or more explicitly
⎡
⎢
⎢
⎢
⎢
⎣
y
11
y
21
 y
m1
y
12
y
22
 y
m2

y
1n
y
2n
 y
mn
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
w
11
w
12
 w
1n
w
21
w
22
 w
2n

w
n1
w
n2
 w
nn
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
x
11
x
21
 x
m1
x
12
x
22
 x
m2

x
1n
x
2n
 x
mn
⎤
⎥
⎥
⎥
⎥
⎦
(12.27)
To obtain the covariance of the features in Y based on the features in X, we can substitute c
Y
and c
T
Y
in the definition of the covariance matrix as
Y
=
1
m

Wc
T
X
−E
Wc
T
X

c
X
W
T
−E
c
X
W
T

(12.28)
By factorizing W, we have:
Y
=
1
m
W
c
X
−
X
T
c
X
−
X
W
T
(12.29)
or
Y
=W
x
W
T
(12.30)
Thus, to transform feature vectors, we can use this equation to find the matrix W such that
Y
is diagonal. This problem is known in matrix algebra as matrix diagonalization.
12.6 Inverse transformation
In the previous section we defined a transformation from the features in X into a new set
Y whose covariance matrix is diagonal. To map Y into X we should use the inverse of the
transformation. However, this is greatly simplified since the inverse of the transformation is
equal to its transpose. That is,
W
−1
=W
T
(12.31)
This definition can been proven by considering that
X
=W
−1
Y
W
T
−1
(12.32)
But since the covariance is symmetric,
x
=
T
x
and
W
−1
Y
W
T
−1
=
W
−1
T
Y
W
T
−1
T
(12.33)
which implies that
W
−1
=
W
−1
T
and
W
T
−1
=
W
T
−1
T
(12.34)
These equations can only be true if the inverse of W is equal to its transpose.
Thus, to obtain the features in X from the Y we have that c
T
Y
=Wc
T
X
can be written as
W
−1
c
T
Y
=W
−1
Wc
T
X
(12.35)
That is,
W
T
c
T
Y
=c
T
X
(12.36)
390 Feature Extraction and Image Processing

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