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Wavelets Question

Started by aadeb March 21, 2011
Hi 

I know this question may be rather simple but I am sturggling to
understand

I am trying to understand wavelets and I have the following question I am
looking at various DWT schemes and if I understood correctly 

The Image is filtered through the low and high frequencies along the rows
and coloumns respectively to produce the 4 combinations LL,HL,LH,HH. After
each stage say after high pass filtering along the coloumns you perform a
downsampling of factor 2 along the rows. Now my question is this.

I totally understand th subsampling issue in the LL case but what about the
HH case for example were you run the high pass filter along the coloumns
and the rows. You still have freuqency components in the upper part of your
spectrum along both dimensions of the image. How can you down sample that
(In time domain) and still not get aliasing ?

Am I missing something fundamental ?

Thanks and Regards
Aaron 


On Mar 21, 11:06&#2013266080;am, "aadeb" <aaronddebattista@n_o_s_p_a_m.gmail.com>
wrote:
> Hi > > I know this question may be rather simple but I am sturggling to > understand > > I am trying to understand wavelets and I have the following question I am > looking at various DWT schemes and if I understood correctly > > The Image is filtered through the low and high frequencies along the rows > and coloumns respectively to produce the 4 combinations LL,HL,LH,HH. After > each stage say after high pass filtering along the coloumns you perform a > downsampling of factor 2 along the rows. Now my question is this. > > I totally understand th subsampling issue in the LL case but what about the > HH case for example were you run the high pass filter along the coloumns > and the rows. You still have freuqency components in the upper part of your > spectrum along both dimensions of the image. How can you down sample that > (In time domain) and still not get aliasing ? > > Am I missing something fundamental ? > > Thanks and Regards > Aaron
Get a copy of Gil Strang's book: Wavelets and Filter Banks. In the applications chapter he addresses image and video compression, showing how the scan is performed for the DWT and for block transformation.
On Mar 21, 11:06&#2013266080;am, "aadeb" <aaronddebattista@n_o_s_p_a_m.gmail.com>
wrote:
> Hi > > I know this question may be rather simple but I am sturggling to > understand > > I am trying to understand wavelets and I have the following question I am > looking at various DWT schemes and if I understood correctly > > The Image is filtered through the low and high frequencies along the rows > and coloumns respectively to produce the 4 combinations LL,HL,LH,HH. After > each stage say after high pass filtering along the coloumns you perform a > downsampling of factor 2 along the rows. Now my question is this. > > I totally understand th subsampling issue in the LL case but what about the > HH case for example were you run the high pass filter along the coloumns > and the rows. You still have freuqency components in the upper part of your > spectrum along both dimensions of the image. How can you down sample that > (In time domain) and still not get aliasing ? > > Am I missing something fundamental ? > > Thanks and Regards > Aaron
See also Jensen and Cour-Harbo's book: Ripples in Mathematics - The Discrete Wavelet Transform. Chapter 6 covers two-dimesional Transforms. This book can give you some good insights into the DWT. After that, get a copy of a journal paper on the subject, and look at the references.
Thanks for the book suggestions will check them out. I was looking at "A
Primer on Wavelets and their Scientific Applications" but I suppose I need
something more in depth

Also I think I figured this out and I think ot is all based on the Scaling
property of the fourier transform.

Thanks 
On Mar 21, 12:06&#2013266080;pm, "aadeb" <aaronddebattista@n_o_s_p_a_m.gmail.com>
wrote:
> Hi > > I know this question may be rather simple but I am sturggling to > understand > > I am trying to understand wavelets and I have the following question I am > looking at various DWT schemes and if I understood correctly > > The Image is filtered through the low and high frequencies along the rows > and coloumns respectively to produce the 4 combinations LL,HL,LH,HH. After > each stage say after high pass filtering along the coloumns you perform a > downsampling of factor 2 along the rows. Now my question is this. > > I totally understand th subsampling issue in the LL case but what about the > HH case for example were you run the high pass filter along the coloumns > and the rows. You still have freuqency components in the upper part of your > spectrum along both dimensions of the image. How can you down sample that > (In time domain) and still not get aliasing ? > > Am I missing something fundamental ? > > Thanks and Regards > Aaron
You are correct - there is aliasing. However the high pass filter has removed the low frequency components. Therefore when the high pass components get aliased down - there isn't anything there to alias onto. A simple way to see this is to draw a spectrum of the high pass signal with a sampling rate of fs. Note that the spectrum will be periodic every fs Hz. Note also that the high pass spectrum will essentially zero out the frequencies from 0 to fs/4 (assuming a real signal i.e. not complex). Also the components from -fs/4 to 0 will also be zero - due to symmetry in a real signal. Now if the sampling rate is reduced to fs/2, then the spectrum will now be periodic every fs/2 Hz, but the original spectrum is still there. In order to keep these 2 conditions, you will see that the spectrum gets alias onto the frequency range 0 to fs/4. Hope that helps. Cheers, Dave
Yep .... that pretty much sums it up and confirms what I was thinking

Thanks for your reply
Aaron