Often, in signal processing applications, it is useful and convenient to have on the hand flexible tools providing time-frequency information from the wavelet coefficients. This means the well known wavelet packets techniques. The main purpose of this work is to explore and discuss several alternatives in this field, related to orthogonal spline wavelets.
In this work we use α-bi-Lipschitz transformation of signals both from empirical and theoretical sources to obtain new tests for the accomplishment of the multifractal formalisms associated with many methods (Wavelet Leaders, Wavelet Transform Modulus Maxima, Multifractal Detrended Fluctuation Analysis, Box Counting, and other) and we give improvements of the present algorithms that result numerically more trustworthy. Moreover the multifractal spectrum does not change in the theory, but as the numeric implementation of the computations may differ for discrete series so we can analyze its variation to study the stability of the proposed algorithms to compute it.
In addition some single coefficients that have been proposed to quantify the whole irregularity of the signal are preserved by enough high α-bi-Lipschitz transformations.
We exhibit the performance of the tests and the improvements of this methods not only in signals generated from deterministic (or sometimes random) numerical processes performed with the computer but also against series from empirical sources in which the multifractal spectrum and the irregularity coefficient were proven of utility both from the analysis and the segmentation of the signal in significant parts as series of Longwave outgoing radiation of tropical regions (and the consequent forecasting applications of precipitations) and certain series of EEG (from patients with crisis of brain absences for instance) and the ability to distinguish (and perhaps to predict) the beginning of the consecutive stages.
In this work we attempt to analize the structure of the classes of deficient spline functions, that is, the ones generated by traslations on the integers of the truncated power functions. Since these classes are correlated with multiresolution structures, the main pourpose of this presentation is to design vector scaling functions, with minimal support. For this, we do not apply Fourier techniques, but elemental properties of the truncated power functions. The double-scale or refinement relationship is demonstrated again from the autosimilarity property of these functions.
In this work we will generalize results linking multiresolution analysis structures and vectorial spaces generated from integer shifts of self-similar or radial basis functions. This connection results of a remarkable relation between causal scaling and causal radial functions, recently exposed by T. Blu and M. Unser for the unidimensional case. Here, we will detail some definitions and will enunciate the main theorems for the r dimensional case.
We propose an approximation scheme on representation spaces which elements are piecewise polynomial functions, with deficient regularity on a pre-established grid of knots. We characterize these spaces, expose relevant properties and define appropriate bases. We design approximations methods based on orthogonal projections of a given signal, restricted to certain conditions, according with the regularity that is wished. We suggest applications for this procedure in the context of signal processing.
In this work we analyze the existence of single scaling functions embedded in a multiresolution analysis structure generated by a multiscaling function. Particularly, we consider the case of spline functions.
In this work we expose some interesting properties of multiresolution structures generated from multiscaling functions. Particularly, we explore relations between different families of spline multiscaling functions embedded in a common multiresolution analysis.
In this work we display and multiresolution analysis scheme restricted on the interval [0,N]. This scheme is developed for the case of Hermite spline functions but it can be implemented in more general cases. Embedded in this scheme we construct semiorthogonal multiwavelets. Also we expose several methods and algorithms for signal processing applications.
Periodic spline wavelets provide an efficient tool to analyze periodic signals. Wavelet analysis gives its best performance when it is applied to detect transients or local events in the signal. However, it is not well suited to characterize stationary phenomena. To overcome this problem we propose a new family of periodic spline functions, capable of playing the role of trigonometric wave-forms. They lead us to an orthogonal decomposition of the signal into quasi-monochromatic spline waves. Further, a full collection of periodic spline wavelet packets is also proposed. These elemental functions can be organized in a large library of orthonormal bases. Thus, one can analyze any periodic signal in accordance with a well adapted strategy.