Page 274 - Digital Analysis of Remotely Sensed Imagery
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236 Cha pte r S i x
create any new information in the components. Instead, it
simply redistributes the available information among the
output components.
• The correlation coefficient between the two newly created
bands is zero after the transformation. In other words, the
information content of component (band) 1 does not overlap
with that of component (band) 2 anymore. For this reason,
the eigen value matrix is usually written as a one-dimensional
T
(1D) matrix [7.36 2.64] .
• Prior to image transformation band 1 carries 6/(6 + 4), or
60 percent, of the total amount of information. After the
transformation, this figure rises to 7.36/(7.36 + 2.64), or
73.6 percent. On the other hand, the information content
of band 2 drops from 4/(6 + 4), or 40 percent, before the
transformation to 2.64/(7.36 + 2.64), 26.4 percent, after
the transformation. Therefore, the first component is
much more informative after the transformation than
before. The opposite is true for the second component. In
fact, the information content of the components decreases
drastically from the first to the last.
In order to project the information content of the input bands
X into new component images Y, it is necessary to determine the
transformation matrix G. The purpose of image transformation is
to construct a new feature space in which the covariance between
any components equals zero.
Y = GX (6.18)
G is a rotation matrix in the following form:
⎡g g g ⎤
⎢ 11 12 1n ⎥
G = ⎢ g 21 g 22 g 2 n ⎥
⎢ ⎥
⎢g g g ⎥
nm⎦
⎣ n1 n2 ⎦
g ⎛ g ⎞
12
In this particular case G = ⎜ g ⎝ 11 g ⎠ ⎟
21 22
G must meet the following condition:
−1
T
G = G (6.19)
This is to say, the inversed matrix equals its transposed matrix
so that pixel values are not artificially enlarged or reduced after
transformation.