Page 100 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

88 Chapter 3

The element a 1 in the above matrix gives the information about how the
first elements of the collected vectors are correlated to each other. Similarly
c 1 gives the information about how the first elements are correlated with
third element. Thus if the matrix is the diagonal matrix, then the elements of
the vectors collected are made independent to each other.

1.1 Diagonalization of the Matrix ‘CM’

1. Compute the Eigen values and the corresponding Eigen vectors of the
matrix ‘CM’
Eigen values are computed by solving the determinants | CM-λI| = 0.
Number of Eigen values obtained is equal to the size of the matrix. For
instance the size of the above mentioned matrix is 4 and hence number of
Eigen values = 4. Let it be λ 1, λ 2, λ 3 and λ 4

2. Compute the Eigen vector ‘X 1’corresponding to the Eigen value ‘λ 1’
by solving the equation [CM. - λ 1] X 1=0.The size of the Eigen vector ‘X 1’
is 1X4 for the above example.

3. Similarly Eigen vectors X 2, X 3 and X 4 are computed.

4. Eigen vectors are arranged rowwise in the matrix to form Transforma-
tion Matrix (say ‘TM’)

5. Transformed Vector ‘Y 1’ is obtained using the transformation matrix
as mentioned below.
Y 1= TM*( X 1- µ x).Similarly Y 2,Y 3, …Y m are obtained. If covariance
matrix is computed for the transformed set of vectors [(i.e.) Y 1, Y 2, Y 3 …
Y m] the covariance matrix is almost diagonal.

1.2 Example

Let us consider the Binary image of size 50x50. Let us collect the positions
of the Black pixel in the image as the two dimensional vectors. Xposition
and Yposition are the elements of the vector considered. Hotelling
transformation is applied as described above to get another set of vectors
called as transformed positions. The Binary image with all pixels valued ‘1’
(i.e.) white is created. Replace the pixel value of the transformed positions in