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issues by smoothing the observed trajectories. Moreover, as we will see in Section
3, low level feature values used to describe trajectories may be composed of velocity
values that require computation of differential values. Using a kernel approximation
of the trajectories provides a simple way to get such differential values.
Let us consider a trajectory
T
k
composed of a set of
l
k
points corresponding to
the temporal successive positions of the moving object in the image plane,
i.e.
,
T
k
=
{
(
x
1
,
k
,
y
1
,
k
)
, ..,
(
x
l
k
,
k
,
y
l
k
,
k
)
}
. A continuous approximation of the trajectory
T
k
is defined by
{
(
u
t
,
k
,
v
t
,
k
)
}
t
∈
[
1;
l
k
]
where:
t
−
j
h
t
−
j
h
l
j
2
l
j
2
1
e
−
(
)
1
e
−
(
)
u
t
,
k
=
∑
x
j
,
k
v
t
,
k
=
∑
y
j
,
k
=
=
,
.
t
−
j
t
−
j
l
k
j
2
l
k
j
2
1
e
−
(
)
1
e
−
(
)
∑
∑
h
h
=
=
First and second order differential values
u
t
,
k
,
v
t
,
k
,
u
t
,
k
and
v
t
,
k
are easily obtained
using standard derivation formulas. Yet simple, such calculations leads to long ana-
lytic expressions that are not detailed here.
Trajectory data used for experiments in Sections 3 and 4 are quite precise and
does not need a denoising pre-processing, However, such a pre-processing is con-
sidered in Section 3 since differential values are needed for squash video pro-
cessing. On the contrary, feature representation for handball video processing (see
Section 4) does not require such a procedure.
Computation time reduction
To reduce computation time, a grouping procedure of the feature values may be
performed. For the
n
considered feature vectors
V
S
j
, groups of
k
group
consecutive
values are formed. For each of these groups, the mean values are computed and
used to construct
n
new feature vectors
V
S
i
j
(having sizes
k
group
times smaller than
the original feature vectors).
2.3
Model Parameter Estimation
The entire set of parameter associated with a HPaSMM is finally given by
θ
=
{
and is estimated by a supervised learning stage. Training videos are con-
sidered and used to compute any parameter of
A
,
φ
,
ψ
}
is further used to perform tem-
poral activity phases recognition using a Viterbi algorithm (see Section 2.4).
A
is the upper level HPaSMM state transition matrix. It is computed using ob-
served transitions between activity phases in the training videos.
In the proposed HPaSMM, mixtures of Gaussian models (GMMs) are used to
model durations of the activity phases
S
i
denoted by
sd
i
. The corresponding set of
parameter
θ
.
θ
is obtained by fitting GMMs using “forward-backward” procedures.
Initializations are here performed using a classical k-means algorithm.
PaHMMs sets of parameters are obtained using “Expectation-Maximization”
procedures as it is described in [25].
ψ
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