To cite this version:Jean-Louis Merrien, Tomas Sauer. From Hermite to stationary subdivision schemes in one and several variables. Advances in Computational Mathematics, Springer Verlag, 2012, 36 (4)
AbstractVector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive derivatives they differ by the "renormalization" of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so-called spectral condition to a vector subdivision scheme. These factorizations are natural extensions of the "zero at π" condition known for the masks of refinable functions. Moreover, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is conceptionally more intricate than the univariate one. Finally, we give some examples of such factorizations.
Abstract-The use of coherence is a well-established standard approach for the analysis of biomedical signals. Being entirely based on frequency analysis, i.e., on spectral properties of the signal, it is not possible to obtain any information about the temporal structure of coherence which is useful in the study of brain dynamics, for example. Extending the concept of coherence as a measure of linear dependence between realizations of a random process to the wavelet transform, this paper introduces a new approach to coherence analysis which allows to monitor time-dependent changes in the coherence between electroenecphalographic (EEG) channels. Specifically, we analyzed multichannel EEG data of 26 subjects obtained in an experiment on associative learning, and compare the results of Fourier coherence and wavelet coherence, showing that wavelet coherence detects features that were inaccessible by application of Fourier coherence.
In this paper we focus on Hermite subdivision operators that act on vector valued data interpreting their components as function values and associated consecutive derivatives. We are mainly interested in studying the exponential and polynomial preservation capability of such kind of operators, which can be expressed in terms of a generalization of the spectral condition property in the spaces generated by polynomials and exponential functions. The main tool for our investigation are convolution operators that annihilate the aforementioned spaces, which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator.
In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of non-stationary bivariate subdivision schemes, which allow to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.
The paper gives an extension of Prony's method to the multivariate case which is based on the relationship between polynomial interpolation, normal forms modulo ideals and H-bases. Though the approach is mainly of algebraic nature, we also give an algorithm using techniques from Numerical Linear Algebra to solve the problem in a fast and efficient way.
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