Image and Video Compression

53Image Transforms

This will allow yEyy x

H

|||

|[]||||

22

==

, where

||

⋅

||

is a norm:

yy

i

N

i

|

|||

2

0

1

2

∑

=

=

−

Now the problem for optimal transform can be formulated as follows:

maxE

yy

H

.[ ]

s.t. yaxaaa

ai

j

iiii ij

,1

,0

;

**

==

=≠

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

a

i

*

are the

eigenvectors of R

x

;

Ra

a

xi

ii

**

=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 ...