Dear all,

I want to design a my own transform basis matrix.

For 2D case, forward transform is: Y=A*X*B, inverse transform is:
Z=C*Y*D... let's say X, Y, Z, A, B, C, D are 8x8 matrices...

Can anybody tell me what relationship should A, B, C, D have?

1)Orthognality? 
2)A=B'? 
3)C=D'?
4)A=B^(-1)? 
5)C=D^(-1)? 
6)B=C?
7)A=D?
...

Thank you very much if you can also point me to some resources for
reference?

Thanks a lot,

-Mizhael

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Request for Question Clarification by livioflores-ga on 21 Aug 2003 20:17 PDT

Hi!!

I need some clarifications:
When you say "let's say X, Y, Z, A, B, C, D are 8x8 matrices", are
these matrices satisfies both equations:
Y=A*X*B
Z=C*Y*D  ?


In the affirmative, the answers does not depend on the "size" of the
matrices, the answer will be general for nxn matrices. So why you are
asking specifically for 8x8 matrices?

What "For 2D case" means?

What A' means? (please clarify the notation, is it the transposed
matrix?)

Thank you.
livioflores-ga

Sufficient conditions are

C=A ^ (-1)
B=D ^ (-1)

  Derivation:

 Z= C * Y * D = C * A * X * B * D

When conditions are satisfied,  C*A=I  and B*D =I

 therefore Z= I * X * I = X, so that transform is inverse.

I is unit matrix here.

References:

for matrix calculations in general
http://www.math.duke.edu/education/ccp/materials/linalg/index.html
and
http://www.numbertheory.org/book/

Mathematics for image compression
http://home.olemiss.edu/~lcao/bm_content.html

Image compression - techniques
http://academic.mu.edu/phys/matthysd/web226/L0423.htm


for advanced techniques:
http://www.geoffdavis.net/dartmouth/wavelet/wavelet.html

SEARCH TERMS
 none for the question,


 for references:

matrix algebra
2D linear filters
mathematics , image compression
linear algebra tutorial


hedgie

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Request for Answer Clarification by mizhael-ga on 28 Aug 2003 09:36 PDT

Hey, Hedgie,

Yeah, image compression is the direction. The thing is that I cannot
use any A and D, right? There must be some properties that A and D
should meet, right?

I saw most books talk about orthonomal/orthogonal matrix for A. But
why? I know orthogonality/unitarility can simplify the design, but is
that absolutely neccesary?

Given A is orthogonal, how do you design this 8x8 matrix? DCT2D is a
good option, but do we have others?

Given A is not orthogonal, how to design this matrix? from a11, a12,
a13, to... a88, then what are the B, C, D's?

Again, this is for image compression, having a quantization after the
transform...

---

Clarification of Answer by hedgie-ga on 13 Sep 2003 16:43 PDT

hi mizhael

Nice overview of the Image compression is here

   http://www.acm.org/crossroads/xrds6-3/sahaimgcoding.html

It covers the older DCT and evolution to the newer DWT.

 I would need more context if additional clarification is needed.

hedgie

There are a few other properties that you might want to include for
image compression transform.

Piotr Wasilewski: Image Processing
http://server.eletel.p.lodz.pl/~piotrwas/transforms.pdf

CS 6723 Image Processing, Lecture 8: Matrix Transformations 
http://vip.cs.utsa.edu/classes/cs6723f2001/lectures/lecture8.html

Image Compression and Packet Video
http://www.apl.jhu.edu/Notes/Beser/525759/icpvf02lect9-handout.pdf
The transform should
– The transform coefficients should have little correlation, so that
scalar quantization can be employed without losing too much in
coding efficiency compared to vector quantization
– Compact the energy into as few coefficients as possible

Video Communications
http://www.ece.umd.edu/class/enee631.F2001/ref/kjrliu_MPEGchapt1.pdf
(textbook chapter about video compression)

Hey Ulu,

The links are good... thanks for posting them to me... I printed them
out and digested them,

here is what I want, in fact I am trying to design my own transform
basis matrix, other than DCT, DHT, DWT, etc...

seems from all books orthogonality is a must, but why? there are good
reasons that the matrix cannot be othorgonal, say in implementation,
we will have round-off error.

if you take a DCT matrix, multiply it with a scaling factor, then
round off to integer, the matrix is no longer orthogonal... then I
need to modify here and there to restore some of its good properties
in order to do good image compression comparable to DCT compression...

now, let's generalize, we don't take DCT, just any matrix, how to
construct a 8X8 transform basis matrix that can also do image
compression(note after that transform we will have a quantization
step)...

I looked on the webpage, but found no way of designing such a11, a12,
a13, ... a88...

can you help me on that?