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.
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