View source for Talk:Wavelet

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As is the case with most engineering topics, this is dense in terminology bordering on buzzword-iness and unnecessary complication in order to sound "mathy." As a mathematician, let me tell you that engineers are the butt of our jokes for this reason.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/69.212.51.181|69.212.51.181]] ([[User talk:69.212.51.181|talk]]) 22:59, 20 September 2008 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:: I agree; this article starts off with non-useful generalizations and then goes into theoretical math that seems either obfuscated or not written by someone's who ever created a practical implementation of a wavelet.  Was someone just summarizing a math textbook?  I even understand FFTs, but this article did not help me to understand wavelets.  It seems to be some technique of DSP in the time domain, but it's utterly incomprehensible beyond that.  Math for the sake of math is not helpful.

Is this sentence for real?
:''The wavelets forming a CWT are subject to Heisenberg's uncertainty principle and, equivalently, discrete wavelet bases may be considered in the context of other forms of uncertainty principle.''
- [[User:Dcljr|dcljr]] 07:07, 4 Sep 2004 (UTC)

Okay, I did a little research and I see there is a connection to the HUP. The way it's mentioned in this article made it seem like a spurious reference. Whatever... - [[User:Dcljr|dcljr]] 03:11, 5 Sep 2004 (UTC)

:The same thing applies to the [[STFT]] as well.  It is pretty important, though I don't really understand it.  Apparently it means that you can't measure the instantaneous freuqency of a signal.  See [[Talk:STFT]] - [[User:Omegatron|Omegatron]] 17:12, July 18, 2005 (UTC)

Heisenberg uncertainty is a physics thing and has nothing to do with wavelet (or Fourier) transforms. What's needed is the equivalent to the [[Nyquist–Shannon sampling theorem]], I think. --[[User:David Cooke|David Cooke]] 21:03, 6 July 2006 (UTC)

:Actually, the Heisenberg uncertainty principle is mentioned in wavelet literature, and as discussed, it is a lot less mysterious than what has been presented in popular physics literature.  As Omegatron says, it does deal with the fact that there is no such thing as "instantaneous frequency" and its actually a very important consideration no matter what type of analysis you consider (Wavelet or Fourier).  The time/frequency representation utilized depends upon the type of signal you are looking for.

:Hmmmm, but maybe the literature is wrong.  [http://www.wavelet.org/phpBB2/viewtopic.php?p=11430 On this page on Wavelet.org] someone claims that the Uncertainty Principle of analysis and the Heisenberg Uncertainty Principle are indeed different.  But the explanations I've read tend to have (IIRC) similar if not the same formalizations.  Perhaps this is just getting into the nitty-gritty semantics and connotation of the Uncertainty Principle in signal analysis, whether it means just the formalization (which from my very uneducated eyes appears to be related to the related and derived Robertson-Schrödinger relation) or that of the original mathematical theory derived by Heisenburg without the implications of the physics involved.  ATM, I'm only using the wikipedia article (as reference) on or HUP or UP, which also globs it all into the same article.

:I feel like Clinton here.... 'It depends upon what your definition of "IS" is.'  [[User:Root4(one)|Root4(one)]] 21:37, 18 November 2006 (UTC)

:: Mallat's "A wavelet tour of signal processing", p.31 claims: <cite>Theorem 2.5 (Heisenberg Uncertainity): The temporal variance and the frequency variable of <math>f \in L^2(\mathcal{R})</math> satisfy <math>\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}</math>.</cite> I think the reference is pretty kosher. - [[User:Sesse|Sesse]] 23:10, 4 December 2006 (UTC)

Hmmm... It's a tricky one the HUP really refers to Quantum Mechanics / QFT. And the article it is right in that sense. And again it's true that this is often stated as one reason for "turning to wavelet analysis". However, in reality (and for applied areas), like has been stated the Nyquist-Shanon sampling rate is key, I.e. that bandwidth limited signals can be represented perfectly given they are sampled at a sufficiently fine rate. From a Physics point of view (and I guess more generally in a theoretical statistics PoC) the point in the article is fine. Though perhaps the article should stress that? Personally I think there should be more of a statistics slant on this article, wavelets are of growing importance in the subject, a mention or discription of non-parametric signal estimation. For those reading with no background in the area, it basically means if you do a DWT of some signal, say a vector <X>, you get back a set of discrete wavelet coefficients, say <d_{j,k}>, basically then you threshold the coefficients (so a cofficient is less than some limit set it to 0, or keep it as it is otherwise). Doing the inverse transform gives a "noise free" representation of the signal.  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/62.30.156.106|62.30.156.106]] ([[User talk:62.30.156.106|talk]]) 17:04, 7 February 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->


