A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...

/pdf/ensemble-empirical-mode-decomposition-a-noise-assisted-data-4htvj7tcuy.pdf

Ensemble empirical mode decomposition: a noise-assisted data analysis method

1 Introduction to Wavelets 2 A Multiresolution Formulation of Wavelet Systems 3 Filter Banks and the Discrete Wavelet Transform 4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases 5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients 6 Regularity, Moments, and Wavelet System Design 7 Generalizations of the Basic Multiresolution Wavelet System 8 Filter Banks and Transmultiplexers 9 Calculation of the Discrete Wavelet Transform 10 Wavelet-Based Signal Processing and Applications 11 Summary Overview 12 References Bibliography Appendix A Derivations for Chapter 5 on Scaling Functions Appendix B Derivations for Section on Properties Appendix C Matlab Programs Index

Introduction to Wavelets and Wavelet Transforms: A Primer

We present the lifting scheme, a simple construction of second generation wavelets; these are wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included.

The lifting scheme: a construction of second generation wavelets

The article provides arguments in favor of an alternative approach that uses splines, which is equally justifiable on a theoretical basis, and which offers many practical advantages. To reassure the reader who may be afraid to enter new territory, it is emphasized that one is not losing anything because the traditional theory is retained as a particular case (i.e., a spline of infinite degree). The basic computational tools are also familiar to a signal processing audience (filters and recursive algorithms), even though their use in the present context is less conventional. The article also brings out the connection with the multiresolution theory of the wavelet transform. This article attempts to fulfil three goals. The first is to provide a tutorial on splines that is geared to a signal processing audience. The second is to gather all their important properties and provide an overview of the mathematical and computational tools available; i.e., a road map for the practitioner with references to the appropriate literature. The third goal is to give a review of the primary applications of splines in signal and image processing.

/pdf/splines-a-perfect-fit-for-signal-and-image-processing-50x3vrh3wj.pdf

Splines: a perfect fit for signal and image processing

[1] Data analysis has been one of the core activities in scientific research, but limited by the availability of analysis methods in the past, data analysis was often relegated to data processing. To accommodate the variety of data generated by nonlinear and nonstationary processes in nature, the analysis method would have to be adaptive. Hilbert-Huang transform, consisting of empirical mode decomposition and Hilbert spectral analysis, is a newly developed adaptive data analysis method, which has been used extensively in geophysical research. In this review, we will briefly introduce the method, list some recent developments, demonstrate the usefulness of the method, summarize some applications in various geophysical research areas, and finally, discuss the outstanding open problems. We hope this review will serve as an introduction of the method for those new to the concepts, as well as a summary of the present frontiers of its applications for experienced research scientists.

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2007RG000228

A review on Hilbert‐Huang transform: Method and its applications to geophysical studies

We consider solutions of a system of refinement equations written in the form formula math where the vector of functions Φ = (Φ 1 ,...,Φ r ) T is in (L p (R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Q a defined on (L p (R)) r by Q a f:= Σ α ∈ z a(α)f(2.-α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Q a n f) n=1,2... in the L p -norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L 2 -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

/pdf/vector-subdivision-schemes-and-multiple-wavelets-ckhsqz97kd.pdf

Vector subdivision schemes and multiple wavelets

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal functionL
 λ(x)
 
$$\sum\nolimits_j { \in z} ^{C_j } \exp ( - \lambda (x - j)^2 ),$$

 x∈R, satisfying the interpolatory conditionsL
 k = δ0k,k∈Z
 . One objective of this paper is to derive several additional properties ofL
 λ. For example, it is shown thatL
 λ possesses the signregularity property sgn[L
 λ(x)]=sgn[sin(πx)/(πx)],x∈R, and that |L
 λ (x)|≤2e
 8 min {(⌊|x|⌋+1)-1,exp(-λ⌊|x|⌋)},x∈R. The analysis is based on a simple representation formula forL
 λ and employs some methods from classical function theory. A second consideration in the paper is the Gaussian cardinal-interpolation operatorL
 λ, defined by the equation (L
 λy)(x):=
 
$$\sum\nolimits_{k \in z} {yk^L \lambda } (x - k)$$

 ,x∈R, y=(yk)k∈Z. On account of the exponential decay of the cardinal functionL
 λ,L
 λ is a well-defined linear map froml
 ∞ (Z) intoL
 ∞ (R). Its associated operatornorm ‖L
 λ‖ is called the Lebesgue constant ofL
 λ. The latter half of the paper establishes the following estimates for the Lebesgue constant: ‖L
 λ‖≍1, λ→∞, and ║Lλ║≍log(1/λ), λ→0+. Suitable multidimensional analogues of these results are also given.

On cardinal interpolation by Gaussian radial-basis functions: Properties of fundamental functions and estimates for Lebesgue constants

The Birkhoff interpolation polynomial $P(X,t)$, $0 \leqq t \leqq 1$ of a given function depends upon the knots of interpolation, $X:0 \leqq x_1 < x_2 < \cdots < x_m \leqq 1$. There is a natural extension of $P(X,t)$ to the case when some of the knots coincide. This extension is proved to be continuous. The method of proof is by “decoalescence”.

Continuity of the Birkhoff Interpolation

This paper presents a general construction of multidimensional interpolatory subdivision schemes. In particular, we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a three-direction box spline Br,r,r of equal multiplicities. The resulting mask for the interpolatory subdivision exhibits all the symmetries of the three-direction box spline and with this increased symmetry comes increased smoothness. Several examples are computed (for r = 2,...,8). Regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness.

Multidimensional Interpolatory Subdivision Schemes

Weak interpolation in Banach spaces

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