File:Continuous wavelet transform.gif

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Summary

Description
English: A (continuous) wavelet transform is given by the integral of the product between signal s and the kernel (wavelet) w, where the kernel is both shifted and stretched, allowing to extract information at different scales.
Date
Source https://twitter.com/j_bertolotti/status/1224296204173021184
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 11.0 code

wavelet[x_] := E^(-(x^2/2)) Cos[\[Pi] x]; (*Define your wavelet*)
signal = 1.3 E^(-((x + 3)^2/(2 0.2))) Cos[1.6 \[Pi] x] + 0.5 E^(-((x - 1)^2/(2 2))) Cos[0.5 \[Pi] x] - 0.5 E^(-((x + 0)^2/(2 4))) Cos[2 \[Pi] x]; (*Define your signal*)
tmp = Table[{\[CapitalDelta], \[Sigma], NIntegrate[signal*wavelet[(x - \[CapitalDelta])/\[Sigma]], {x, -20, 20}]}, {\[CapitalDelta], -6, 6, 0.05}, {\[Sigma], 0.1, 2.4, 0.2}]; (*Calculate the wavelet transform for a discrete set of shifts and stretchings*)
p1 = Table[
   Grid[{{
      Plot[{signal, wavelet[(x - tmp[[All, m]][[All, 1]][[l]])/tmp[[All, m]][[All, 2]][[1]]], signal*wavelet[(x - tmp[[All, m]][[All, 1]][[l]])/tmp[[All, m]][[All, 2]][[1]]]}, {x, -7, 7}, PlotRange -> All, 
       PlotStyle -> {Directive[Black], Directive[Red], Directive[Thin, Purple]}, Filling -> {3 -> Axis}, ImageSize -> Large, Ticks -> None, Axes -> {True, False}]
      ,
      Show[
       Table[
        ListPointPlot3D[If[n < m, tmp[[All, n]], tmp[[1 ;; l, n]]] , PlotRange -> {{-6, 6}, {0, 2.35}, {-0.7, 0.7}}, PlotStyle -> Directive[Black], Filling -> Bottom, FillingStyle -> Directive[White, Thickness[0.005] ], ViewPoint -> {1.3, 2.4, 2.}] /. Point[a___] :> {Thickness[0.015], Line[a]}
        , {n, 1, m}]
       , PlotLabel -> "\[Integral]s(x)*w(\!\(\*FractionBox[\(x - \[CapitalDelta]\), \(\[Sigma]\)]\)) dx", LabelStyle -> {Black, Bold, FontSize -> 18}, AxesLabel -> {"\[CapitalDelta]", "\[Sigma]", ""},        ImageSize -> Large ]
      }}]
   , {m, 1, Dimensions[tmp][[2]], 1}, {l, 1, Dimensions[tmp][[1]], 10}];
ListAnimate[Flatten[p1]]

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

Captions

The coninuous wavelet transform of a signal can be calculated by convolving the signal with a kernel whose argument is rescaled.

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3 February 2020

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current22:36, 3 February 2021Thumbnail for version as of 22:36, 3 February 2021892 × 373 (3.59 MB)Bürgerentscheidresized to fit 100 MP limit
08:59, 4 February 2020Thumbnail for version as of 08:59, 4 February 20201,161 × 486 (9.19 MB)BertoUser created page with UploadWizard
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