----

merge [[wavelet]] and [[wavelet transform]]?  pretty related... - [[User:Omegatron|Omegatron]] 20:15, Sep 29, 2004 (UTC)

----

In my last edit, I formulated a distinction between a wavelet transform, which acts on functions or continuous signals, and its implementation in the [[fast wavelet transform]] via [[filter bank]]s, which acts on coefficient sequences.--[[User:LutzL|LutzL]] 06:56, 31 May 2005 (UTC)

==Updating wavelet content==

Have merged in [[wavelet transform]] to this article.  I'm also attempting a revamp of the all the wavelet content on wikipedia as i feel there is a lot of missing information and not very easy to understand.  Having just completed a masters project on the subject (and knowing nothing about them when i started) i feel i'm in a good position to do this.

If anyone wants to help i've put together a list of what i think need doing (feel free to add to it).  In particular i don't have much knowledge on wavelets with continuous signals or complex wavelets so if someone could help with that.

[[User:Johnteslade|Johnteslade]] 10:24, 15 July 2005 (UTC)

{{todo}}

Just to warn you, I have already made two major sets of corrections to this page (the first in July 2004) and they have since evolved back into incorrect forms. I do not think it is worth being rigorous or careful in Wikipedia articles as your hard work will only be undone by someone who doesn't know what they're talking about. Just go for a rough guide (i.e. minimal maths) on the main page and give references to relevant books and papers that will discuss things properly and have been peer reviewed by accomplished individuals. It may be worth being more rigorous on side pages (my definitions of continuous wavelets are still intact, for example). - Jon Harrop (Masters, PhD and job in wavelets).

:: Just thinking out loud here. For a piecewise rectilinear function (e.g., an irregular saw-toothed function) defined over a finite domain, is there a dense, complete, orthonormal set of coordinate functions, also piecewise rectilinear, that can be used as basis for expansions?  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/69.158.109.40|69.158.109.40]] ([[User talk:69.158.109.40|talk]]) 13:51, 7 October 2007 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

The only sticking point of your program IMHO is that, you assume non-academics need to understand wavelets.
They donot have to know it, and if you water down the quality of an article, it would turn off people. Wikipedia
mathematics sections are quite technical, but donot compromise on the same "make lay person understand it" mission.
I would suggest we maintain wavelet article upto good technical quality, and also have a short 1-2 paragraph introduction
about applications of wavelets. I think most people are not able to understand wavelets without having any idea of
transforms or signal processing. We shouldn't be wasting much valuable time arguing on this, IMHO.
--Parasakthi ( பராசக்தி ) 15:04, 14 April 2009 (UTC)

== Loose terminology ==
Can a wavelet be ''any'' kind of time-limited oscillation?  If you just generate a sinusoid and multiply it by a [http://www.c3.lanl.gov/wibarf/shane.demo/webfiles/images/hanning.jpg cosine window] unrelated to any kind of transform, is that considered a wavelet, too?
:Theroretically yes, but using any waveform would probably make the output useless - with most of the coefecients just in one scale.  In the discrete transform, the wavelet is generally defined by the filters used and the wavelet is designed this way.  [[User:Johnteslade|john'''SLADE''']] [[User_talk:Johnteslade|(talk)]] 17:25, 18 July 2005 (UTC)
::I'm not talking about wavelet transforms.  I'm just talking about the word "wavelet" itself.  
::I guess a better thing to ask: Did the word "wavelet" exist as a reference to any time-limited oscillation before the transform was invented/discovered? - [[User:Omegatron|Omegatron]] 18:24, July 18, 2005 (UTC)
:::The word is due to Morlet/Grossmann in the early 80ies. However, they didn't call it wavelet, but french "ondelette", which means small wave. A little later it was transformed into english by translating "onde" into "wave", so you get wavelet. They used it first for the continuous transform, a version of the mathematical Calderon-Zygmund-theory from the middle of the century. In this context, any function in <math>L^1\cap L^2</math> that is a derivative of a function in the same space can be used as wavelet, even some more irregular ones. As indicated, for [[Multiresolution analysis|MRA]]-based diskrete WT, one needs functions of this space that are connected to a function satisfying a refinement equation. The [[Haar wavelet]] was invented by Alfred Haar in 1909, but obviously he didn't call it "wavelet".--[[User:LutzL|LutzL]] 06:16, 19 July 2005 (UTC)
:::See: I. Daubechies: [http://www.princeton.edu/~icd/publications/ "Where do wavelets come from?---A personal point of view"], article 74 in list, published in the Proceedings of the IEEE Special Issue on Wavelets 84 (no. 4), pp. 510--513, April 1996.--[[User:LutzL|LutzL]] 06:39, 19 July 2005 (UTC)

