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Fig. 2
Illustration of the modified Viterbi algorithm used to decode the semi-Markovian up-
per level sequence of activity state
S
. This is a simple example with three upper level state
S
i
and eight frames
T
k
. Top of the image: decoded upper level state sequence
S
for the eight
frames. For the three following tables, numbers on the vertical axis correspond to upper level
states
S
i
numbering and the horizontal axis denotes the frames
T
k
. Upper table:
P
(
y
,
S
|
θ
)
values. Middle table: time index of the corresponding previous change of upper level state.
Down table: value of the corresponding previous upper level state. Red lines corresponds to
the decoding process, from the final frame
t
8
to
t
1
. Couples of values (
S
i
,
t
j
) between up and
middle table denotes the previous upper state
S
i
and previous time index of upper state change
t
j
corresponding to current decoding frame.
a
d
−
1
ii
p
(
d
i
)=
(
1
−
a
ii
)
,
where
a
ii
is the probability of staying in state
S
i
from time
t
to
t
1.
Comparison between HPaSMM and HPaHMM architectures will highlight the
importance of modeling the upper level state durations provided by HPaSMMs.
Fig. 3 contains the HPaHMM version of the HPaSMM presented in Fig. 1.
Hence, estimation of the set of HPaHMM parameters
+
is similar to
the HPaSMMs one except for matrix
A
. Indeed, since they specifically model upper
level state durations, HPaSMMs training procedure computes
A
by looking at the
transitions between activity phases only at times
q
r
(the time index of the end-point
of the
r
th
segment defined in the Section 2.4). For HPaHMMs training procedure,
every time instant (
i.e.
, every frames) transitions of activity phases are considered to
learn
A
. To retrieve activities using a HPaHMM, a simple Viterbi algorithm is here
needed. The likelihood
P
θ
=
{
A
,
φ
,
ψ
}
to maximize is here defined, for an observation
sequence
y
of size
K
and a sequence of upper level state
S
, by:
(
y
,
S
|
θ
)
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