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,
(
...
...
)
denote the set of weight vectors as
w
c
1
c
C
, where
C
is the total number of
(
)
,
(
...
...
)
SSOM nodes. The input vectors
x
can then be transformed from
a series of postures to a series of map units based on their best matching units
(BMUs), i.e.,
t
1
t
T
S
=
Q
(
G
)=(
u
1
,...,
u
t
,...,
u
T
)
,
t
∈
[
1
,
T
]
(11.5)
u
t
=
arg min
c
(
||
x
(
t
)
−
w
c
||
)
(11.6)
where
Q
(
G
)
is the quantization operation and
u
t
is the index of the BMU of the
input
x
.
For each posture sample from an input gesture, the transformation involves
finding the BMU and using this node to index the input sample. After transforming
a temporal sequence of postures onto the map, an output sequence of indices results.
The transformation can be described as a sequence of node indices,
S
, or a trajectory
of individual node positions,
Tr
, on the spherical surface (defined in a 3D co-
ordinate system).
Given the sequence or trajectory traced on the SSOM, we consider a number of
alternative descriptors for a gesture instance and class: posture sparse codes, posture
occurrence, posture transitions, and posture transition sparse codes.
(
t
)
11.5.1
Sparse Code of Spherical Self-organizing Map
The sparse code (SC) method has been utilized for structuring the coding labels of
the hierarchical SOM [
343
,
351
,
352
]. This method is adopted in the current work,
and compared to the newly proposed methods. During mapping of posture vectors,
the weight vectors
w
c
,
(
are labeled as the activated nodes if they are the
winning nodes according to Eq. (
11.6
). Each node has a state
S
c
describing whether
it is a winner for a gesture element or not, and the whole state of the nodes are used
as the output,
SC
1
...
c
...
C
)
=(
S
1
,...,
S
c
,...
S
C
)
.The
S
c
is defined as follows:
1
,
if
c
=
u
t
|
t
∈
[
1
,
T
]
S
c
=
(11.7)
0
,
otherwise
S
c
is regarded as a sparse code which represents an activated pattern of winner
nodes for a gesture element. The sparse code only represents the existence of a
set of postures, and not their frequency of occurrence. For instance, if a particular
gesture involves a set of five postures, some of which are held for a length of time,
then the sparse code will only indicate that they occurred, and won't consider the
duration. This offers a time invariant measure of posture existence, and is useful
when detecting gestures that may be performed at different speeds.
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