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The horizontal axis and the vertical axis are the same as those of Figure 3, respec-
tively. The parameters (
ˆ
β
1
,
ˆ
β
2
,
ˆ
β
3
,
ˆ
β
4
) were obtained by least-squares estimation.
We used the following inter-frame prediction with motion compensation for
rate evaluation, since representative video coding algorithms like H.264/AVC and
MPEG-2 adopt inter-frame prediction with motion compensation. Each frame was
divided into blocks, and each block was predicted from its previous frame by motion
compensation. The number of references was one. The block size used for motion
compensation was 16
±
8 [pixels] at 1000 [frames/sec]. The search range decreased according as the frame-
rate increased. For example, we set
×
16 [pixels]. The search range of motion estimation was
16 [pixels] at 500 [frames/sec]. The motion
estimation scheme was full search algorithm. The criterion of motion estimation
was the sum of absolute differences (SAD) between current block and reference
block. Namely, selected displacement minimized SAD between current block and
reference block. Figure 5 shows bit-rate of motion vectors of the original sequences
and the temporally sub-sampled sequences, and Figure 6 shows those of the original
sequences and the temporally down-sampled sequences. The horizontal axis is the
frame-rate [frames/sec] and the vertical axis is the bit-rate [bits/pixel] which is the
sum of entropy of the two elements of the motion vector.
As shown in Figure 3, the results of the experiments well agree with the values
yielded by the proposed model. In other words, equation (10) and equation (14) well
model the relationship between the bit-rate of prediction error and frame-rate. Table
1 and table 2 show residual sum of squares (RSS) between the results of rate evalua-
tion experiments, and the theoretical values from the proposed model, as a measure
of the fitness of the proposed model. Table 1 shows the results for the temporally
sub-sampled sequences and Table 2 shows those for the temporally down-sampled
sequences. In these tables, we compare our model with the conventional model in
[21] which is expressed as follows:
±
I
(
F
)=
a
1
a
2
(1
−
exp(
−
1
/
(
a
2
F
)))
(15)
where
a
1
and
a
2
are constants that depends on the video signal. In this experiment,
parameters (
a
1
and
a
2
) were obtained by least-squares estimation. As shown in Table
1 and 2, our model achieved smaller RSS than the conventional model.
Ta b l e 1
Residual sum of squares (RSS) of temporally sub-sampled sequences
model RSS RSS RSS
(golf) (tennis) (toy)
proposed 3.39e-03 2.96e-04 2.60e-03
conventional 2.24e-01 1.29e-01 1.62e-01
Ta b l e 2
Residual sum of squares (RSS) of temporally down-sampled sequences
model RSS RSS RSS
(golf) (tennis) (toy)
proposed 2.60e-04 6.33e-04 5.46e-03
conventional 1.27e-01 3.09e-01 1.37e-01
