scholarly journals Introduction to redundancy rules: the continuous wavelet transform comes of age

2018 ◽  
Vol 376 (2126) ◽  
pp. 20170258 ◽  
Author(s):  
Paul S. Addison
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
State Of The Art ◽  
Structural Characteristics ◽  
Continuous Wavelet ◽  
Theme Issue ◽  
Current State ◽  
Analysis Methods ◽  
Wide Range ◽  
New Form

Redundancy: it is a word heavy with connotations of lacking usefulness. I often hear that the rationale for not using the continuous wavelet transform (CWT)—even when it appears most appropriate for the problem at hand—is that it is ‘redundant’. Sometimes the conversation ends there, as if self-explanatory. However, in the context of the CWT, ‘redundant’ is not a pejorative term, it simply refers to a less compact form used to represent the information within the signal. The benefit of this new form—the CWT—is that it allows for intricate structural characteristics of the signal information to be made manifest within the transform space, where it can be more amenable to study: resolution over redundancy. Once the signal information is in CWT form, a range of powerful analysis methods can then be employed for its extraction, interpretation and/or manipulation. This theme issue is intended to provide the reader with an overview of the current state of the art of CWT analysis methods from across a wide range of numerate disciplines, including fluid dynamics, structural mechanics, geophysics, medicine, astronomy and finance. This article is part of the theme issue ‘Redundancy rules: the continuous wavelet transform comes of age’.

scholarly journals Quantitative feature analysis of continuous analytic wavelet transforms of electrocardiography and electromyography

2018 ◽  
Vol 376 (2126) ◽  
pp. 20170250 ◽  
Author(s):  
Mark P. Wachowiak ◽  
Renata Wachowiak-Smolíková ◽  
Michel J. Johnson ◽  
Dean C. Hay ◽  
Kevin E. Power ◽  
...  
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Wavelet Transforms ◽  
Continuous Wavelet ◽  
Theme Issue ◽  
Practical Applications ◽  
Time Frequency ◽  
New Family ◽  
Physiological Processes ◽  
Analytic Wavelet

Theoretical and practical advances in time–frequency analysis, in general, and the continuous wavelet transform (CWT), in particular, have increased over the last two decades. Although the Morlet wavelet has been the default choice for wavelet analysis, a new family of analytic wavelets, known as generalized Morse wavelets, which subsume several other analytic wavelet families, have been increasingly employed due to their time and frequency localization benefits and their utility in isolating and extracting quantifiable features in the time–frequency domain. The current paper describes two practical applications of analysing the features obtained from the generalized Morse CWT: (i) electromyography, for isolating important features in muscle bursts during skating, and (ii) electrocardiography, for assessing heart rate variability, which is represented as the ridge of the main transform frequency band. These features are subsequently quantified to facilitate exploration of the underlying physiological processes from which the signals were generated. This article is part of the theme issue ‘Redundancy rules: the continuous wavelet transform comes of age’.

scholarly journals Continuous wavelet transform and higher-order spectrum: combinatory potentialities in breath sound analysis and electroencephalogram-based pain characterization

2018 ◽  
Vol 376 (2126) ◽  
pp. 20170249 ◽  
Author(s):  
Leontios J. Hadjileontiadis
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Higher Order ◽  
Continuous Wavelet ◽  
Time Axis ◽  
Theme Issue ◽  
Third Order ◽  
The Third ◽  
Order Spectrum ◽  
Electroencephalogram Eeg

The combination of the continuous wavelet transform (CWT) with a higher-order spectrum (HOS) merges two worlds into one that conveys information regarding the non-stationarity, non-Gaussianity and nonlinearity of the systems and/or signals under examination. In the current work, the third-order spectrum (TOS), which is used to detect the nonlinearity and deviation from Gaussianity of two types of biomedical signals, that is, wheezes and electroencephalogram (EEG), is combined with the CWT to offer a time–scale representation of the examined signals. As a result, a CWT/TOS field is formed and a time axis is introduced, creating a time–bifrequency domain, which provides a new means for wheeze nonlinear analysis and dynamic EEG-based pain characterization. A detailed description and examples are provided and discussed to showcase the combinatory potential of CWT/TOS in the field of advanced signal processing. This article is part of the theme issue ‘Redundancy rules: the continuous wavelet transform comes of age’.

The continuous wavelet transform with rotations

2005 ◽  
Vol 4 (1) ◽  
pp. 45-55
Author(s):  
Jaime Navarro ◽  
Miguel Angel Alvarez
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Continuous Wavelet

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

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