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where
Δx
local
,i
and
Δy
local
,i
are the difference in the
x
-and
y
-coordinates of the
two local kernel centres respectively, and
σ
x
local
,i
and
σ
y
local
,i
are the standard
deviations of the local kernel in the
i
th historical linked pair of kernels. A second
spatial similarity,
s
foreign
,i
, is similarly computed for the two foreign kernels.
A temporal similarity,
s
i
, between the current and
i
th historical linked pairs
of kernels is computed and is given by,
s
i
=
exp
,σ
t
t
)
2
σ
t
(
t
−
−
=0
.
(9)
1
,
σ
t
=0
where
t
is the most recent transit time associated with the current linked pair
of kernels, and
t
and
σ
t
are the mean and standard deviation respectively of the
set of transit times associated with the historical linked pair of kernels.
2.3 Determining Correspondences between Pairs of Tracks
Based on the spatial and temporal similarities computed so far for the cur-
rent linked pair of kernels, a
track link weight
is calculated. This represents the
likelihood that the local and foreign tracks in the most recent track snapshots
associated with the current linked pair of kernels represent the same real-world
object. This link is represented by the track IDs of the respective tracks. If a
track link weight has not yet been initialised between these two tracks, then it
is initialised as follows,
init
=
w
s
i
w
tr
s
local
,i
·
s
foreign
,i
·
s
i
·
w
i
,
(10)
and if a track link weight already exists between the two tracks, then it is incre-
mented as follows,
init
=
w
t
old
+
w
s
i
w
tr
s
local
,i
·
s
foreign
,i
·
s
i
·
w
i
,
(11)
where
w
i
is the kernel link weight of the
i
th historical linked pair of kernels. By
summing the spatial and temporal similarities, we effectively assemble a weighted
kernel density estimator, where each weight is given by a kernel link weight. In
addition, a track link counter,
m
, of the number of times the track link weight
has been increased, is kept. The track link counter provides a measure of the
consistency of the evidence used to determine the track link weight.
As visual, spatial and temporal evidence is accumulated that these two tracks
represent the same real world object, their track link weight and track link
counter increases. Once both are suciently large, we can be confident there is a
correspondence
between the tracks. Specifically, a correspondence between two
tracks is declared if in any frame their track link weight crosses a first threshold,
w
t
min
, and if their track link count crosses a second threshold,
m
min
, i.e.,
C
=
True
,
if
w
tr
>w
t
min
and
m>m
min
(12)
False
,
otherwise
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