::::Gotcha.  So the original definition of the word was implicitly associated with the transform. - [[User:Omegatron|Omegatron]] 14:02, July 19, 2005 (UTC)

:::::Is the word sometimes used in another context then the transform? If not, can anyone motivate why two different articles are needed?
[[User:Mastomer|Mastomer]] ([[User talk:Mastomer|talk]]) 13:43, 20 January 2013 (UTC)

Is it possible to distill this discussion down and give a yes or no answer to the original question? That is: Can a wavelet be ''any'' kind of time-limited oscillation? Apologies, but I get a bit lost in some of the technical discussion, here. I'm also wondering if a sinusoid multiplied by a square function, so giving a finite duration sinusoid without tapering, can be considered to be a "wavelet". If so, then the opening sentence would have to be revised. Thanks, [[User:Isambard Kingdom|Isambard Kingdom]] ([[User talk:Isambard Kingdom|talk]]) 14:54, 14 August 2016 (UTC)

== QMF (Quadrature Mirror Filterbank?) ==

Having googled, I have found some wavelet definitions given as low-pass filterbanks, and I've found a definition of QMF for an even length filterbank (HP(n) = (-1)^n . LP(N-1-n), n = 0,1,...,N-1). But this could do with being included on this page or as another page called QMF. It would be nice to have a definition for an odd length filterbank too - I can't find that anywhere.

:Hi, QMF is not entirely correct, since it only applies, in the strict sense of the definition, to the Haar-Wavelet. Wavelet filter banks are CQF, conjugate quatrature filterbanks, since the analysis filterbank is the adjoint (wrt. the canonical hermitian structure on the spaces of discrete signals) to the synthesis filter bank. The difference, as I understand, is that in QMF, the same filters are used in analysis and synthesis, whereas in CQF, the analysis filters are time reversed wrt. the synthesis filters (and conjugate if complex).--[[User:LutzL|LutzL]] 06:38, 5 September 2005 (UTC)

== Mother Wavelet ==

Any chance of anyone simplifying the mathematical jargon used in this para so normal people (like me) can understand It?--[[User:Light current|Light current]] 22:56, 27 September 2005 (UTC)

On Mother wavelets, these are the functions that are "stretched" and "moved", in the Orthogonal case lets say you have two wavelet coefficients, you calculate the "inner product" between those two points and the first two points (x(1) and x(2))  in your data vector, then "shift" your wavelet you calculate the inner product of those too coefficients against x(3) and x(4)...and so on, that your first "finest resolution level". Next you stretch your wavelet so now you might have four wavelet coefficients, you you calculate the inner product of those against x(1):x(4), and so on, you do this for as along as you can, and you have your wavelet transform (discrete). So the mother wavelet is the non-shifted non-scaled "starting" wavelet.  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/62.30.156.106|62.30.156.106]] ([[User talk:62.30.156.106|talk]]) 17:18, 7 February 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

'''Using wavelet theory.'''
Does anyone know what the second half of this para means? Im sure there's some useful info there if someone can bring it out by explaining more simply.--[[User:Light current|Light current]] 20:51, 28 September 2005 (UTC)

== Jargon ==

WTF are L^1 and L^2?  Why assume that the reader has seen this as "L" before?
:Because they are standard notations? If the reader doesn't know about these, then he will also not understand the integrability condition they imply.--[[User:LutzL|LutzL]] 11:51, 4 October 2005 (UTC)
::More importantly, why bother using the proper notation when the actual content (the admissibility criterion from the resolution of the identity) is actually incorrect? - Jon Harrop
:::Could You please explain in broader terms which property is missing or wrongly stated? One could forget about the L1-condition, but only if one explains (here or elsewhere) that the Fourier transform as a unitary transform on L2 cannot always be expressed with the Fourier integral. Given this, the zero condition should read <math>\hat\psi(0)=0</math> and one could also give teh admissibility criterion. Or were You refering to the discrete transform, since I know such a "partition of unity" condition only in that case. You see, I'm slightly confused about where to place Your criticism. But I should be concerned, since most of the additions to the original (and incompletely stated) "mother wavelet" part were done by me. Perhaps You could visit the Daubechies wavelet article to point out mistakes or critical omissions there as well?--[[User:LutzL|LutzL]] 12:59, 1 November 2005 (UTC)
::::I believe you need to discuss the resolution of the identity and, in the context of a specific resolution of the identity, define the corresponding admissibility condition. IIRC, zero mean is a loose, formally incorrect but practically useful and commonly cited form of admissibility condition. Daubechies' "Ten Lectures on Wavelets" gives a few resolutions of the identity and discusses the corresponding admissibility criterions. Perhaps the utility of these different forms of resolution could also be discussed. It is probably also worth discussing the implications of using wavelets that fail to satisfy the admissibility condition, or only satisfy it approximately (e.g. the Morlet wavelet as Goupillaud used it), specifically that the transform is not likely to reverse well and time-frequency interpretations will suffer from a systematic error in the form of a "DC bias". Also, as time-frequency analysis is perhaps the most common application of the CWT, perhaps it would be better to list applications that benefit from vanishing moments rather than claim that "most situations" require vanishing moments. The statement "one prefers continuously differentiable functions with compact support" is surprising because compact support has little effect on the asymptotic algorithmic complexity of the transform but comes at a grave cost in terms of time-frequency uncertainty. I think it is important to avoid tainting the description with your personal bias. I tried to be unbiased but it is very difficult, of course, because wavelets are such an all-encompassing subject and people tend to approach wavelets from rather specific angles. - Jon Harrop

