The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
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among images (Kilic, S. et al, 2006). This method firstly
calculates a standard transformation matrix. Then, the new
image, namely principal component data, is obtained from the
input image based on the transformation matrix. At last, the
results of change detection may be acquired using image
difference method.
Figure 1. The constitution of the change polygon system
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Suppose a vector of n dimension image X = [*,, x 2 , • • •, x n ] 7 ,
which is performed using a linear transformation. Namely,
Y = Q T X
(1)
Where, Q = [Q,,Q 2 ,-.
. q ] is an orthogonal
matrix. The
covariance matrix is:
*
II
*
-E(X)I(X-E(X)r}
(2)
Because G. is a real and Symmetrie matrix, an orthogonal matrix
Q =[ß,,ß 2 , .ßj
consequentially exists,
and makes
0GQ become a diagonal matrix. That is:
orthogonal transformation is performed on original image, the
principal component Y = \Y ,Y ,-,Y ] T is obtained. The
12 M
change detection may be used to the second components
because the change targets are mainly displayed in the second
components transformed.
Canonical correlation analysis is a kind of the statistical
analysis methods that analyzes the linear relationship between
two random variables (Chen Lei et al, 2007). The idea means
that canonical transformation is applied to the image using the
different linear combination transformation, in which a
transformation of the largest correlation coefficient can be
found. In fact, the method can be converted to the problem of
solving eigenvalues and eigenvectors.
Suppose that the covariance of the random variable [x Y]
is£, andf/ = a T X V = b T Y. X ma Y be separated and written
as:
Whe
prob
the <
detei
QCO =
x i 0
X
(3)
Solve the eigenvalues of the image matrix G,
and calculate the correspond eigenvectors Q . After the
(4)
Canonical correlation analysis means that the linear
transformation is performed to the random variable [jf Y\,
and make the coefficient vectors a and b of U = a T X and
V = b T Y meet: