Talk:Wavelet

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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.

- 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... - 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 - 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. --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. 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.' Root4(one) 21:37, 18 November 2006 (UTC)
Mallat's "A wavelet tour of signal processing", p.31 claims: Theorem 2.5 (Heisenberg Uncertainity): The temporal variance and the frequency variable of f \in L^2(\mathcal{R}) satisfy \sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}. I think the reference is pretty kosher. - 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. —Preceding unsigned comment added by 62.30.156.106 (talk) 17:04, 7 February 2008 (UTC)


merge wavelet and wavelet transform? pretty related... - 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 banks, which acts on coefficient sequences.--LutzL 06:56, 31 May 2005 (UTC)

Contents

[edit] 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.

Johnteslade 10:24, 15 July 2005 (UTC)

To-do list for Wavelet: edit · history · watch · refresh

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? —Preceding unsigned comment added by 69.158.109.40 (talk) 13:51, 7 October 2007 (UTC)

[edit] Loose terminology

Can a wavelet be any kind of time-limited oscillation? If you just generate a sinusoid and multiply it by a 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. johnSLADE (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? - 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 L^1\cap L^2 that is a derivative of a function in the same space can be used as wavelet, even some more irregular ones. As indicated, for 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".--LutzL 06:16, 19 July 2005 (UTC)
See: I. Daubechies: "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.--LutzL 06:39, 19 July 2005 (UTC)
Gotcha. So the original definition of the word was implicitly associated with the transform. - Omegatron 14:02, July 19, 2005 (UTC)

[edit] 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).--LutzL 06:38, 5 September 2005 (UTC)

[edit] Mother Wavelet

Any chance of anyone simplifying the mathematical jargon used in this para so normal people (like me) can understand It?--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. —Preceding unsigned comment added by 62.30.156.106 (talk) 17:18, 7 February 2008 (UTC)

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.--Light current 20:51, 28 September 2005 (UTC)

[edit] 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.--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 \hat\psi(0)=0 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?--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 L1 or L2 you could name the sets in english (p-power integrable functions) and link to lp space so people can wiki that. anonymous post on 10 Dec 2005
Anonymous poster above is right! 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.--LutzL 09:37, 12 December 2005 (UTC)

[edit] 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. - 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 unusual views by posting them on 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.

  1. Please sign your posts with four tildes, like this: ~~~~
  2. 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. linas 17:38, 10 December 2005 (UTC)

[edit] 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.--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. 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.--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. —Preceding unsigned comment added by 62.30.156.106 (talk) 17:23, 7 February 2008 (UTC)

[edit] 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.
  • 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.

--LutzL 14:31, 24 April 2006 (UTC)

[edit] A sufficient condition for reconstruction (L^2(\R))

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

A sufficient condition for the reconstruction of any signal x of finite energy by the formula

x(t)=\sum_{m\in\Z}\sum_{n\in\Z}\langle x,\,\psi_{m,n}\rangle\cdot\psi_{m,n}(t)
is that the functions \{\psi_{m,n}:m,n\in\Z\} form a tight frame of L^2(\R).

Thanks. --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.

Proof. The definition of tight frame gives

\langle x,x\rangle=\sum_{m,n}\langle x,\psi_{m,n}\rangle^2

for any x\in L^2(\R). By using this equation for x + y with x,y\in L^2(\R), we get

\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.

On the other hand, we have

\langle x+y,x+y\rangle=\langle x,x\rangle+\langle y,y\rangle+2\langle x,y\rangle,

and using the definition again, we infer

\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,

for any x,y\in L^2(\R). Now an application of the Riesz representation theorem completes the proof. Temur 16:30, 7 June 2007 (UTC)

[edit] 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. 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 Introduction to wavelets paper. The intro goes from general to specific, and so much becomes clear from the first 4 sentences! -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. —Preceding unsigned comment added by 62.30.156.106 (talk) 17:27, 7 February 2008 (UTC)