:You guys, people wont go to Wikipedia if they already know the subject. When you use <math>L^1</math> or <math>L^2</math> you could name the sets in english (p-power [[integrable function]]s) and link to [[lp space]] so people can wiki that. <small>anonymous post on 10 Dec 2005</small>

::Anonymous poster above is right! [[User:Linas|linas]] 17:38, 10 December 2005 (UTC)
:::WTF don't you write in the first place that the links are missing? And why aren't you putting them in place youselves, if you know which ones they are? 'Cause, 'tis wikipedia, you know.--[[User:LutzL|LutzL]] 09:37, 12 December 2005 (UTC)

== Commercial Abuse ==

The "Wavelet based time-frequency analysis in Mathematica" external link is a link to a commercial for a small Mathematica worksheet that was placed on this page by the author of that worksheet. This presumably is a clear violation of the [[Wikipedia:What_wikipedia_is_not]] guidelines. Similar violations by the same author also have occurred in the [[OCaml]] and [[Ray_tracing]] articles, cf. the associated discussion pages. - [http://www.cip.physik.uni-muenchen.de/~tf/ Thomas Fischbacher]

That link is to a tutorial page that demonstrates the application of time-frequency analysis to example signals from several classical subjects. The actual analyses are done using commercial products but the contents of the page are educational in their own right. Fischbacher's objections to my other contributions have now been ignored, perhaps because he has since abused them in an attempt to justify his [http://www.cip.physik.uni-muenchen.de/~tf/raytracer/ unusual views] by posting them on [http://groups.google.co.uk/group/comp.lang.lisp/msg/131c6a851987fb6a?hl=en& comp.lang.lisp] (where they were also ignored). See MarkSweep's comments in the history of the [[OCaml]] article. - Jon Harrop



Gee. Thanks for putting in my name and that reference to my web page, Jon. I must admit that I initially did not bother, but maybe should have. But, even more thanks for actually changing your previous commercial link that pointed to the actual product salesblurb to a more neutral one:

http://en.wikipedia.org/w/index.php?title=Wavelet&diff=27016379&oldid=25759232

However, don't you *think* it's just a little bit dishonest to do so and nevertheless try to give the impression on this discussion page as if this link always had pointed to that other tutorial page, and not to the merchandising page? Well, actually, it's not overly clever, at least. Wikipedia has version control (see above), you know.

As for the links you provided, should someone really be interested, I'd advise some googling to get more context, which is bound to give interesting insights into other out-of-context-quoting and history re-writing.

-- T.F.

#Please sign your posts with four tildes, like this: <nowiki>~~~~</nowiki>
#The external links section should be cleaned up and merged with the references section, i.e. have, author, title, date, at a minimum. This will discourage advert links. [[User:Linas|linas]] 17:38, 10 December 2005 (UTC)

== Wavelets are not necessarily orthonormal ==

From the article:
 In formal terms, this representation is a wavelet series, which is the coordinate representation of a square integrable function with respect
 to a complete, orthonormal set of basis functions for the Hilbert space of square integrable functions. Note that the wavelets in the JPEG2000
 standard are biorthogonal wavelets, that is, the coordinates in the wavelet series are computed with a different, dual set of basis functions.

Why bother to say that wavelets form an orthonormal basis, and then proceed to point out that there are wavelet families which are not orthogonal?

