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In this paper, we obtain the unified error estimate for some weak biorthogonal greedy algorithms with respect to dictionaries in Banach spaces by using some kind of K-functional. From this estimate, we derive the sufficient conditions for the convergence and the convergence rates on sparse classes induced by the K-functional. The results on convergence and the convergence rates are sharp. [ABSTRACT FROM AUTHOR]

Wavelets have generated significant interest from both theoretical and applied researchers over the last few decades. We proposed biorthogonal wavelet-based multigrid method for the numerical solution of elliptic partial differential equations. The performance of the proposed scheme is better than the existing ones in terms of superconvergence with low computational time. Some of the illustrative examples demonstrate the applicability of the method. [ABSTRACT FROM AUTHOR]

In this paper, Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasidefinite matrix of bivariate generalized functions with a well-defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel-type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss-Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation in which a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses are performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel-Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular, a comparison of between both methods is presented. The nonspectral method is applied to unimodular Christoffel perturbations, and a simple example for a degree one massless Geronimus perturbation is given. [ABSTRACT FROM AUTHOR]

This paper presents biorthogonal wavelet-based multigrid (BWMG) and full approximation scheme (FAS) for the numerical solution of parabolic partial differential equations (PPDEs), which are working horse behind many commercial applications like finger print image compression. Performance of the proposed schemes is better than the existing ones in terms of super convergence with less computational time. Some of the test problems are taken to demonstrate the applicability and validity of the method. [ABSTRACT FROM AUTHOR]

It is well known that the system of translates in L 2 (ℝ) has been characterized as Bessel sequences, frames and Riesz bases in terms of the Fourier transform. The aim of this paper is to obtain similar types of characterization for the wavelet system emerging out of integer translations and dyadic dilations. The existence of the biorthogonal dual system and nonredundancy properties of the wavelet system are also investigated here. [ABSTRACT FROM AUTHOR]

By using biorthogonal wavelets, Reynaud-Bouret, Rivoirard and Tuleau-Malot provide the adaptive and optimal L 2 -risk estimation for density functions (not necessarily having compact support) in a Besov space B r , q s (ℝ) [P. Reynaud-Bouret, V. Rivoirard and C. Tuleau-Malot, Adaptive density estimation: A curse of support?, J. Stat. Plan. Inference141(1) (2011) 115–139]. The authors pose an open problem: Can L p -risk (1 ≤ p < ∞) estimation be given in their setting? In this paper, we try to solve that problem for p ∈ [ 2 , + ∞) by using wavelet estimators. [ABSTRACT FROM AUTHOR]

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorization of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics. [ABSTRACT FROM AUTHOR]

The objective of this work is to propose a novel idea of transforming temporal redundancies present in videos. Initially, the frames are divided into sub-blocks. Then, the temporally redundant blocks are grouped together thus generating new frames with spatially redundant temporal data. The transformed frames are given to compression in the wavelet domain. This new approach greatly reduces the computational time. The reason is that the existing video codecs use block matching methods for motion estimation which is a time consuming process. The proposed method avoids the use of block matching method. The existing H.264/AVC takes approximately one hour to compress a video file where as the proposed method takes only one minute for the same task. The experimental results substantially proved that the proposed method performs better than the existing H.264/AVC standard in terms of time, compression ratio and PSNR. [ABSTRACT FROM AUTHOR]

It is well-known that the Gabor expansions converge to identity operator in weak* sense on the Wiener amalgam spaces as sampling density tends to infinity. In this paper, we prove the convergence of Gabor expansions to identity operator in the operator norm as well as weak* sense on as the sampling density tends to infinity. Also we show the validity of the Janssen's representation and the Wexler-Raz biorthogonality condition for Gabor frame operator on . [ABSTRACT FROM AUTHOR]

In this paper, we describe an approach to construct of biorthogonal N-periodic discrete MRA wavelets on the basis of generalized Walsh functions. The proposed method is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

