Databases Reference
In-Depth Information
The total reconstruction error is given by
M
2
r
1
α
k
2
−
2
R
k
2
θ
k
σ
=
σ
(53)
k
=
The objective of the bit allocation procedure is to find
R
k
to minimize (
53
) subject to the
constraint of (
51
). If we assume that
α
k
is a constant
α
for all
k
, we can set up the minimization
problem in terms of Lagrange multipliers as
R
M
M
1
M
2
−
2
R
k
2
θ
J
=
α
σ
−
λ
−
R
k
(54)
k
k
=
1
k
=
1
Taking the derivative of
J
with respect to
R
k
and setting it equal to zero, we can obtain this
expression for
R
k
:
2
log
2
M
2
log
2
2
θ
k
1
1
2
R
k
=
α
ln 2
σ
−
(55)
Substituting this expression for
R
k
in (
51
), we get a value for
λ/
M
:
M
2
k
1
M
M
=
2
θ
2
−
2
R
α
ln 2
σ
(56)
k
=
1
Substituting this expression for
λ/
M
in (
55
), we finally obtain the individual bit allocations:
2
θ
σ
1
2
log
2
k
k
=
1
(σ
R
k
=
R
+
(57)
1
M
2
θ
k
)
Although these values of
R
k
will minimize (
53
), they are not guaranteed to be integers or
even positive. The standard approach at this point is to set the negative
R
k
s to zero. This will
increase the average bit rate above the constraint. Therefore, the nonzero
R
k
s are uniformly
reduced until the average rate is equal to
R
.
If we substitute the value of
R
k
fromEquation (
57
) into the expression for the reconstruction
error variance for the
k
th quantizer shown in Equation (
52
), we obtain
2
θ
k
k
=
1
(σ
σ
1
2
log
2
M
)
θ
k
2
−
2
(
R
+
1
2
θ
k
)
2
r
k
2
σ
=
α
k
σ
2
θ
k
k
=
1
(σ
σ
θ
k
2
−
2
R
2
−
log
2
1
M
2
θ
k
)
2
=
α
k
σ
θ
k
k
=
1
(σ
M
2
−
2
R
2
2
α
k
σ
θ
k
)
=
2
θ
σ
k
M
1
M
2
−
2
R
2
=
α
k
1
(σ
θ
k
)
(58)
k
=