:Someone with the time to defend the changes could formulate that in most textbooks on wavelet theory there are only orthogonal wavelets. The theory of orthogonal wavelet series is, because of the relatively simple theory of orthonormal bases in separable Hilbert-spaces, much simpler than the theory of frames and stable bases that lies behind biorthogonal wavelets. But JPEG2000 is the most widely known application of wavelets, so a note of caution to the different nature of their wavelets should turn up in the introduction.--[[User:LutzL|LutzL]] 09:31, 9 January 2006 (UTC)

::Wrong. Computer scientists use wavelets for signal coding and tend to stick to orthonormal wavelet bases and discrete wavelets. Scientists and engineers often used the continuous wavelet transform (not orthogonal) for time-frequency analysis. This is described in detail in my PhD thesis but I am not allowed to cite it (the link was deleted as spam) because I am affiliated with my own PhD thesis. I was going to fix the article but the Wikipedia admins just convinced me that it would be a waste of my time. If you want accurate information on this (or any other) subject, I suggest that you do not read Wikipedia. [[User:Jdh30|Jon Harrop]] 04:48, 9 April 2007 (UTC)

:::Wrong to the square. The FBI fingerprint wavelet as well as the reference wavelets of JPEG2000, CDF 3/5 and 7/9, are biorthogonal. Computer scientists in image-processing prefer linear phase over orthogonality. For audio compression things are different, but then there is no widely known audio compression standard using wavelets. Wavelet based OFDM, where orthogonality is already in  the name, is a nice experiment, but has no widely used application at this time. Disclaimer: I do not live in Japan.--[[User:LutzL|LutzL]] 07:31, 30 April 2007 (UTC)

They are generally introduced as orthonormal transforms (well at least that's how I was introduced to them), but in reality the effects of shift invariance. So for example in statistics the MODWT and NDWT (Maximum Overlap, Non-Decimated respectively) transforms are often prefered.  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/62.30.156.106|62.30.156.106]] ([[User talk:62.30.156.106|talk]]) 17:23, 7 February 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Reorganization of the wavelet articles ==

Johnteslade announced something like that last year, there was lately some activity in restructuring content by HenningThielemann, believed to be identic to 134.102.210.237, but all not very convincing. I was giving someone on www.wavelet.org the advice to check the pages here for details of the wavelet theory. But had to admit (to myself) later that for someone that only know some calculus and wants a quick glance on how wavelet transforms are implemented, this information is unaccessibly hidden in the wikipedia-articles. So I would propose:
* To move this article to [[wavelet theory]] and 
** remove the focus on wavelet functions. Instead, it should be a short overview on the parts of wavelet theory.
** move and join the part of the [[continuous wavelet transform]] to its own article. It is ridiculous that this overview it has more detail on this topic than the specialized article.

* reformulate the [[discrete wavelet transform]] to be also an overview on this topic, 
** using [[wavelet series|wavelet basis]] and [[dual wavelet|dual wavelet basis]] to explain orthogonal and biorthogonal wavelet transforms
** pointing out that most practical transforms are constructed via a [[multiresolution analysis]] (MRA),
** that from this one gets a nice and fast implementation via the [[fast wavelet transform]] (FWT) algorithm.
** Then comes some difficult part where one should explain that working backwards from the requirements of the FWT one gets conditions on the filterbanks that can have solutions which also lead to a MRA. There should be an extra article [[Construction of wavelet scaling functions]], which re-joins the theoretical contents of the [[orthogonal wavelet]] and [[biorthogonal wavelet]] articles.
** next come the links to the examples [[Haar wavelet]], [[Daubechies wavelet]], [[Coifflet wavelet|Coiffman wavelet]] etc. for orthogonal transforms and [[Cohen-Daubechies-Feauveau wavelet]] etc. for symmetric biorthogonal transforms.

*The article on the [[fast wavelet transform]] should be enhanced by ready-to-implement pseudocode for both orthogonal and biorthogonal transforms. Obviously, this code should be removed from the overview on diskrete wavelets.

--[[User:LutzL|LutzL]] 14:31, 24 April 2006 (UTC)

== A sufficient condition for reconstruction (<math>L^2(\R)</math>) ==

I'd appreciate a reference or proof sketch for the following claim:

<blockquote>
A sufficient condition for the reconstruction of any signal ''x'' of finite energy by the formula
:<math> x(t)=\sum_{m\in\Z}\sum_{n\in\Z}\langle x,\,\psi_{m,n}\rangle\cdot\psi_{m,n}(t)</math>
is that the functions <math>\{\psi_{m,n}:m,n\in\Z\}</math> form a [[frame of a vector space|tight frame]] of <math>L^2(\R)</math>.
</blockquote>

Thanks. --[[User:Rezarob|Reza Rob]] 03:31, 4 January 2007 (UTC)

:I included the following proof in the article but [[User:Gareth_Owen]] removed it because it was "not really appropriate for an encyclopedia article". If you are still interested here is the proof.

