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In-Depth Information
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1111
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1
√
8
11
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1
−
111
−
1
−
1
H
=
(50)
1
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1
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1, the transform operation
consists simply of addition and subtraction. For this reason, this transform is useful in situations
where minimizing the amount of computations is very important. However, the amount of
energy compaction obtained with this transform is substantially less than the compaction
obtained by the use of the DCT. Therefore, where sufficient computational power is available,
DCT is the transform of choice.
Because the matrix without the scaling factor consists of
±
13.5 Quantization and Coding of Transform
Coefficients
If the amount of information conveyed by each coefficient is different, it makes sense to assign
differing numbers of bits to the different coefficients. There are two approaches to assigning
bits. One approach relies on the average properties of the transform coefficients, while the
other approach assigns bits as needed by individual transform coefficients.
In the first approach, we first obtain an estimate of the variances of the transform coeffi-
cients. These estimates can be used by one of two algorithms to assign the number of bits used
to quantize each of the coefficients. We assume that the relative variance of the coefficients
corresponds to the amount of information contained in each coefficient. Thus, coefficients
with higher variance are assigned more bits than coefficients with smaller variance.
Let us find an expression for the distortion, then find the bit allocation that minimizes the
distortion. To perform the minimization, we will use the method of Lagrange [
194
]. If the
average number of bits per sample to be used by the transform coding system is
R
and the
average number of bits per sample used by the
k
th coefficient is
R
k
, then
M
1
M
R
=
R
k
(51)
k
=
1
where
M
is the number of transform coefficients. The reconstruction error variance for the
k
th
quantizer
2
r
k
2
σ
is related to the
k
th quantizer input variance
σ
θ
k
by the following:
r
k
=
α
k
2
−
2
R
k
2
θ
k
σ
σ
(52)
where
α
k
is a factor that depends on the input distribution and the quantizer.
