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˜
where
j
is also an image point index. The maximum log-likelihood estimate of
φ
t
is
explicitly defined as follows
˜
˜
˜
L
(
φ
)
≡
log
p
(
β
|
φ
)
.
(10)
t
t
t
The correspondences between features are not known a-priori from frame to frame
in spite of the capture of a
concrete
gait model to generate
˜
φ
t
, which is used to gener-
ate a 3-D model ˜
α
t
. However, the conditional logarithm of the posterior probability,
˜
˜
, can instead be computed over the previous estimates of
˜
Q
t
. Substituting
Eq. (9) into Eq. (10) and considering the Markovian properties, one can re-write the
conditional log-likelihood as
(
φ
|
φ
)
φ
t
t
−
1
˜
˜
˜
φ
t
−
1
)=
∑
i
˜
∑
j
˜
˜
Q
(
φ
t
|
p
(
α
t
j
|
˜
β
t
i
,
φ
t
−
1
)
log
p
(
β
t
i
|
α
t
j
,
˜
φ
t
)
.
(11)
4.2
EM Algorithm
As applied here, the EM algorithm starts from an initial “guess" of the scene struc-
ture, which is derived from the motion parameters provided by the gait model and
previous feature positions, and then projects the perspective 3-D “template" to a 2-D
image using the features to be matched across two frames of the sequence using the
following iterative steps:
1.
E-step
:
In this step, we formulate the
a posteriori
probability of the incomplete data set,
˜
˜
p
(
α
t j
|
˜
β
t
i
,
φ
t
−
1
)
, contained in Eq. (11). Bayes' rule is again applied to obtain
˜
˜
˜
˜
˜
p
(
α
t j
|
φ
t
−
1
)
p
(
β
t
i
|
α
t j
,
φ
t
−
1
)
˜
˜
p
(
α
t j
|
˜
β
t
i
,
φ
t
−
1
)=
φ
t
−
1
)]
.
(12)
˜
˜
˜
∑
k
[
p
(
α
t k
|
˜
φ
t
−
1
)
p
(
β
t
i
|
α
t k
,
˜
˜
˜
˜
To pursue solutions for
p
(
α
t j
|
˜
φ
t
−
1
)
and
p
(
β
t
|
α
t j
,
˜
φ
t
−
1
)
in Eq. (12), the probability
˜
p
(
α
t j
|
˜
φ
t
−
1
)
may be written as
1
N
˜
˜
∑
i
˜
p
(
α
t j
|
˜
φ
t
−
1
)=
p
(
α
t j
|
˜
β
t
i
,
φ
t
−
1
)
,
(13)
where
N
is the number of the features. This indicates that the posterior probabil-
ity
p
˜
˜
˜
˜
˜
(
α
t j
|
φ
t
−
1
)
is determined by the individual joint densities
p
(
α
t j
|
β
t
i
,
φ
t
−
1
)
over
˜
˜
the feature points considered. To compute
p
(
α
t j
|
φ
t
−
1
)
, it is necessary to take into
˜
˜
(
˜
|
,
)
account the initial estimate of
p
, which seriously affects the character-
istics of convergence, e.g. accuracy and efficiency, for final correspondence. Here,
we preset it to be
α
β
φ
t
−
1
t
j
t
i
1
N
. In other words, the posterior probability of registration of each
feature is uniform in the first instance.
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