Databases Reference
In-Depth Information
The various quantization and binary coding techniques have been described at some length
in previous chapters, so we will spend the next section describing various transforms. We will
then discuss quantization and coding strategies in the context of these transforms.
13.3 The Transform
All of the transforms we deal with will be linear transforms; that is, we can get the sequence
{
θ
n
}
from the sequence
{
x
n
}
as
N
−
1
θ
n
=
x
i
a
n
,
i
(9)
i
=
0
This is referred to as the
forward transform
. For the transforms that we will be considering,
a major difference between the transformed sequence
{
θ
n
}
{
x
n
}
and the original sequence
is
θ
that the characteristics of the elements of the
sequence are determined by their position
within the sequence. For instance, in Example 13.2.1 the first element of each pair of the
transformed sequence is more likely to have a large magnitude compared to the second element.
In general, we cannot make such statements about the source output sequence
. Ameasure
of the differing characteristics of the different elements of the transformed sequence
{
x
n
}
{
θ
n
}
is
n
of each element. These variances will strongly influence how we encode
the transformed sequence. The size of the block
N
is dictated by practical considerations.
In general, the complexity of the transform grows more than linearly with
N
. Therefore,
beyond a certain value of
N
, the computational costs overwhelm any marginal improvements
that might be obtained by increasing
N
. Furthermore, in most real sources, the statistical
characteristics of the source output can change abruptly. For example, when we go from a
silence period to a voiced period in speech, the statistics change drastically. Similarly, in
images, the statistical characteristics of a smooth region of the image can be very different
from the statistical characteristics of a busy region of the image. If
N
is large, the probability
that the statistical characteristics change significantly within a block increases. This generally
results in a larger number of transform coefficients with large values, which in turn leads to a
reduction in the compression ratio.
The original sequence
the variance
σ
{
x
n
}
can be recovered from the transformed sequence
{
θ
n
}
via the
inverse transform
:
N
−
1
x
n
=
0
θ
i
b
n
,
i
(10)
i
=
The transforms can be written in matrix form as
θ
=
Ax
(11)
x
=
B
θ
(12)
where
A
and
B
are
N
×
N
matrices, and the
(
i
,
j
)
th element of the matrices is given by
[
A
]
i
,
j
=
a
i
,
j
(13)
[
B
]
i
,
j
=
b
i
,
j
(14)
