the theory, design and multiplier-less (ML) realization of a class of perfect reconstruction (PR) low-delay biorthogonal nonuniform cosine-modulated filter banks (CMFBs). It is based on a recombination (or merging) structure previously proposed by the authors. By relaxing the original CMFB and the recombination transmultiplexer (TMUX) in the recombination structure to be biorthogonal, nonuniform CMFBs with lower system delay can be obtained. This also increases the possible choices of the prototype filters to meet different design objectives. A matching condition is introduced to suppress the spurious response resulting from the mismatch in the transition bands of the two biorthogonal CMFBs. A complete factorization of biorthgonal CMFB using the lifting scheme is employed to obtain structurally PR biorthogonal nonuniform filter banks (FBs), which are robust to coefficient quantization. In addition, by approximating the lifting coefficients and the modulation matrices by the sum of powers-of-two (SOPOT) coefficients, ML realization with very low implementation complexity is obtained. Design examples and comparison are given to illustrate the effectiveness of the proposed method.