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Invariant feature values for players motion characterization
The invariant feature representation of the activity embedded in a single moving
players is here presented. Following the notations introduced in Section 2.2, we
consider the continuous representation of a trajectory
T
k
and more precisely their
first and second order differential values
u
t
,
k
,
v
t
,
k
,
u
t
,
k
and
v
t
,
k
. The relevant invariant
representation of a single trajectory
T
k
is defined by the feature value
˙
γ
t
,
k
such that:
v
t
,
k
u
t
,
k
−
u
t
,
k
v
t
,
k
˙
γ
t
,
k
=
=
κ
t
,
k
.
w
t
,
k
u
t
,
k
+
v
t
,
k
v
t
,
k
u
t
,
k
)
where
γ
t
,
k
=
arctan
(
corresponds to the local orientation of the trajectory
T
k
,
v
t
,
k
u
t
,
k
−
u
t
,
k
v
t
,
k
2
is
u
t
,
k
+
v
t
,
k
)
κ
t
,
k
=
is the curvature of the trajectory
T
k
and
w
t
,
k
=(
2
u
t
,
k
+
v
t
,
k
)
(
the velocity of point
.
Such a feature value characterizes both the dynamic (through velocity informa-
tion) and the shape (through curvature information) of
T
k
. Moreover, it has been
shown that the ˙
(
u
t
,
k
,
v
t
,
k
)
γ
t
,
k
feature value is invariant to translation, rotation and scale in the
images [9]. Hence, the feature vector used to characterize a given activity
S
i
of a
player
P
k
described by the trajectory
T
k
is the vector containing the successive val-
ues (one for each image time index in activity
S
i
)of ˙
S
i
γ
t
,
k
:
S
i
k
˙
=[
˙
,
˙
, ...,
γ
n
i
−
1
,
k
,
˙
γ
n
i
,
k
]
,
˙
γ
γ
γ
1
,
k
2
,
k
where
n
i
is the size of the processed trajectories in activity
S
i
.
Invariant feature values for players interaction characterization
Interaction between players is of a crucial interest in order to have an efficient rep-
resentation of complex activities in sports videos. To this aim, the spatial distance
(see Fig. 4) between the two players
P
1
and
P
2
is considered to characterize their
interaction. Hence, at each successive time
j
, the distance between the two squash
players trajectories
T
1
and
T
2
is defined by:
=
(
−
)
2
+(
−
)
2
.
d
j
x
j
,
1
x
j
,
2
y
j
,
1
y
j
,
2
More specifically, the normalized distance is computed,
i.e.
:
d
j
=
/
.
d
j
d
norm
The feature value
d
j
is trivially a translation and rotation invariant feature in
the 2D image plane. Moreover, to provide a scale invariant feature, a contextual
normalizing factor
d
norm
has to be computed in the images. In the processed squash
videos, the normalizing factor is the distance between the two sides of the court. The
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