Understanding Sports Video Using Players Trajectories - Intelligent Video Event Analysis and Understanding - page 129

2.1 Upper Layer: Semi-Markovian Activity Modeling

Hence, in our HPaSMM framework, the upper level layer is dedicated to the model-
ing of activities. Each HPaSMM states S i actually corresponds to an activity phase.
This upper level layer is constructed using a semi-Markovian scheme [6] which has
the advantage to explicitly model state durations in the upper level states S i . Indeed,
Markovian models stand upon the hypothesis that state durations follow geometrical
laws. Yet, upper level states ( i.e. , activities) do not necessarily follow simple geo-
metrical laws and require more complex modelings. So, in the proposed HPaSMM,
mixtures of Gaussian models (GMMs) are used to model the durations of the ac-
tivity phases denoted by sd i . In the illustration in Figure 1, the upper level layer is
surrounded in red and is composed of four upper level states S 1 , S 2 , S 3 ,and S 4 .
The set of parameters involved in the upper level layer is composed of A and

ψ is the upper level HPaSMM state transition matrix at the time

index of activity changes (see [6]), and

,where A = { a i , j }

ψ denotes the set of parameter related to

GMMs state duration models.

2.2 Lower Layer: Parallel Markovian Feature Modeling

Lower level layer of the HPaSMM scheme is dedicated to the modeling of feature
values computed on the trajectories. For a given sport, a number n of feature value
v j is chosen. They have to describe players movements and their respective inter-
actions in each video frame. Moreover, to process a large variety of sports videos,
such feature values will also have to stand invariant to appropriate transformations.
Description of squash and handball feature representations are further provided in
Sections 3 and 4.
The set of successive feature values v j (one for each video frame) corresponding
to a upper level state S i are placed into a vector denoted by V S i

j . For each state of
the upper level S i , the corresponding feature vectors V S i j are modeled using a n-layer
PaHMMs, where one HMM is dedicated to each feature vector V S j . The PaHMMs
are here defined in a similar way than those presented in [25], where conditional
probabilities B of observations are fitted by GMMs. In the illustration in Figure 1,
the lower level layer is surrounded in green. For each considered activity S i ,three
feature vectors V S i j are considered and modeled by a 3-layer PaHMM.
The set of PaHMMs parameters is denoted by

is composed of B , A (state

φ . φ

transition matrix) and π (initial state distribution) of the n -layer PaHMMs for each

activity phase S i .

Kernel approximation

We describe here a pre-processing that may be computed on the raw trajectory data
to obtain continuous representations of the trajectories. Two advantages are implied
in the choice of such a pre-processing. First of all, if trajectory data obtained from
tracking procedure is noise corrupted, the kernel approximation help handling such