We present a simple non-Hermitian model to describe the phenomenon of asymmetric tunneling between two energy-degenerate sites coupled by a non-reciprocal interaction without dissipation. The system was described using a biorthogonal family of energy eigenvectors, the dynamics of the system was determined by the Schrödinger equation, and unitarity was effectively restored by proper normalization of the state vectors. The results show that the tunneling rates are indeed asymmetrical in this model, leading to an equilibrium that displays unequal occupation of the degenerate systems even in the absence of external interactions. [ABSTRACT FROM AUTHOR]

The Szegő-Askey polynomials are orthogonal polynomials on the unit circle. In this paper, we study their asymptotic behavior by knowing only their weight function. Using the Riemann-Hilbert method, we obtain global asymptotic formulas in terms of Bessel functions and elementary functions for z in two overlapping regions, which together cover the whole complex plane. Our results agree with those obtained earlier by Temme [Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986) 369-376]. Temme's approach started from an explicit expression of the Szegő-Askey polynomials in terms of an 2F1-function, and followed by integral methods. [ABSTRACT FROM AUTHOR]

Wavelet-based image coding has been adopted in the international standard JPEG 2000 for its efficiency. It is well-known that the orthogonality and symmetry of wavelets are two important properties for many applications of signal processing and image processing. Both can be simultaneously realized by the wavelet filter banks composed of a complex allpass filter, thus, it is expected to get a better coding performance than the conventional biorthogonal wavelets. This paper proposes an effective implementation of orthonormal symmetric wavelet filter banks composed of a complex allpass filter for lossy to lossless image compression. First, irreversible real-to-real wavelet transforms are realized by implementing a complex allpass filter for lossy image coding. Next, reversible integer-to-integer wavelet transforms are proposed by incorporating the rounding operation into the filtering processing to obtain an invertible complex allpass filter for lossless image coding. Finally, the coding performance of the proposed orthonormal symmetric wavelets is evaluated and compared with the D-9/7 and D-5/3 biorthogonal wavelets. It is shown from the experimental results that the proposed allpass-based orthonormal symmetric wavelets can achieve a better coding performance than the conventional D-9/7 and D-5/3 biorthogonal wavelets both in lossy and lossless coding. [ABSTRACT FROM AUTHOR]

We provide a procedure for constructing biorthogonal multiwavelets from a family of biorthogonal multiscaling functions compactly supported on [-1,1]. The scaling vectors and the associated multiwavelets are piecewise continuously differentiable, symmetrical and possess approximation order three. The construction of scaling vectors is accomplished using quadratic fractal interpolation functions. The filters corresponding to scaling vectors possess certain properties which enable us to construct a new pair of biorthogonal scaling vectors and associated multiwavelets with different regularity and approximation order, related to the old ones by differentiation. The old and new biorthogonal multiwavelet systems give rise to compactly supported biorthogonal multiwavelet basis for the space of divergence-free vector fields on the upper half plane with the Navier boundary condition. [ABSTRACT FROM AUTHOR]

Recorded medical images often represent a degraded version of the original scene due to imperfections in electronic or photographic medium used. The degradations may have many causes, but two dominant degradations are noise and blur. Restoration of blurred and noisy medical images is of fundamental importance in several medical imaging applications. Most of the medical image denoising techniques need removal of blur before the denoising. Denoising of medical images in presence of blur is a hard problem. Most of the wavelet transform-based denoising techniques use the orthonormal wavelets and suitable for image corrupted with only additive white Gaussian noise. In the present work, we have proposed a denoising algorithm for medical images based on the lifting-scheme and linear phase characteristic of biorthogonal wavelet transform. A level-dependent soft thresholding function has been used which is based on the standard deviation, the absolute mean and the absolute median of the wavelet coefficients. The linear phase characteristic of the biorthogonal filters used in denoising reduces the distortions at edge points of image. Also, the lifting schemes of the biorthogonal wavelet filters make the algorithm efficient and applicable in real time. Experimental results show that the proposed denoising method outperform standard wavelet, complex wavelet and curvelet-based denoising techniques in terms of the SNR and PSNR (in dB) and it offers effective noise removal from noisy medical images while maintaining sharpness of objects in the image. [ABSTRACT FROM AUTHOR]