<i>Proof</i>. The definition of tight frame gives
:<math>\langle x,x\rangle=\sum_{m,n}\langle x,\psi_{m,n}\rangle^2</math>
for any <math>x\in L^2(\R)</math>.
By using this equation for <math>x+y</math> with <math>x,y\in L^2(\R)</math>, we get
:<math>\langle x+y,x+y\rangle=\sum_{m,n}\langle x+y,\psi_{m,n}\rangle^2
=\sum_{m,n}\langle x,\psi_{m,n}\rangle^2+\langle y,\psi_{m,n}\rangle^2+2\langle x,\psi_{m,n}\rangle\langle y,\psi_{m,n}\rangle.
</math>
On the other hand, we have
:<math>\langle x+y,x+y\rangle=\langle x,x\rangle+\langle y,y\rangle+2\langle x,y\rangle,</math>
and using the definition again, we infer
:<math>\langle x,y\rangle=\sum_{m,n}\langle x,\psi_{m,n}\rangle\langle y,\psi_{m,n}\rangle=\langle y,\sum_{m,n}\langle x,\psi_{m,n}\rangle\psi_{m,n}\rangle,</math>
for any <math>x,y\in L^2(\R)</math>.
Now an application of the [[Riesz representation theorem]] completes the proof.
[[User:Temurjin|Temur]] 16:30, 7 June 2007 (UTC)

== The Introduction ==

Shouldn't the Introduction actually do some "introducing" for the layman? It seems a little stilted in its language - it should be a broader statement with a brief description of the capabilities of Wavelet Theory.

::The use of continuous wavelets in science and engineering is covered in detail in my PhD thesis. This article is (was) a very cut down version of that. Refer to my thesis for a good introduction to that aspect of this subject. [[User:Jdh30|Jon Harrop]] 04:50, 9 April 2007 (UTC)

::: There should be a more accessible introduction in general, not only of the applications but what a wavelet is, why it is useful and how the transform works. Compare with the [http://web.archive.org/web/20120201024041/http://www.amara.com/IEEEwave/IEEEwavelet.html Introduction to wavelets] paper.  The intro goes from general to specific, and so much becomes clear from the first 4 sentences!  -[[User:Pgan002|Pgan002]] 23:41, 29 April 2007 (UTC)


:::: The best introduction to wavelets (IMO) is the MIT open ware course on Engineering mathematics that has video lectures. After that, really you want to get guidance from your academic supervisor. Wavelets theory is taught VERY differently depending on your subject area (wavelets know span everything from CS - Geography). This article is pretty technical, but saying that, it's pretty decent too.  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/62.30.156.106|62.30.156.106]] ([[User talk:62.30.156.106|talk]]) 17:27, 7 February 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Problem in Introduction ==

The introduction asserts that wavelets divide a signal into different frequency components. This is incorrect. That's what Fourier analysis does. Wavelets cut up a signal into different "scale" components, not frequency. Frequency is only defined in terms of complex exponentials, any link between scales and frequencies is intuitive and heuristic, not precise. More information can be found at M.B.Priestly's 1995 paper titled "Wavelets and Time dependent spectral analysis".  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/203.167.251.186|203.167.251.186]] ([[User talk:203.167.251.186|talk]]) 22:52, 8 January 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:Then why didn't you change it? It's a perfectly reasonable point. One could even say that a challenge in the design part of wavelet theory is to come up with transforms where the scale components behave approximately as frequency components. Note that a frequency component always spans a certain frequency range.--[[User:LutzL|LutzL]] ([[User talk:LutzL|talk]]) 12:14, 9 January 2009 (UTC)

== A worthy source for non-mathematicians ==

... at least, was as article in Scientific American specifically about wavelets. It was clear, and taught me enough to hold me for more than a decade. Sorry to say, I don't know when it was published, but Gerard Piel was alive and well, just about sure. Back then, Sci. Am. presumed that people had minimal difficulty reading and attention spans with durations of several minutes at least instead of (more likely) a few seconds at best. Nevertheless, graphics were not neglected.

Consider that most Wikipedia readers might not have any idea what "orthogonal" means, and fewer might think of lines in a plane that are at right angles. However, when one refers to orthogonal functions, you lose a lot of people.

Another source I found very helpful was the sample illustrations in the data sheet for an Analog Devices complex IC devoted (iirc!) to wavelet decomposition and reconstruction of images. It was most interesting to see, as graphic images, the different "levels" of wavelets created by wavelet analysis. (I'm probably mistaken in assuming similarity between decomposition and analysis.)

