Image Processing Reference
In-Depth Information
U
0
U
1
„
11
11
«
„
−
11
−
11
«
U
2
U
3
„
11
−
1
−
1
«
„
−
11
1
−
1
«
F
F
„
1
/
2 1
0
/
2
«
„
1
/
2 1
0
«
/
2
Fig. 3.5. The discrete image
F
can be expanded as the sum of the basis vectors
U
0
,
···
,
U
3
weighted with the expansion coefficients
c
i
. The image intensities, representing matrix values,
vary uniformly from black to white as the matrix values change from
−
1 to 1. The matrix
F
is obtained by summing
c
i
U
i
and should be identical to
F
Example 3.3. Assume that we have a discrete image of size 2
×
2
F
=
0
.
51
00
.
5
(3.40)
and the four orthogonal basis vectors as
U
0
=
11
,
U
1
=
−
11
,
U
2
=
11
,
U
3
=
−
11
11
−
11
−
1
−
1
1
−
1
which are illustrated by Fig. 3.5.
We calculate the coefficients
c
i
when
F
is expanded in the basis
U
i
as:
F
=
i
c
i
U
i
(3.41)
The space of 2
2 images is a Hilbert space with the scalar product given by Eq.
(3.38). The coefficients are obtained according to
×
c
i
=
U
i
,F
(3.42)
U
i
,
U
i
