Recognition of facial expressions - Knowledge Based Systems ...

The term σ
ij
is the covariance between the pixel i and the pixel j. The relation between
the covariance coefficient and the correlation coefficient is:
Equation 9
r ij
=
σ
ii
ij
σ ⋅σ
jj
The correlation coefficient is a normalized covariance coefficient. By making the
covariance matrix of the new components to be a diagonal matrix, each component
becomes uncorrelated to any other. This can be written as:
Equation 10
Y
= Y ∗Y
T
=
σ
Y
11
0
...
0
σ
0
Y
22
...
0
...
...
...
...
σ
0
0
...
Y
w*
h,
w*
h
In the previous relation, X is the matrix containing the images of a given facial area and
Y is the matrix containing the column image vectors.
The form of the diagonal covariance matrix assures the maximum variance for a variable
with itself and minimum variance with the others.
The principal components are calculated linearly. If P be the transformation matrix,
then:
Equation 11
Y = P
∗ X
X = P ∗Y
The columns of P are orthonormal one to each other and:
T
Equation 12
P = P
−1
P
T
∗ P = I
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