Databases Reference
In-Depth Information
A
200
B
190
72
Height (in)
F I GU R E 8 . 4
Compression scheme for encoding height-weight pairs.
Let us examine the distortion for this scheme a little more closely. As the conditional
probability for this scheme is not of the form of Equation (
50
), we can no longer write the
distortion in the form of Equation (
52
). Recall that the general form of the distortion is
N
−
1
M
−
1
D
=
d
(
x
i
,
y
j
)
P
(
x
i
)
P
(
y
j
|
x
i
)
(53)
i
=
0
j
=
0
Each term in the summation consists of three factors: the distortion measure
d
(
x
i
,
y
j
)
,the
source density
P
. The distortion measure is a
measure of closeness of the original and reconstructed versions of the signal and is generally
determined by the particular application. The source probabilities are solely determined by
the source. The third factor, the set of conditional probabilities, can be seen as a description
of the compression scheme.
Therefore, for a given source with some
pdf
(
x
i
)
, and the conditional probability
P
(
y
j
|
x
i
)
{
P
(
x
i
)
}
and a specified distortion measure
d
(
·
,
·
)
, the distortion is a function of only the conditional probabilities
{
P
(
y
j
|
x
i
)
}
; that is,
D
=
D
(
{
P
(
y
j
|
x
i
)
}
)
(54)
Therefore, we can write the constraint that the distortion
D
be less than some value
D
∗
as a
requirement that the conditional probabilities for the compression scheme belong to a set of
conditional probabilities
that have the property that
D
∗
}
={{
P
(
y
j
|
x
i
}
such that
D
(
{
P
(
y
j
|
x
i
)
}
)
(55)
