Wavelets question (Specifically Daubechies, understanding scaling functions)

[aespielberg]

Hi all,

I have a question about Daubechies Wavelets. I've recently been trying to teach them to myself from the pdf here:

http://www.google.com/url?sa=t&rct=...xfjlCg&usg=AFQjCNF4Dd-42mv46JszPuQPM8Gui7TFyA

(sorry about the link...requires citeseer access...I tried to upload the pdf but it's too big by 0.5 MB)

I understand Daubechies wavelets are pretty much wavelets obeying certain normalization and orthogonality conditions, and most distinctly, a vanishing moments condition. I've read through that pdf and understand there's a process for getting the transfer functions for the transformed scaling functions.

What I don't understand is, how do you use this to go back to the original untransformed scaling functions? I know they can't be written explicitly, but what's the process (in other words, how do I get psi and phi for a given j, k pair)?

If anyone can explain it to me or point me to a good resource, I'd be really grateful.

 

[chiro]

Hey aespielberg and welcome to the forums.

When you say scaling functions I'm a little confused, but this is just because the notation I used was a little different.

Basically the way I was taught is that you have scaling functions which you use to get your wavelet function which you use to project your function to each basis function.

When you say unscaled are you referring to the 'original scaling function' and by transformed you mean the 'wavelet function'? If so does this mean you want to understand how to go from 'wavelet' back to 'scaling function' (using my terminology)?

Also in relation to your question, are you familiar with the detailed steps of a Multi-Resolution-Analysis (MRA)? I think that this will clear up anything you want to know with regards to your question (and will help answer other questions you may have).

 

[aespielberg]

Thanks for the welcome! I don't know why I haven't posted here earlier, I might start making this one of my more active forums.

I don't know, maybe I have my terminology messed up. As far as I understand, wavelets have:

-A series of basis functions, phi and psi, at a certain resolution level j. phi and psi span orthogonal spaces s.t. the union of those spaces are the space spanned by phi at j + 1. I thought that the phi were called scaling functions and that the coefficients were filter coefficients. It's possible I'm getting my terminology messed up

-A transform. phi has a phi-hat which is the Fourier transform of phi. The same goes for psi. It also has a coefficient which I thought was called the transfer function.

Anyway, the reason I'm asking actually is because I'm trying to understand MRA and the wavelet transform. I understand it's a discrete grid with k and j values, j representing the resolution and k the points on that grid. I guess I'm stuck understanding exactly what the steps are. Would you like to outline them for me or point me to a resource that explains it at a high level (and preferably has more detail as well after the outline)? A main confusion is that for the Haar wavelet, the basis functions are well defined, but for the Daubechies, they are not, so I don't know how one derives them.

Thanks for your help.

 

[chiro]

Thanks for the welcome! I don't know why I haven't posted here earlier, I might start making this one of my more active forums.

I don't know, maybe I have my terminology messed up. As far as I understand, wavelets have:

-A series of basis functions, phi and psi, at a certain resolution level j. phi and psi span orthogonal spaces s.t. the union of those spaces are the space spanned by phi at j + 1. I thought that the phi were called scaling functions and that the coefficients were filter coefficients. It's possible I'm getting my terminology messed up

-A transform. phi has a phi-hat which is the Fourier transform of phi. The same goes for psi. It also has a coefficient which I thought was called the transfer function.

Anyway, the reason I'm asking actually is because I'm trying to understand MRA and the wavelet transform. I understand it's a discrete grid with k and j values, j representing the resolution and k the points on that grid. I guess I'm stuck understanding exactly what the steps are. Would you like to outline them for me or point me to a resource that explains it at a high level (and preferably has more detail as well after the outline)? A main confusion is that for the Haar wavelet, the basis functions are well defined, but for the Daubechies, they are not, so I don't know how one derives them.

Thanks for your help.

The MRA is concerned with first building a scaling function with certain properties and then using that to generate the wavelet function that you already know how to use (exactly like you use for example the Haar wavelet).

With MRA, the idea is that you start with a 'coarse' version of your scaling function and then you fill in the gaps using the MRA identities.

The MRA tells you how to generate the scaling parameters and when you combine those with with the scaling function itself, you get your wavelet.

In terms of deriving the Debauchies wavelet, there is one page I recall that had a pretty good article on deriving the Debauchies wavelet: note that you can't derive it analytically! You get the first values by solving an eigenvalue problem and then you use the MRA formulas to get the rest of the scaling and hence the wavelet functions. Here is the page:

http://rip94550.wordpress.com/2009/05/07/the-dyadic-expansion-of-daubechies-d4-scaling-function/

The notes I have for wavelets are from the university and I can't distribute them freely unfortunately. Hopefully though at least for the Debauchies wavelet enquiry, the above will help.

 

[aespielberg]

Wow, that blog post cleared (almost) everything up! Thank you so much for linking to that!

I'll have to re-read the long windy derivation on how the h's are derived. Just to clarify, it's just the only set of h that allow for vanishing moments, while fulfilling other basic properties of wavelets, right?

Also, I guess the only other thing that doesn't make sense to me is...when you perform a wavelet decomposition, I was under the impression that what you were effectively doing was taking a function, and determining which coefficients would allow you to represent it most accurately using the wavelet basis functions (sort of projecting it onto the wavelet basis functions). If you have all the h values from the original decomposition, all the phi values, and all the psi values, what's left to solve for?