Method for data hiding based on LeGall 5/3 of Cohen-Daubechies-Feauveau: CDF 5/3 wavelet with data compression and random scanning of secret imagery data together with steganography is proposed. Invisibility of secret imagery data is evaluated based on Peak Signal to Noise Ratio: PSNR with SIDBA standard image database. The experimental results show that PSNR of LeGall based wavelet utilized data hiding ranges from 43.82 to 46.9 while that of Daubechies based method ranges from 44.33 to 44.75 when the coded secret imagery data is inserted in the first 3 digits from Least Significant Bit: LSB of the original image. Data compression ratio for the secret imagery data ranges from 1.3 to 19.4 which depends on the complexity of the secret imagery data. Meanwhile, PSNR of data hidden image ranges from 46.83 to 47.41. Consequently, the proposed data hiding method is permissive because PSNR is over 40 dB results in satisfaction on invisibility of the secret imagery data in the data hidden image. [ABSTRACT FROM AUTHOR]

We present a new family of triangular biorthogonal wavelets that is defined on a triangular lattice by introducing a new operation to generalize two-dimensional lifting, which we call twist. The resulting filters inherit several remarkable features of the early triangular biorthogonal wavelet filters such as the hexagonal symmetry of low-pass filters, symmetrical arrangement of three high-pass filters on the lattice, and that the wavelet decomposition produces uniform energy distributions over three detail components, preserving the isotropy of decomposed images. Additionally, these filters are a biorthogonal set of truly nonseparable two-dimensional wavelet filters that have much larger support, which provides much larger portions of the total energy to three detail components of decomposed images. We show that this plays a crucial role when extracting the edge structure of an image. [ABSTRACT FROM AUTHOR]

In this paper, we present an explicit method to construct directly in the x-domain compactly supported scaling functions corresponding to the wavelets adapted to a sum of differential operators with constant coefficients. Here the adaptation to an operator is taken to mean that the wavelets give a diagonal form of the operator matrix. We show that the biorthogonal compactly supported wavelets adapted to a sum of differential operators with constant coefficients are closely connected with the representation of the null-space of the adjoint operator by the corresponding scaling functions. We consider the necessary and sufficient conditions (actually the Strang-Fix conditions) on integer shifts of a compactly supported function (distribution) f ∈ S'(ℝ) to represent exactly any function from the null-space of a sum of differential operators with constant coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient biorthogonal wavelet construction for the ternary Loop subdivision based on the lifting scheme, and give the corresponding decomposition and reconstruction formulas. We apply the proposed algorithm to the progressive transmission and the denoising of 3D graphics, and show its sufficient stability and good performance by some numerical experiments. Since the usual input mesh does not possess the property of subdivision connectivity, while the multiresolution analysis based on subdivision surface requires the processed initial mesh to be semi-regular, we further propose a new rapid algorithm of constructing the initial mesh which not only possesses ternary subdivision connectivity but also approximates the given input mesh. [ABSTRACT FROM AUTHOR]

This paper deals with the construction of spline wavelets on the interval [0, 1], which have zero boundary values and vanishing moments. One begins with some primal scaling functions and their biorthogonal duals. Then desired biorthogonal wavelets are given by the method of Dahmen, Kunoth and Urban. Since the structure of those wavelets looks complicated, one tries to construct those types of wavelets only in primal side finally, without using any dual informations. Some numerical experiments show good effects, although the uniform stability remains to be proved theoretically. [ABSTRACT FROM AUTHOR]

In this paper, we describe an algorithm for computing biorthogonal compactly supported dyadic wavelets related to the Walsh functions on the positive half-line ℝ+. It is noted that a similar technique can be applied in very general situations, e.g., in the case of Cantor and Vilenkin groups. Using the feedback-based approach, some numerical experiments comparing orthogonal and biorthogonal dyadic wavelets with the Haar, Daubechies, and biorthogonal 9/7 wavelets are prepared. [ABSTRACT FROM AUTHOR]

