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and lower quantization errors can be achieved. However, the size of the map also
increases. In contrast, Fig.
11.10
a shows the plot of the prototype vectors generated
from four sub-codebooks (each sub-codebook has the size of 2
2) overlaying in
the same input space. Each sub-codebook is trained separately by the input samples
belonging to the same class, and the Voronoi cells with the associated members are
shown in Fig.
11.10
b-e. In this case, four of the 2
×
2 SOM produce 16 Voronoi
cells if each SOM is trained separately. The variances computed on each Voronoi
cell, Var
×
in each sub-codebook, are less than that of the single codebook. As a
result, the summation of the quantization errors [according to Eq. (
11.18
)] from the
quantization by the sub-codebooks in Fig.
11.10
b-e, is less than that of the vector
quantization by the single-codebook in Fig.
11.9
b, 2.4 dB.
{
x
|
j
}
11.7.2
The Hidden Markov Models of Gesture
As in Eq. (
11.5
), the transformation of the gesture
G
i
with the use of
C
-node
SSOM can be expressed as
S
. This is considered
as a transformation of the continuous trail to a sequence of
C
discrete symbols,
which defines the finite states needed to build first order Markov chain models. The
transformation
N
=
Q
(
G
)=
{
u
1
,...,
u
t
,...,
u
T
}
as discussed in Eq. (
11.11
) replaces consecutive equal values
for symbols
u
with a single value, and outputs the trajectory
Tr
for the gesture
G
on the SSOM. This results in zeroing the self transition probability values in the
Markov transition probability matrix, and thus a loss of information regarding the
duration of a particular state. However, this information is not critical to gesture
recognition.
A Markov model, for each of the
K
classes in the gesture data set, is created from
the training data, i.e.,
(
S
)
Tr
1
,
Tr
2
,...,
Tr
N
i
}ₒ
ʻ
k
{
(11.20)
where
Tr
i
is the trajectory of the
i
-th gesture instance in the
k
-th gesture class, and
N
is the number of instances. All
Tr
i
,
N
i
are obtained from the
k
-th sub-
codebook SSOM. The sequence of the
u
m
values in the trajectory
Tr
i
i
=
1
...
of the training
N
i
i
Tr
i
}
set
{
1
, will be used for the calculation of the transition probability of the
=
model
ʻ
k
describing class
k
. This results in a set of
K
Markov models,
Tr
1
,
Tr
2
,...,
Tr
N
i
}ₒ
ʻ
k
ʻ
=
{
ʻ
1
,...,
ʻ
k
,...,
ʻ
K
}
:
{
(11.21)
The transformation of a gesture instance to the SSOM and the Markov model is
intuitively depicted in Fig.
11.11
.
In order to conduct isolated gesture recognition, the following steps are per-
formed:
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