The statistical correlation with invariant moments for
Geometric correction improvement
2. Statistical Correlation Measure
Let f1 and f2 are search and window images that come form the same scene. The elements of f1 (j,k) and f2(j,k) will be highly correlated spatially. So, the conventional correltion measure, it is relatively difficult to distinguish the peak of R(u,v). With the spatially filtering to decorrelate or "whiten", this problem will be removed. Let the column vector Q and P represent the image function f1(j,k) and f2(j,k), repectively, scaned in a vertical raster fashion.
Equation -2
So the whitening filtered images matrix are
Equation - 3
where HQ and HP are obtained by a factorization of the image covariance matrices.
Equation - 4
Hence,
Equation - 5
where ?Q and ?P are diagonal matrices containing eigen values along the diagonal, EQ and EP are composed of eigen vactors arranging in corresponding column form on each eigen values in ?Q and ?P The basic correlation operation (eq. 1) is now preformed on the whitened vector A and B, yielding the statistical correlation measure.
Equation - 6
which can be reduced to
Equation - 7
Where
Under the assumption of Markov Process Image, that is , the row and column image elements are assumed to be samples of Markov process (Pratt, 1972). Hence, the image covariance matrix, C is given by
Equation - 8
where JxK is size of window area (Figure 1) and .is the adjacent element correlation. The eigen values and eigen vectors can be found recursively and whitening filter become to (Arcess, 1970):
Equation - 9
Multiplication of the image vector(Q) by whitening filter (G) is equivalent to convolving the image f1(j,k) with the two dimensional function (Pratt, 1974)
Equation - 10
If the images are complety spatially unrelated, the .= 0. the whitening filter becomes
Equation - 11
Hence, the statistical measure reduce to the simple correlation measure. At the other extreme, is correlation factor .= 1, thus
Equation - 12
whereas, form as spatial discrete differentiation operation. Thus, while the images are highy correlated, the statistical correlation measure concentrates on the edge outline comparison between the two scenes.