Wavelets are the powerful tool for signal processing especially bio-signal processing. Wavelet transform is used to represent the signal to some other time frequency representation better suited for detecting and removing redundancies. In this paper, electrocardiogram (ECG) signal coding using biorthogonal wavelet-based Burrows-Wheeler Coder is discussed. Biorthogonal wavelet transform is used to decompose the ECG signal. Then the Burrows-Wheeler Coder is applied in order to compress the decomposed ECG signal. The Burrows-Wheeler Coder is the combination of Burrows-Wheeler Transformation (BWT), Move-to-Front (MTF) coder and Huffman coder. Compression Ratio (CR) and Percent Root mean square Difference (PRD) are used as performance measures. ECG signals/records from MIT-BIH arrhythmic database are used to evaluate the performance of this coder. This algorithm is tested with 25 different records from MIT-BIH arrhythmia database and obtained the average PRD as 0.0307% to 3.8706% for the average CR of 3.6362 : 1 to 280.48 : 1. For record 117, the CR of 8.1638 : 1 is achieved with PRD 0.1652%. This experimental results show that this coder outperforms other coders such as Djohn, EZW, SPIHT, Novel algorithm etc. that exist in the literature in terms of coding efficiency and computation. [ABSTRACT FROM AUTHOR]

An algorithm is presented for constructing a pair of high approximation order biorthogonal multiscaling function with dilation factor a in terms of any given pair of biorthogonal multiscaling function. The special case that a = 2 is discussed. If the dilation factor a = 2, then a biorthogonal multiwavelet pair is constructed explicitly. Finally, examples are given. [ABSTRACT FROM AUTHOR]

We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were first systematically studied by Berndt and Zaharescu where their proofs involved complex contour integration. We show how to prove most of Berndt–Zaharescu's and some new identities by elementary methods of discrete Fourier analysis. [ABSTRACT FROM AUTHOR]

The atmospheric greenhouse effect, an idea that many authors trace back to the traditional works of Fourier (1824), Tyndall (1861), and Arrhenius (1896), and which is still supported in global climatology, essentially describes a fictitious mechanism, in which a planetary atmosphere acts as a heat pump driven by an environment that is radiatively interacting with but radiatively equilibrated to the atmospheric system. According to the second law of thermodynamics, such a planetary machine can never exist. Nevertheless, in almost all texts of global climatology and in a widespread secondary literature, it is taken for granted that such a mechanism is real and stands on a firm scientific foundation. In this paper, the popular conjecture is analyzed and the underlying physical principles are clarified. By showing that (a) there are no common physical laws between the warming phenomenon in glass houses and the fictitious atmospheric greenhouse effects, (b) there are no calculations to determine an average surface temperature of a planet, (c) the frequently mentioned difference of 33° is a meaningless number calculated wrongly, (d) the formulas of cavity radiation are used inappropriately, (e) the assumption of a radiative balance is unphysical, (f) thermal conductivity and friction must not be set to zero, the atmospheric greenhouse conjecture is falsified. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new family of perfect reconstruction (PR) complex filter banks, named interleaved discrete Fourier transform modulated filter banks (Interleaved DFT-FBs). In the filter banks, the analysis filters are generated by interlaced exponential modulating two different analysis prototype filters, and the synthesis filters are generated by two different synthesis prototype filters via the same manner. The filter banks have a simple polyphase structure similar to DFT modulated filter banks (DFT-FBs). More importantly, the proposed Interleaved DFT-FBs can achieve critically sampled PR complex filter bank with FIR analysis and synthesis filters, which is impossible for DFT-FBs. We give and prove the PR condition for 2M-band Interleaved DFT-FBs. Utilizing the result, the design procedure of the prototype filters is presented. In addition, by the theoretic analysis and numerical examples, it is shown that the analysis and synthesis filters cannot simultaneously provide good stopband attenuation for the critically sampled PR Interleaved DFT-FBs. Although the limitation always exits, the filter banks can find applications in some subband coding systems of high bit rate. [ABSTRACT FROM AUTHOR]