Databases Reference
In-Depth Information
for some
j
such that
l
−
1
t
(
j
)
k
n
(
i
)
k
i
=
0
As shown in Chapter 13, the operational rate distortion function can be obtained by min-
imizing the Lagrangian functional
D
. We assume the existence of an additive
distortion metric such that for each quality level
Q
l
the corresponding distortion
D
l
(λ)
+
λ
R
(λ)
is given
by
D
n
i
i
where
i
ranges over the code-blocks and
n
i
is based on the available truncation points. Given
a rate constraint for each of the quality levels
R
l
max
we can set up the problem of finding the
number of bits contributed by each code-block in a quality level as the problem of finding the
set of
n
i
that minimize
D
l
=
D
n
i
,λ
i
+
λ
R
i
,λ
i
If for some
we can find the truncation points that satisfy the rate constraint, then this set of
truncation points will provide the minimum distortion for the given rate. From the discussion
on operational rate distortion functions in Chapter 13 we know that
λ
λ
is the negative of the slope of the distortion rate function.
1.
2.
The slope is identical for each coefficient for an optimum bit allocation.
Therefore, for a given
we can restrict our attention to individual code-blocks and for each
code-block find the truncation point that minimizes the quantity
D
n
i
,λ
i
λ
R
i
,λ
. Taubman
and Marcellin [
220
] propose the following simple algorithm for finding an optimum set of
truncation points for a given value of
+
λ
λ
:
1.
Initialize
n
opt
i
=
0.
2.
For
t
=
1
,
2
,...
T
i
R
n
opt
D
n
opt
R
(
t
)
i
D
(
t
)
i
(a)
Compute
R
=
−
i
and
D
=−
(
−
i
)
.
i
i
set
n
opt
i
(b)
If
D
/
R
>λ
=
t
.
Note that at the end of the Tier I coding pass the set of valid truncation points are known for
each of the code-blocks. This procedure is then a simple enumerative procedure for a given
value of
λ
. The particular values of
λ
itself can be obtained using the bisection procedure
described in Chapter 13.
16.5.7 JPEG 2000 bitstream
One of the major contributions of the JPEG 2000 standard is its flexibility [
220
]. We have
seen that with EBCOT each subband of each component is partitioned into code-blocks. The
bitplanes corresponding to the subband coefficients are encoded using a three-pass procedure
