Re: [WM]: DCT energy compression

[Cagatay Candan <gte479k@prism.gatech.edu>]

>Robert Gray "Toeplitz and Circulant Matrices: A review" document 
>available at www-isl.stanford.edu/~gray/toeplitz.pdf might give several

>clues for a theoretical proof. It gives an understandable explanation 
>of Szego works for "engineers lacking the background to attack 
>mathematical literature on the subject" like me.
>
>As far as I remember:
>Weakly stationary time series have toeplitz covariance matrices, 
>diagonal via KLT. These are asymptotically equivalent (in a certain
>sense) to circulant matrices, diagonal via Fourier transform.
>
>Hope this helps.
>Teddy

Hi Teddy,  I believe you have more than enough maths maturity for image 
processing,
if you can read/write things you did in the second paragraph.

I just want to emphasize a few fine points, which may confuse someone
here 
and there:

**KLT=PCA >> Diagonalizes the random processes (diagonalization means 
making process uncorrelated by a linear transform) applicable to all 
stationary processes. But the problem is diagonalization matrix depends
on 
the input random process.

**Yes, DFT diagonalizes circulant matrices.  But DCT does not have that 
feature. (Also covariance matrices are not circulant in general)

**Instead, DCT ``almost" diagonalizes covariance matrix of AR(1) process

with high correlation coefficient. (see Jain's book for the why/how 
question on this)

**AR(1) process with high correlation coefficient>> defined as 
y[n]=(corr_coeff)*y[n-1]+noise[n];
a sample run can be n={0,1,2,3,4}; corr_coef=0.9;  noise[n]={ 5,0,10,5}
=> 
y[n]={100,95,85,86,82}
so AR(1) produces slowly changing sequence of numbers which may
approximate 
the behavior of the pixels of an image in a small size block.

**Unfortunately DCT does not diagonalize this process for all
corr_coef's, 
but as corr_coeff ==> 1, DCT
gets better and better at approximation.

**This is the fundamental reason (I guess) DCT with 8x8 block size is 
selected for JPEG.

**finally, there can be other transforms for the ``approximate 
diagonalization", which may have better features for image coding 
(mainly,  less blockiness at low bit rates). These ones are also studied
in 
depth,  they lack the fast computation advantage of DCT, but
consistently 
beat DCT at almost any kind of comparison.

I hope I have managed to clarify a few things without messing up
something 
else.
Entropy as you know dictates that you can tidy up everything at the same

time,  :))

best wishes,
cagatay



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