Databases Reference
In-Depth Information
10
5
80
90 100 110
120
130
140 150
160 170 180 190 200 210 220 230
−5
−
10
F I GU R E 13 . 2
The transformed sequence.
where
x
is the two-dimensional source output vector
x
0
x
1
x
=
(2)
x
0
corresponds to height and
x
1
corresponds to weight,
A
is the rotation matrix
cos
φ
sin
φ
A
=
(3)
−
sin
φ
cos
φ
φ
is the angle between the
x
-axis and the
y
=
2
.
5
x
line, and
θ
0
θ
1
θ
=
(4)
is the rotated or transformed set of values. For this particular case, the matrix
A
is
0
.
37139068 0
.
92847669
A
=
(5)
−
0
.
92847669 0
.
37139068
and the transformed sequence (rounded to the nearest integer) is shown in Table
13.2
.(Fora
brief review of matrix concepts, see Appendix B.)
Notice that for each pair of values, almost all the energy is compacted into the first element
of the pair, while the second element of the pair is significantly smaller. If we plot this sequence
in pairs, we get the result shown in Figure
13.2
. Note that we have rotated the original values
by an angle of approximately 68 degrees
(
.
)
.
Suppose we set all of the second elements of the transformation to zero, that is, the second
coordinates of the sequence shown in Table
13.2
. This reduces the number of elements that
need to be encoded by half. What is the effect of throwing away half the elements of the
sequence? We can find that out by taking the inverse transform of the reduced sequence. The
inverse transform consists of reversing the rotation. We can do this by multiplying the blocks
of two of the transformed sequences with the second element in each block set to zero with
the matrix
arctan 2
5
cos
φ
−
sin
φ
A
−
1
=
(6)
φ
φ
sin
cos
We obtain the reconstructed sequence shown in Table
13.3
. Comparing this to the original
sequence in Table
13.1
, we see that, even though we transmitted only half the number of
