# Talk:Continuous wavelet transform

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## Some issues

There are multiple formulations of the continuous wavelet transform and of the resolution of the identity and corresponding admissibility criteria. The two most popular are Goupillaud's and Delprat's. The resolution of the identity is not an "inverse transform". I think the page should mention applications of the CWT and approaches to computation. -- Jon Harrop

"CWT-based time-frequency analysis has many benefits over other time-frequency methods (such as the short-time or windowed Fourier transform, Wigner-Ville and Choi-Williams distributions)." -- Why? This statement has not been supported this entry

My statement requires too much background to clarify in this article so I have added a citation. Jon Harrop 11:25, 9 April 2007 (UTC)
First, please demonstrate why this particular fact needs citation and how your own thesis as a source would meet WP:A. Femto 12:09, 9 April 2007 (UTC)
The "fact" needs citation because it is being questioned. I'll add references to random other pieces of work that back this up. Jon Harrop 09:07, 17 April 2007 (UTC)
I've replaced the deleted citation to one referencing Addison's book. It isn't as good but hopefully it won't get deleted... Jon Harrop 09:11, 17 April 2007 (UTC)

## Consistency with the Wavelet article

Hi, all. Is it ok if I change the notation in the article on continuous wavelet transform to make it consistent with that in the article on wavelets? The symbols used for the location and scale parameters in one article have been swapped with respect to those used in the other. If I don't get replies arguing against the change in a couple of days I'll just go ahead with it, although I am not sure whether this is proper. I will welcome being educated on the correct procedure. Regards, Prsmendonca (talk) 14:24, 31 January 2008 (UTC)

Feel free to edit. It's been a long time since substatial changes have been made in the whole wavelet complex.--LutzL (talk) 09:54, 1 February 2008 (UTC)
Done, sorry for the delay. Prsmendonca (talk) 14:24, 29 April 2008 (UTC)

## grammar question

The second sentence, "Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization," is ambiguous, in that it is impossible to know, without prior knowledge, whether the object of the adjectival phrase "offers very good time and frequency localization" is the time frequency representation of a signal, or the signal itself. Please clarify.

## Usefulness of the CWT being continuous ??

This article needs to make much clearer what is the value and relevance and usefulness of continuous wavelet transforms, as opposed to discrete wavelet transforms.

At the moment it is full of rather general statements like CWT allows time-frequency localisation; or CWT is used in image processing. But there is rather little about what precisely the value of the transform being continuous is. Jheald (talk) 12:01, 12 July 2011 (UTC)

One use I suppose is as a probe, to be able to pull out features with a particularly marked local freqency occurring at particular times/locations; and/or to look at a histogram for the amplitude of such features, binned across the whole time line (or space). And of course there's the visual scaleogram.
Whereas DWT is very useful for compression into a small number of components, then reassembly (synthesis). As our wavelet article puts it: "Generally, an approximation to DWT is used for data compression if signal is already sampled, and the CWT for signal analysis"; which I suppose is a good summary, though it would be nice to add more examples, detail and perspectives, if people have them. Jheald (talk) 17:14, 12 July 2011 (UTC)

## Duality formula

As far as I understand, the duality formula is missing a complex conjugate. If forward transform is

${\displaystyle X_{w}(a,b)={\frac {1}{\sqrt {|a|}}}\int _{-\infty }^{\infty }x(t)\psi ^{\ast }\left({\frac {t-b}{a}}\right)\,dt}$

and duality condition is

${\displaystyle \int _{0}^{\infty }\int _{-\infty }^{\infty }{\frac {1}{|a^{3}|}}\psi ^{\ast }\left({\frac {t_{1}-b}{a}}\right){\tilde {\psi }}\left({\frac {t-b}{a}}\right)\,db\ da=\delta (t-t_{1})}$

then, the inverse formula is derived trivially.

Am I right? Arkadi kagan (talk) 21:21, 25 November 2011 (UTC)

Aren't dual wavelets only for discrete wavelets? Why are they needed? Just normalize the original wavelet properly.Pierrecurie (talk) 00:11, 19 April 2014 (UTC)

## History

The article about George Zweig claims that he discovered the CWT. Should we have a detailed section on the history of this transform? Hamsterlopithecus (talk) 17:42, 24 May 2013 (UTC)

## Examples

I was hoping to see a list of examples and maybe a brief discussion of their applications and theoretical properties. There are three listed here: Wavelet#Continuous_wavelet_transforms_.28continuous_shift_and_scale_parameters.29. There are several more listed: http://www.ljmu.ac.uk/GERI/98293.htm and here Continuous_wavelet. Thanks in advance for whoever puts forth the time and effort to consolidate all this information onto one page. (Mouse7mouse9 22:09, 1 May 2014 (UTC)) — Preceding unsigned comment added by Mouse7mouse9 (talkcontribs)