# Wavelets Question

Started by 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

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