Workshop proceeding - final.pdf - Faculty of Information and ...

where we are using the same symbol Γ for the redefined merit function of (42). Note that similar to
(34), the merit function in (42) also consists of two parts, the measurement information part
(represented by the first term on the right side) and the transition information part (represented by the
last three terms on the right side). Equation (42) differs (34) only in the measurement information part,
where the former has to perform maximization over the space of nuisance parameters b
k + 1
. We now
consider the ML estimate of the nuisance parameters. The logarithm of the likelihood function in (42)
can be written as
2 sk+ 1+
uk+
1
0 0
( ) { ( )}
( i
ln , l ) 2
p zk+ 1| xk+ 1, bk+ 1
=−Nlnσ
− + ln I
2
0
2 uk+ 1zk+
1
/ σ
(43)
σ
(,) il
where s = ∑ S +
; ( i0, l0)
denotes the corresponding cell when a target locates at x k +1
.
k+ 1
z
(,) il∈
k 1
We evaluate the partial derivatives of the logarithm likelihood function as
( i0, l0) 2
( )
I ( )
( 0, 0)
1
2
1 1
/
i l
z
uk+ zk+
σ
k+ 1
xk+ 1 k+ 1 1
k+
1
=− +
2
2
∂u ( i0, l0) 2
k+ 1
σ I ( )
1
0
2 u
1 1
/
k
k
zk
σ σ u
+
+ +
( i0, l0) 2
I (
( )
0, 0)
1
2 u
1 1
/ i l
k+ zk+
σ
k+ 1
xk+ 1 k+ 1
2 z
1 1
k
u
k+ k+
k
= − −
2
2
2 2
( i0, l0) 2 2
2
∂σ
( σ ) σ I0( 2 uk+ 1zk+
1
/ σ ) ( σ )
∂ln p | , b 1
z
( z
)
∂ ln p | , b s + u N
+ 1 + 1
where I() = I() ′ is the first-order modified Bessel Function. Equating (44) to zero, we obtain
1 0
( i0, l0) ( uˆk+ zk+
2
σˆ
)
( i0, l0) ( uˆk+ zk+
2
σˆ
)
I1 2
1 1
/
uˆ
k + 1
( i0, l0)
I 1
0
2
1 1
/ zk
+
= (46)
Substituting the result to (45), and equating it to zero, we have
2
uˆk
+ 1= s k + 1− Nˆσ
(47)
By solving (46) and (47) jointly, we can find the MAP estimate of the unknown
2
parameter bˆ 2
ˆ ˆ
k+ 1
( σ , uk+
1)
. Notice that both ˆ σ and u ˆk + 1
enter the function of (46) in a nontrivial way,
therefore, solving the function will be an iterative numerical process, which is computationally
expensive.
Now we consider a simple way of estimating (
2
, ). The target signature only affects the cell it
σ u k + 1
locates, as a result, if a target is present and locates in cell ( i0, l0)
at k + 1st frame, the pixel ( i0 , l0
z )
k + 1
is a
(,) il
noncentral chi-square variable, while the other pixels z
k + 1
, (, il)
∈ S , (, il) ≠ ( i0, l0)
, are exponentially
distributed variables. Both the noncentral chi-square and exponential variables share the same
2
parameterσ , therefore, we can estimateσ 2 (mean of the exponential distribution) by averaging the
exponentially distributed pixels
( i0, l0)
2 1
( , ) 1 1
ˆ σ il sk − zk
= ∑ z
+ +
(,) il ,(,) il ( i 1
0, l0)
k +
=
∈ ≠
N −1 S
(48)
N −1
Substituting (48) into (47) gives
( i0, l0)
Nzk+
1
− sk+1
uˆ
k + 1
=
(49)
N −1
Substituting (48) and (49) into (42) and dropping the terms independent of ( xk+ 1,
mk+
1)
, we end up with
a modified merit function
Γ ( x , m )
k+ 1 k+ 1 k+
1
( i0, l0)
( i0, l0) ( i0, l0)
(2N−1)
z 2 ( N 1)( N z
1
k 1
sk)
z
k+
−s
⎧
⎫
k ⎪
⎛ −
+
− ⎞
k+
1 ⎪
=− +ln I
( i 0
0, l0) ⎨ ⎜
⎟
( i0, l0)
⎬
sk −z ⎜
k+ 1
sk −z
⎟
k+
1
⎩⎪
⎝
⎠⎭⎪
+ max ln ( | , ) + ln ( | , ) +Γ ( , )
xk,
mk
{ p x x m p m m x x m }
k+ 1 k k k+
1 k k k k k
where we are using the same symbol Γ for the redefined merit function of (50).
Once the merit function of last frame has been calculated, the last state and mode estimates are
obtained using
(44)
(45)
(50)
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