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2.4
Activity Recognition by Log-Likelihood Maximization
We give here details of the modified Viterbi algorithm when applied to log-likelihood
maximization of HPaSMM. We consider here an upper level state sequence S con-
taining R successive segments. Each segment corresponds to an activity associated
with an upper level state S i . We denote by q r the time index of the end-point of the
r th segment. The sequence of observations that characterize a segment r is given
by y ( q r 1 + 1 , q r ] =
,...,
= ... =
is supposed known
and is used to perform recognition of successive activity phases using the modified
Viterbi algorithm as follows.
The algorithm provides the decoded sequence of upper level HPaSMM states
S that maximizes the log-likelihood, i.e. , such that S
y q r 1 + 1
y q r
such that S q r 1 + 1
S q r .
θ
=
arg max S log P
(
y
,
S
| θ )
.The
likelihood P
is defined, for an observation sequence y and a sequence of
upper level state S , by:
(
y
,
S
| θ )
R
r = 1 P ( S r | S r 1 )
P
(
y
,
S
| θ )=
R
r = 1 P ( sd i = q r q r 1 | ψ ; S q r )
×
R
r = 1 P ( y ( q r 1 + 1 , q r ] | φ ; S q r ) .
×
A simple example that illustrates the modified Viterbi algorithm for HPaSMM is
presented in Figure 2. For each upper level state and from beginning frame t 1 to end-
ing frame t N , three tables are first filled. As described in Figure 2, one is composed
of the P
| θ )
values, the second table contains time index of the previous change
of upper level state and the last table corresponds to numbering of the current upper
level state. Similarly to classical Viterbi algorithm, the decoding process goes from
the last frame to the first frame. The maximum P
(
y
,
S
value at the last frame is
selected. Then, second and third tables enable to find the current upper level state S i
and its phase beginning frame t j . The corresponding value i is kept in S for the de-
coded frames (see Fig. 2). The process ends when S is completely filled, i.e. ,when
the beginning frame is reached.
(
y
,
S
| θ )
2.5
A Comparison Method: Hierarchical Parallel Hidden Markov
Models
The method considered for comparison purposes is based upon a hierarchical par-
allel hidden Markov models (HPaHMMs) build with a similar architecture than
the HPaSMMs previously described. The only difference is that HPaHMMs do not
model specifically upper level state durations. In a HPaHMM, the upper level state
duration is supposed to follow a simple geometric law given by:
 
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