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understanding wavelet transform in image processing....

Started by brettcooper September 5, 2012
Dear Forum,

I am learning about wavelets and image processing. I have seen how the
wavelet transform of an image looks like: it is a bunch of subimages that
seem low pass and high pass filtered versions of the original image....

That is not what I expected the wavelet transform to look like after
studying the continuous one-dimensional wavelet transform, which represents
the coefficients of the various wavelets once the mother wavelet is
chosen....

Does anyone have any hint on how to interpret the wavelet transform of an
image and all its subimages?

Is each of those filtered subimages representing wavelet coefficients? 
In the case of the Fourier transform, the transform itself does not look
anything like a filtered version of the original image itself...same goes
for the 1D continuous wavelet transform...

thanks
brett
On Wednesday, September 5, 2012 11:58:55 AM UTC-5, brettcooper wrote:
> Dear Forum, I am learning about wavelets and image processing. I have see=
n how the wavelet transform of an image looks like: it is a bunch of subima= ges that seem low pass and high pass filtered versions of the original imag= e.... That is not what I expected the wavelet transform to look like after = studying the continuous one-dimensional wavelet transform, which represents= the coefficients of the various wavelets once the mother wavelet is chosen= .... Does anyone have any hint on how to interpret the wavelet transform of= an image and all its subimages? Is each of those filtered subimages repres= enting wavelet coefficients? In the case of the Fourier transform, the tran= sform itself does not look anything like a filtered version of the original= image itself...same goes for the 1D continuous wavelet transform... thanks= brett Brett, If you're looking at images and wavelets, then you are using the discrete w= avelet transform, which is not the same "animal" as the continuous wavelet = transform. Where the continuous is a series of translations and dilations, = the discrete is a series of filter banks with disimation. The filter banks = form a low-pass and high-pass filter.=20 Look at block diagram of the discrete transform, and I think what you are = seeing will make sense to yo. Maurice
Thanks Maurice,
I followed your advice. But the continuous and the discrete wavelet
transforms must be somehow similar, right?

In 1D, I am able to see the relationship and how the discrete version
approximates the continuous WT. But in 2D it seems that the discrete
wavelet transform is just filtering at different scales...

Brett
On Thursday, September 6, 2012 9:29:16 AM UTC-5, brettcooper wrote:
> Thanks Maurice, I followed your advice. But the continuous and the discre=
te wavelet transforms must be somehow similar, right? In 1D, I am able to s= ee the relationship and how the discrete version approximates the continuou= s WT. But in 2D it seems that the discrete wavelet transform is just filter= ing at different scales... Brett The DWT is a series of filters. First do a low-pass and high-pass disimatin= g each ouypuy by 2. Store the high-pass. You now have 2 of your original im= age, each i/2 the size of the original. Now do the same thing again, but on= ly on the low-pass output. Again you have 2 images, but this time they are = 1/4 the size of the original. At this point, you have 3 images, one that is= 1/2 the original size, and 2 that are 1/4 the original size. Keep doing th= is for as loag as you want or can. The CWT and DWT are not the same animal. Get a copy of Mallot's paper on mu= lti-resolution. Does this help to show what you are seeing?
On Wednesday, September 5, 2012 6:58:55 PM UTC+2, brettcooper wrote:
> Dear Forum, > > > > I am learning about wavelets and image processing. I have seen how the > > wavelet transform of an image looks like: it is a bunch of subimages that > > seem low pass and high pass filtered versions of the original image.... > > > > That is not what I expected the wavelet transform to look like after > > studying the continuous one-dimensional wavelet transform, which represents > > the coefficients of the various wavelets once the mother wavelet is > > chosen.... > > > > Does anyone have any hint on how to interpret the wavelet transform of an > > image and all its subimages? > > > > Is each of those filtered subimages representing wavelet coefficients? > > In the case of the Fourier transform, the transform itself does not look > > anything like a filtered version of the original image itself...same goes > > for the 1D continuous wavelet transform... > > > > thanks > > brett
There is some history, wavelet poped up in the mind of various engineers, for different needs. One such is Morlet, for seismical echoes data for searching for oil. Continuous wavelets. Another is Daubechie, Orthogonal or biorthogonal wavelet, after the maths guys went cleaning the house and put common vocabularies on top of it. Oh she is one of them, BTW. Orthogonal or Bi-orthogonal also means a minimum basis able to reconstruct the signal with no or only a residue error you can still continue to encode. So you end-up with the dyadic grid. It all depends on the end needs.
On Wednesday, September 5, 2012 6:58:55 PM UTC+2, brettcooper wrote:

> > Is each of those filtered subimages representing wavelet coefficients?
yes
> > In the case of the Fourier transform, the transform itself does not look > > anything like a filtered version of the original image itself...same goes > > for the 1D continuous wavelet transform...
It's a to restricitve view of thinking of the filter function of wavelet as a brickwall filter or lengthy sinc for the lowpass, i.e., the design of the filter function is very specific for wavelet, comes with a different "cahier des charges" in french. Lowpass, highpass, is a shortcut word, a tad to approximative. There's a connection, at least in audio, btwn filter banks and wavelet. They are some prerequist conditions, though.
> thanks > > brett
Philippe,

Orthogonality and projections, except for 3D space is somewhat hard to get an intuition. For functions, we know all the smooth sine and cosine, but beyond that, also, intuition is somewhat lacking.

There's a brilliant* book that came out on wavelet recently:

wavelet
a concise guide
by amri-homayoon najmi
john hopkins university press

on page 111 and 112 you'll see an explanation using the haar function & wavelet. This is the departure in the book from the part about the continuous wavelet (morlet & co) to the orthogonal (mallat, daubechie & co) wavelets. Departure in the book, a connection, intersection about the whole wavelet topic, or both families.

Also, about orthogonality of functions, you may play a bit with the hilber xform under matlab on a compact test signal.


(* and thinner than the mallat one - a must have to impress the girls at the beach :^)


On Tuesday, September 11, 2012 5:08:53 PM UTC+2, kelvin....@gmail.com wrote:
> On Wednesday, September 5, 2012 6:58:55 PM UTC+2, brettcooper wrote: > > > > > > > > Is each of those filtered subimages representing wavelet coefficients? > > > > yes > > > > > > > > In the case of the Fourier transform, the transform itself does not look > > > > > > anything like a filtered version of the original image itself...same goes > > > > > > for the 1D continuous wavelet transform... > > > > It's a to restricitve view of thinking of the filter function of wavelet as a brickwall filter or lengthy sinc for the lowpass, i.e., the design of the filter function is very specific for wavelet, comes with a different "cahier des charges" in french. > > > > Lowpass, highpass, is a shortcut word, a tad to approximative. > > > > There's a connection, at least in audio, btwn filter banks and wavelet. They are some prerequist conditions, though. > > > > > thanks > > > > > > brett
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