Of course, copyright is most important in Wikipedia, but what can't be copyrighted is an understanding and an approach to explaining a topic.

To the author[s], I'd say please try to keep in mind what a reader who is curious about the topic is likely to know. Sad to say, even Fourier transforms are probably something totally unfamiliar. ("What's an FFT?")

I had higher hopes of learning something more than I did.

The reason I came to this article was the BBC news item about Google using image similarity, also being aware of the Linux application imgSeek, which apparently uses wavelet decomposition of images (and similarity of the wavelets?) to locate similar images.

I'm pondering whether wavelets and solitons are related, btw.

I do wish you all well!
Regards, [[User:Nikevich|Nikevich]] ([[User talk:Nikevich|talk]]) 17:23, 21 April 2009 (UTC)

::Hi, You should probably read on under [[Discrete Wavelet Transform]], which deals with image compression etc. Sadly, from a didactical point of view, the collection of wavelet articles is in a lamentable state, even if all the important technical information is present somewhere. Is "orthogonal" for "at right angles" really that esoteric? I thought that phrases like "orthogonal to reality" had entered popular culture. I'm not so sure about "perpendicular" as a synonym. It would be nice if You could find references for all your sources.--[[User:LutzL|LutzL]] ([[User talk:LutzL|talk]]) 07:57, 22 April 2009 (UTC)

So, if I'm reading the introduction to this article correctly, a wavelet is a pulse? That's what I think it says; 'starts at zero, rises, falls to zero again'; a pulse, right? So doesn't that description conflict with the subsequent comparison with seismic waves, or EKG, where the amplitude crosses back and forth across zero? Am I again proving to be just too psychically incompetent to correctly divine what's trying to be said?[[User:Trigley|Trigley]] ([[User talk:Trigley|talk]]) 13:50, 7 June 2009 (UTC)
:No and yes. The theory of continuous wavelets allows for a wide class of functions to be basis functions. In practical applications a wavelet has compact support, i.e. it starts at zero, rises, oszillates a small number of times and then falls again to zero. The integral over all oszillations is zero. There are graphs of examples of widely used wavelets somewhere in the article.--[[User:LutzL|LutzL]] ([[User talk:LutzL|talk]]) 10:05, 8 June 2009 (UTC)

::I see your reasoning for modifying the introduction, but still feel your wording was too imprecise, even for a non technical definition. If it begins at zero and then oscillates is there any reason to say it increases and decreases also? Must amplitude always evoke modulation? Is "ripple" not a bit of a colloquialism? I propose the following; "Non-technically a '''wavelet''' is is a curve whose amplitude is zero on either side of a compact region of oscillation, similar to the pulses on a seismograph or ECG." Is this sufficiently descriptive and succinct for the layperson?  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/97.83.161.77|97.83.161.77]] ([[User talk:97.83.161.77|talk]]) 16:31, 9 June 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== head section ==

The head section starts out with a long comparison of wavelets to musical notes. While this might be useful for understanding wavelets I don't think the head section is the place for it. A dedicated section after a more formal introduction would be more appropriate. [[User:Male1979|Ben]]  <sup>[[User_talk:Male1979|T]]</sup>/<sub>[[Special:Contributions/Male1979|C]] </sub> 10:40, 26 April 2010 (UTC)
: I agree. The intro is long and rambling and doesn't really get to the point. [[User:BenFrantzDale|—Ben FrantzDale]] ([[User talk:BenFrantzDale|talk]]) 12:43, 26 April 2010 (UTC)

::Just some comments on the introduction.
:: {{quotation| A wavelet is a wave-like oscillation with an amplitude that starts out at zero(0), increases, and then decreases back to zero.}}
:: I think this sentence excels in vagueness. I don't have any idea what the author is trying to say.

::{{quotation|A set of "complementary" wavelets will deconstruct data without gaps or overlap so that the deconstruction process is mathematically reversible.|| }} 
::So wavelets are items that deconstruct data without gaps or overlap? (I guess this is false but it is just what I read.)

