6 research outputs found

    Description, modeling and forecasting of data with optimal wavelets

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    Cascade processes have been used to model many different self-similar systems, as they are able to accurately describe most of their global statistical properties. The so-called optimal wavelet basis allows to achieve a geometrical representation of the cascade process-named microcanonical cascade- that describes the behavior of local quantities and thus it helps to reveal the underlying dynamics of the system. In this context, we study the benefits of using the optimal wavelet in contrast to other wavelets when used to define cascade variables, and we provide an optimality degree estimator that is appropriate to determine the closest-to-optimal wavelet in real data. Particularizing the analysis to stock market series, we show that they can be represented by microcanonical cascades in both the logarithm of the price and the volatility. Also, as a promising application in forecasting, we derive the distribution of the value of next point of the series conditioned to the knowledge of past points and the cascade structure, i.e., the stochastic kernel of the cascade process.

    From Differential Equations to the Construction of New Wavelet-Like Bases

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    In this paper, an approach is introduced based on differential operators to construct wavelet-like basis functions. Given a differential operator L with rational transfer function, elementary building blocks are obtained that are shifted replicates of the Green's function of L. It is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential B-spline representation. The corresponding B-splines are entirely specified by their poles and zeros; they are compactly supported, have an explicit analytical form, and generate multiresolution Riesz bases. Moreover, they satisfy generalized refinement equations with a scale-dependent filter and lead to a representation that is dense in L2 L _{ 2 } . This allows us to specify a corresponding family of semi-orthogonal exponential spline wavelets, which provides a major extension of earlier polynomial spline constructions. These wavelets are completely characterized, and it is proven that they satisfy the following remarkable properties: 1) they are orthogonal across scales and generate Riesz bases at each resolution level; 2) they yield unconditional bases of L2 L _{ 2 } -either compactly supported (B-spline-type) or with exponential decay (orthogonal or dual-type); 3) they have N vanishing exponential moments, where N is the order of the differential operator; 4) they behave like multiresolution versions of the operator L from which they are derived; and 5) their order of approximation is (N − M), where N and M give the number of poles and zeros, respectively. Last but not least, the new wavelet-like decompositions are as computationally efficient as the classical ones. They are computed using an adapted version of Mallat's filterbank algorithm, where the filters depend on the decomposition level

    Wavelet Theory Demystified

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    In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in the Lp L _{ p } -sense and a sharper theorem stating that smoothness implies order

    Complex B-Splines

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    We propose a complex generalization of Schoenberg's cardinal splines. To this end, we go back to the Fourier domain definition of the B-splines and extend it to complex-valued degrees. We show that the resulting complex B-splines are piecewise modulated polynomials, and that they retain most of the important properties of the classical ones: smoothness, recurrence, and two-scale relations, Riesz basis generator, explicit formulae for derivatives, including fractional orders, etc. We also show that they generate multiresolution analyses of L2 L ^{ 2 } (R) and that they can yield wavelet bases. We characterize the decay of these functions which are no-longer compactly supported when the degree is not an integer. Finally, we prove that the complex B-splines converge to modulated Gaussians as their degree increases, and that they are asymptotically optimally localized in the time-frequency plane in the sense of Heisenberg's uncertainty principle

    On Fresnelets, interference fringes, and digital holography

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    In this thesis, we describe new approaches and methods for reconstructing complex-valued wave fields from digital holograms. We focus on Fresnel holograms recorded in an off-axis geometry, for which operational real-time acquisition setups readily exist. The three main research directions presented are the following. First, we derive the necessary tools to port methods and concepts of wavelet-based approaches to the field of digital holography. This is motivated by the flexibility, the robustness, and the unifying view that such multiresolution procedures have brought to many applications in image processing. In particular, we put emphasis on space-frequency processing and sparse signal representations. Second, we propose to decouple the demodulation from the propagation problem, which are both inherent to digital Fresnel holography. To this end, we derive a method for retrieving the amplitude and phase of the object wave through a local analysis of the hologram's interference fringes. Third, since digital holography reconstruction algorithms involve a number of parametric models, we propose automatic adjustment methods of the corresponding parameters. We start by investigating the Fresnel transform, which plays a central role in both the modeling of the acquisition procedure and the reconstruction of complex wave fields. The study of the properties that are central to wavelet and multiresolution analysis leads us to derive Fresnelets, a new family of wavelet-like bases. Fresnelets permit the analysis of holograms with a good localization in space and frequency, in a way similar to wavelets for images. Since the relevant information in a Fresnel off-axis hologram may be separated both in space and frequency, we propose an approach for selectively retrieving the information in the Fresnelet domain. We show that in certain situations, this approach is superior to others that exclusively rely on the separation in space or frequency. We then derive a least-squares method for the estimation of the object wave's amplitude and phase. The approach, which is reminiscent of phase-shifting techniques, is sufficiently general to be applied in a wide variety of situations, including those dictated by the use of microscopy objectives. Since it is difficult to determine the reconstruction distance manually, we propose an automatic procedure. We take advantage of our separate treatment of the phase retrieval and propagation problems to come up with an algorithm that maximizes a sharpness metric related to the sparsity of the signal's expansion in distance-dependent Fresnelet bases. Based on a simulation study, we suggest a number of guidelines for deciding which algorithm to apply to a given problem. We compare existing and the newly proposed solutions in a wide variety of situations. Our final conclusion is that the proposed methods result in flexible algorithms that are competitive with preexisting ones and superior to them in many cases. Overall, they may be applied in a wide range of experimental situations at a low computational cost
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