Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.

Coset Sum: An Alternative to the Tensor Product in Wavelet Construction

Abstract: A multivariate biorthogonal wavelet system can be obtained from a pair of multivariate biorthogonal refinement masks in multiresolution analysis setup. Some multivariate refinement masks may be decomposed into lower dimensional refinement masks. Tensor product is a popular way to construct a decomposable multivariate refinement mask from lower dimensional refinement masks. We present an alternative method, which we call coset sum, for constructing multivariate refinement masks from univariate refinement masks. The coset sum shares many essential features of the tensor product that make it attractive in practice: 1) it preserves the biorthogonality of univariate refinement masks, 2) it preserves the accuracy number of the univariate refinement mask, and 3) the wavelet system associated with it has fast algorithms for computing and inverting the wavelet coefficients. The coset sum can even provide a wavelet system with faster algorithms in certain cases than the tensor product. These features of the coset sum suggest that it is worthwhile to develop and practice alternative methods to the tensor product for constructing multivariate wavelet systems. Some experimental results using 2-D images are presented to illustrate our findings.

Frequency Warping Biorthogonal Frames

Abstract: Frequency warping is theoretically designed to be a unitary operator of infinite input and output dimensions, thus performing the resolution of identity. In real implementations finite dimensions have to be considered, then perfect reconstruction cannot be fulfilled. The accuracy of reconstruction is particularly compromised in case of non-smooth warping maps, which are more useful for practical applications. In order to overcome this limitation, a new frequency warping biorthogonal frame operator for non-smooth warping maps is introduced in this work. The proposed transformation is based on a mathematical model which has been previously introduced for computational purposes. By adding some redundancy with respect to the truncation of the infinite dimensions operator, the effect of an infinite output dimension can be taken into account in a compressed way, based on an analytical factorization. In the reconstruction process, the additional redundant samples are expanded, thus guaranteeing near perfect reconstruction.

Some Extensions of Askey-Wilson's Q-Beta Integral and the Corresponding Orthogonal Systems

Abstract: A seven-parameter extension of Askey and Wilson's four parameter q-beta integral is written in a symmetric form as the sum of multiples of two very-well-poised balanced basic hypergeometric 10Φ9 series. Two special cases are considered in which the evaluation of the integral gives single terms by the q-Dixon formula in one case and by a special case of the Verma-Jain formula in the other. An orthogonal polynomial system is obtained in the first case and a system of biorthogonal rational function is obtained in the second. It is also shown that the biorthogonal system represents a generalization of Rogers’ q-ultraspherical polynomials.

On wavelet decomposition of spaces of first order splines

Abstract: We consider approximate relations in the form of a system of linear algebraic equations that yield B φ -splines. We construct Lagrange type splines of the first order and give examples of polynomial, trigonometric, hyperbolic, and exponential B φ -splines. We also construct a system of linear functionals biorthogonal to the B φ -splines and resolve an interpolation problem generated by this system. For refined nonuniform grids we establish an embedding of spaces of B φ -splines. The decomposition and reconstruction formulas are obtained. Bibliography: 20 titles.

Even-length biorthogonal wavelets for digital watermarking

Abstract: In wavelet based digital watermarking, when the particular wavelet filter used forms part of a secret key, the diversity and size of a family of filters contributes to the level of security. Diversity has been shown to improve the security of watermarking systems from hostile attacks. This paper presents a family of even length biorthogonal wavelet filters for use in digital watermarking. The filters are explicitly parameterized by two free parameters and can provide further diversity in watermarking. The filters have a prescribed number of vanishing moments ensuring some degree of smoothness in the resulting wavelet function. The study is conducted in the context of a blind watermarking scheme and the robustness of the filters to range of attacks is evaluated.