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

# Wavelets Question

Started by ●March 21, 2011

Reply by ●March 21, 20112011-03-21

On Mar 21, 11:06�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 > AaronGet 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.

Reply by ●March 21, 20112011-03-21

On Mar 21, 11:06�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 > AaronSee 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.

Reply by ●March 21, 20112011-03-21

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

Reply by ●March 22, 20112011-03-22

On Mar 21, 12:06�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 > AaronYou 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

Reply by ●March 22, 20112011-03-22