380 
R(K,P) = E(I; om (Lak 34p m) = 75^, oes "s (k)s(p) (11) 
2.3 Image Transform Statistics 
  
Applying a two dimensional Karhunen-Loeve or Hotelling unitary 
transforamtion of size N? to the image defined in the preceeding 
paragraph, it can be shown (Ref. 1) that the first and second order 
statistics of the transform coefficients, Zn ? are repectively given by: 
O0 for nyt t 0 
E(25 ps (12a) 
2 mN for. n 2$ s 
n,2 0,....,N-1 
: 2 2 
EC(Zn,e-"n,2)(Zr,s-"p,5)? = {GA 8, +01 hhä(n«r) S68-8) 
farn,2= 0,.. 1.81 7 sit (12b) 
where: 
Uno ^ E(Z, 2) 
N-1 
d ; 2 2 2, 2 
nz m Engl / to 
2=0 
N-1 
s 24.2. 2, 2 
JUR. = EU 2" vin /No, 7p A 
What is learned from equation (12) is that: 
a) the transform coefficients are uncorrelated and their variances 
are product separable in row and column indices. 
b) the variance of the transform coefficients due to speckle is the 
same for all coefficients and hence the speckle component can be 
regarded as an additive uncorrelated random variable to the 
coefficient value given by the unspeckled image. 
The properties of the second order statistics given by (12b) are satisfied 
for a two dimensional cosine transform of transform size N — e and the 
variance of the transform coefficients is given by the two dimensional 
Fourier transform of the image covariance function given by (11). Hence, 
2 co oo Kmn mn 
Ya Z ZR exp-i{— *—1 (13) 
E(7, 77 
zm pz» P N N 
n, 
It follows for ^, and 8, defined in (12b): 
  
An = (14a) 
n z TN