
53Image Transforms
This will allow yEyy x
|[]||||
, where
||
⋅
||
is a norm:
i
N
i
|||
2
0
1
2
∑
=
=
−
Now the problem for optimal transform can be formulated as follows:
maxE
H
.[ ]
s.t. yaxaaa
j
iiii ij
,1
;
**
==
Without going into the details of the proof for the above maximization prob-
lem, it can be shown that the energy for y can be maximized if
i
are the
eigenvectors of R
x
;
a
xi
=λ (i.e., transformation matrix A is formed using
the eigenvectors of the covariance matrix).
3.8 The Karhunen-Loeve Transform: Two-Dimensional
Image Transform with a Data-Dependent Basis
The Karhunen-Loeve transform (KLT) is also an example of the unitary ...