🚀 go-pugleaf

Thanks for the responses to my image correlation question which convinced 
me that the NMR article defines the correlation coefficient under the 
assumption of an image with zero mean even though they don't specify it.  
However, such an assumption doesn't hold for the ultrasound images I am 
working with.  

My problem is that when I take a 3rd order linear predictive equation:

U'_i,j = A1 * U_i-1,j + A2 * U_i,j-1 + A3 * U_i-1,j-1

and then minimize the expected value of the square prediction error:

E[(U_i,j - U'_i,j)^2]

I get equations for A1, A2, and A3 which have factors in them of the
form used in the NMR article to define rho(x,y), namely:

E(U_i,j U_i-x,j-y)
------------------
     E(U_i,j^2)

My dilemma then reduces to:

Can I safely use these A1, A2, and A3 with an image with a non-zero
mean???  

If the answer is no, then I understand the discrepancy 
between the two definitions of rho(x,y).  However, if the answer is
yes, which I don't see how it *couldn't* be given that all I am
doing is computing some image statistics, a linear predictor, and
coefficients to minimize the MSE of said predictor, then there is 
a definite discrepancy in the two definitions.

So am I missing something inherent in either my 3rd order linear
prediction equation or the process of minimizing the mean square
prediction error which constrains me to using images of zero mean?

Another thing which I have realized - when I compute rho(x,y) using
the NMR definition, I get numbers greater than 1 which implies gain
exists in the prediction.  When I compute rho(x,y) using the
definition which subtracts the square of the first moment from
numerator and denominator, I get numbers close to but never
exceeding 1.  To me, it is logical that a correlation coefficient
would lie between 0 and 1.  Is that an incorrect conclusion for me
to draw?

Thanks for any help you can give me!

Susan
--
Susan E. Margulies       
HP Imaging Systems       
508-681-2153 (work)      
susanm@hp-and.an.hp.com