An Improvement of Geometric Correlation of Satellite Image
2. Statistical Correlation Measurement
The elements of f
1(j,k) and f
2(j,k) will be highly correlated spatially since f1 and f2 are the same scene and each spatially correlated to a significant extent for natural imagery. So the conventional correlation measurement, it is relatively difficult to distinguish the peak of R (u,v). The problem has been overcome by spatial filtering to decorrelate or "whiten" each image by whitening filter matrixes
[Hq]
-1 and [Hp]
-1 Let the column vector Q and P represent the image function
f
1(j,k) and f
2(j,k), respectively, when the image is scanned in a vertical raster fashion. Thus
So the whitening filter images matrix are
A =[HQ]-1 Q (4)
Bu,v = [Hp]-1 Pu,v (5)
Where H
Q and H
p are obtained by a factorization of the image covariance matrixes
CQ = HQHQT (6)
Cp = HpHpT (7)
Hence, HQ = EQLQ1/2
Hp = EpLp1/2
Where
LD
Q and
LP are diagonal matrix containing eigenvalues along the diagonal,
E
Q and E
P are composed of eigenvectors arranges in column form corresponding on each eigen values in
LQ and
Lp. For the statistical estimation, an image vector f can be considered to be a sample of a random process of know mean f and with a covariance matrix [4].
Where,
So,
The basic correlation operation (Eq. 1) is now performed on the whitened vector A and B, yielding the statistical correlation measurement.
Equation (13) can be reduced to
where G = (C
r)
-1. It should be noted that the decorrelation is now reduced to only one size images filtering with whitening filter matrixes, G. Anywhere, the whitening matrix (G) requires to the computation of two set of eigenvectors and eigenvalues of covariance matrix(C ), that is numerically difficult for all and large of computation.
Under the assumption of Markov Process Image, the row and column image elements are assumed to be samples of Markov process [5]. Hence, the image covariance matrix (C ) is given by
JXK is size of window area (Fig. 1) and p is the adjacent element correlation. The eigenvalues and eigenvectors can be found recursively [3] and whitening filter become to:
The multiplication of the image vector (Q) by whitening filter (G) is equivalent ot convolving the image. f
1(j,k) with the two dimensional function [2]
If the images are completely spatially unrelated, then p =0 the whitening filter becomes
Hence, the statistical measurement reduces to be the simple correlation measure. For the other extreme, the correlation factor p is equal to 1, thus
whereas, form as spatial discrete differentiation operation. Thus, as the images are highly correlated, the statistical correlation measurement concentrates on the edge outline comparison between the two scenes.
