Re: few technical questions

> The current research in wavelet transform area focusses on incorporating
> "directionality" or "multi-directionality" into wavelet transform.
> Traditional 2D wavelet transform is builded by one 1D transform in
> horizontal and one 1D transform in vertical direction. After each transform
> signal is divided into two parts: the part with only directional information
> (vertical or horizontal) in some resolution, and the part without it.
 > Traditionally we treat the parts as a differencies and sums.

Hmm, what you described here is only one step of the complete wavelet 
transform. What it does is indeed a separation of the data into two 
chunks, which may also be called the average section and the detail 
section (sums and diffs is ok, but blurs the interpretation of what is 
really happening).

A full wavelet transform iterates this: The average section is 
transformed again, so you end up with the unchanged detail of say Level 
J-1 (if original image has dimension of about 2^J) and then the averages 
and details of level J-2. And so on till Level K which may be 0. The 
number of subimages you get are (J-K+1)^d (here d(imension)=2), so you 
get 16 with three substeps. They have different sizes and most of them 
are not quadratic. Only if you stop after one substep, they are.

The cool thing is that you can sort the coefficients by size, take as 
many of the largest ones as you want, and you are guaranteed that you 
have picked the right ones.

I apologize if you know all this.

> After wavelet transform of image we have 4 images, sums, vertical
> differencies,  horizontal diferencies, and vertical diferencies of
> horizontal diferencies, each with half of width and height of original.
> I propose an extension of this approach: after 2-level 1D transformation,
>  one - and one | ), the next transform on entire 4-images-block follows:
> exactly the same transform but in \ and / directions. \ transform processes
> data in diagonal direction eg. [0,0] then [1,1], then [2,2], etc. The /
> transform processes data in other-diagonal direction eg. [2,0], then [1,1],
> then [0,2]. They both are exactly normal wavelet transform.

The diagonals all have different lengths unless you continue them 
periodically. Is this intended (-> artefacts)? And I fear that you lose 
some approximation quality you get from a pure wavelet transform.

> In result we have 16 bands each with signal of different directionality.

I am positively curios how your method performs quality-wise against a 
three-level (or full) wavelet transform which also gets you 16 (or more) 
bands (and should be about equally fast) where you strictly go for 
coefficient size. Maybe better for diagonally dominant pictures and 
worse for vert/horiz symmetry (-> test examples)?

Regards,

Carsten


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