Information Technology Reference
In-Depth Information
As stated, we assume a multivariate Gaussian motion model for the conditional
probability
p
˜
˜
˜
. So, its maximum likelihood estimate can be represented
by the continuous density [10] as
(
β
t
i
|
α
t
j
,
φ
t
−
1
)
1
2
2
1
det
ex p
T
D
z
1
2
1
2
εΣ
−
1
˜
˜
˜
p
(
β
t
i
|
α
t j
,
φ
t
−
1
)=
−
ε
,
(14)
π
(
Σ
)
where T indicates transpose;
D
z
= 6 since there are six transformation parameters;
the error-residual
˜
ε
=
β
t
i
−
γ
t
j
,
(15)
˜
which is the Euclidean distance between the matched image feature,
β
t
i
,andthe
projected position derived from the gait model and prior structure,
γ
t
j
. The variance-
Σ
−
1
covariance matrix
Σ
and its inverse
are both 2
N
×
2
N
positive definite sym-
metric because of their elements
σ
ij
from the covariance of
ε
i
and
ε
j
,i.e.
T
Σ
=
E
[
εε
]
.
(16)
Re-writing the logarithmic part of Equation (11) results in the new form as follows:
1
2
˜
˜
˜
˜
˜
∑
i
∑
j
˜
T
Q
(
φ
t
|
φ
t
−
1
)=
−
p
(
α
t
j
|
˜
β
t
i
,
φ
t
−
1
)(
ϑ
−
(
β
t
i
−
γ
t
j
)(
β
t
i
−
γ
t
j
)
)
.
(17)
where
stands for the total summation of other terms after taking logarithms. The
3-D position of
ϑ
γ
t
is
˜
˜
α
t
=
R
(
t
)
α
t
−
1
+
T
(
t
)
,
(18)
where
R
(
t
)
(or
R
afterwards) is a rotation matrix represented by Euler angles, and
T
d
(
(or
T
d
afterwards) is a translation vector which contains
T
x
,
T
y
,and
T
z
. The up-
dated parameters for the tri-axial Euler angles, and displacements can be expressed
as
t
)
⎧
⎨
3
k
=
1
θ
(
t
)=
θ
+
∑
θ
x
k
sin
(
ω
r
x
k
t
+
φ
r
x
k
)
,
x
x
0
3
k
θ
y
(
t
)=
θ
y
0
+
∑
θ
y
k
sin
(
ω
r
y
k
t
+
φ
r
y
k
)
,
=
1
3
k
θ
z
(
t
)=
θ
z
0
+
∑
θ
z
k
sin
(
ω
r
z
k
t
+
φ
r
z
k
)
,
=
1
(19)
3
k
⎩
T
x
(
t
)=
T
x
0
+
∑
1
T
x
k
sin
(
ω
t
x
k
t
+
φ
t
x
k
)
,
=
3
k
=
1
T
y
k
sin
(
)=
+
∑
(
ω
t
y
k
t
+
φ
t
y
k
)
,
T
y
t
T
y
0
3
T
z
(
t
)=
T
z
0
+
∑
k
=
1
T
z
k
sin
(
ω
t
z
k
t
+
φ
t
z
k
)
,
where the overall coefficients, i.e.
ω
r
x
k
, etc., can be estimated using the iteratively-
reweighted least-squares (
IRLS
) method [2] with historic data.
IRLS
continuously
updates a weight function so as to minimise the effects of gross outliers within the
optimization. The motion parameters at time instant,
t
, will become deterministic.
2.
M-step
:
Eq. (17) is iterated in order to find its maximum. This involves finding
θ
x
0
,
˜
φ
t
=
˜
M
(
φ
)
so that
t
−
1
˜
˜
˜
˜
Q
(
φ
|
φ
)
≥
Q
(
φ
|
φ
)
.
(20)
t
t
−
1
t
−
1
t
−
1
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