::{{quotation|In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of a vector space, for the Hilbert space of square integrable functions.}}
::What representation? This sentence is looks like a mathematical nightmare, but it was the first sentence of which I could guess what the author was trying to say. "You can use wavelets to decompose a function into a serie of functions using some abstract mathematical concepts and considering some conditions." Above that he provided a link to a much more useful article about the wavelet transform. Which was helpful.
::So I totaly agree that the head is crappy.
[[User:Mastomer|Mastomer]] ([[User talk:Mastomer|talk]]) 13:19, 20 January 2013 (UTC)

== Wavelet not in plural? ==

This article is too mathematical and doesn't explain someone who doesn't want to go into the details what "waveletS" are, why they return in so many contexts, and what the core ideas to hold onto are. Also, the fact that the title of the article is "Wavelet" instead of "Wavelets" is already a disappointment on its own: In reality, they're always called "Wavelets", and what interests me is what "Wavelets" are about, not what a single "Wavelet", which is a more detailed underlying mathematical concept, is.[[Special:Contributions/193.190.253.144|193.190.253.144]] ([[User talk:193.190.253.144|talk]]) 22:09, 14 June 2010 (UTC)
:I'll agree that it's too mathematical, but they're kindof a pain to describe, so if you know how to improve it, by all means go ahead. They aren't always called wavelets, see for example the [[Haar wavelet]]. And I think the goal should be to describe both the underlying concept and how they're used - think of it like a broad overview article in line with [[Animal]]. [[User:VernoWhitney|VernoWhitney]] ([[User talk:VernoWhitney|talk]]) 22:22, 14 June 2010 (UTC)

== Suggest deleting "Name" section ==

What is the purpose of the section [[Wavelet#Name]]?  As acknowledged in [http://dspace.mit.edu/bitstream/handle/1721.1/43934/37923971.pdf?sequence=1 Thomas Colthurst's 1997 MIT Ph.D. thesis], the word "wavelet" is over two centuries old, you can see it for example in Book VIII of Shelley's [http://knarf.english.upenn.edu/PShelley/mab8.html ''Queen Mab''], "Like the vague sighings of a wind at even That wakes the wavelets of the slumbering sea", or in [http://archive.org/stream/notesonenglishd00colegoog/notesonenglishd00colegoog_djvu.txt Coleridge's ''Literary Remembrances'' III], "You only hide it by foam and bubbles, by wavelets and steam-clouds, of ebullient rhetoric."  In seismology [[Seismic inversion#Wavelet estimation]] concerns the problem of resolving multiple seismic reflections of a pulse generated by an explosive charge in order to identify the surface(s) that the pulse bounced off.  In physics the constituent waves of a [[wave packet]] are sometimes referred to as wavelets, for example as in Nappo's [http://books.google.com/books?id=P3g3HAk-CTkC&pg=PA19&lpg=PA19&dq=%22wave+group%22+wavelet&source=bl&ots=2G7f1pcccl&sig=OA3DZ_T0sejI98t5ebIa91vJIWU&hl=en&sa=X&ei=x5t6UuC8A4rhiAL8zoCoCQ&ved=0CCsQ6AEwADgK#v=onepage&q=%22wave%20group%22%20wavelet&f=false ''An Introduction to Gravity Waves''].

None of these uses, whether in poetry, physics, seismology, etc. have anything to do with using wavelet transforms to analyse an arbitrary signal as a sum of wavelets.  The fact that the word "wavelet" was in the English language (and presumably "ondelette" in French) centuries before the invention of the wavelet transform hardly seems to warrant even a remark, let alone an entire section, which furthermore conveys the impression that wavelets as treated in this article predate 1980, which is simply false. [[User:Vaughan Pratt|Vaughan Pratt]] ([[User talk:Vaughan Pratt|talk]]) 20:06, 6 November 2013 (UTC)

:What version of the article are you referring to? Once there was this questionable content that I was not innocent of. But the two sentences that stand now are, the way I read them, in accord with your arguments. One may merge them into the lede or into the history section, there is no mystery to solve in this name. Until proven otherwise, as mathematical object these functions were first introduced as "ondelettes" in french texts, with the translation of them into english they became, without further ado, "wavelets". But this is one of the minor problem of the whole wavelet section in wikipedia.--[[User:LutzL|LutzL]] ([[User talk:LutzL|talk]]) 22:05, 6 November 2013 (UTC)

Agreed.  A sentence in the history section about the origin of (the French for) "wavelet" should suffice.  [[User:Vaughan Pratt|Vaughan Pratt]] ([[User talk:Vaughan Pratt|talk]]) 06:11, 7 November 2013 (UTC)

== Close Paraphrasing in Applications of WNNs ==

The subsection on the applications of WNNs is closely paraphrased from https://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/AV0809/martinmoraud.pdf. The paper is open access but does not appear to have been released into the public domain. The same edit which contributed the text in question also contributed the first part of the WNN section, though I didn't find any suggestion this is also infringing from a quick search.

[[User:Frankplow|Frankplow]] ([[User talk:Frankplow|talk]]) 16:38, 17 October 2021 (UTC)