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increase in distance between two adjacent nodes in the tree, for which a maximum
allowed value is determined empirically. For each resulting cluster
l
, the
w
eight
w(l)
is set according to the number of points
P
belonging to
l
as
w(l)
√
P
.The
square root is used to constrain the weight for clusters consisting of many points.
For each cluster the mean velocity of all points belonging to it is determined.
=
2.3.4.3 Generation and Tracking of Object Hypotheses
From here on, persons and objects can be represented as a collection of clusters of
similar velocity within an upright cylinder of variable radius. An object hypothesis
R(a)
is represented by a four-dimensional parameter vector
a
(x,y,v,r)
T
, with
x
and
y
being the centre position of the cylinder on the ground plane,
v
denoting
the velocity of the object, and
r
the radius. This weak model is suitable for persons
and most encountered objects.
To extract the correct object positions, we utilise a combination of parameter op-
timisation and tracking. We first generate a number of initial hypotheses, optimise
the location in parameter space, and then utilise the tracking algorithm to select hy-
potheses which form consistent trajectories. Initial object hypotheses are created at
each time step by partitioning the observed scene with cylinders and by including
the tracking results from the previous frame, respectively. Multidimensional un-
constrained nonlinear minimisation (Nelder and Mead,
1965
), also known as the
simplex algorithm, is applied to refine the position and size of the cylinders in the
scene, so that as many neighbouring clusters with similar velocity values as possible
can be grouped together to form compact objects, as shown in Fig.
2.21
b. An error
function
f(
a
)
used for optimisation denotes the quality of the grouping process for
a given hypothesis. Each hypothesis is weighted based on the relative position, rel-
ative velocity, and weight of all clusters
l
within the cylinder
R(
a
)
using Gaussian
kernels according to
=
f
r
(
a
)
l
∈
R(
a
)
f(
a
)
=
w(l)f
d
(l,
a
)f
v
(l,
a
)
(2.37)
r(
a
)
2
2
H
r,
min
r(
a
)
2
with
f
r
(
a
)
=
exp
(
−
)
−
exp
(
−
2
H
r,
max
)
keeping the radius in a realistic range,
2
−
[
s(l)
−
s(
a
)
]
f
d
(l)
=
exp
(
)
reducing the importance of clusters far away from the
2
H
d
2
2
H
v
)
masking out clusters having dif-
fering velocities. The functions
r(
a
)
,
s(
a
)
, and
v(
a
)
extract the radius, the two-
dimensional position on the ground plane, and the velocity of the hypothesis
a
,
respectively. The kernel widths
H
are determined empirically. Figure
2.22
shows
the error function from (
2.37
), parameterised for opposing velocities. Local minima
are centred on top of the objects of interest.
After optimisation, hypotheses with identical parameterisation are merged and
those without any clusters within
R(
a
)
are removed. The remaining hypotheses are
tracked over time using a particle filter, keeping only object hypotheses forming a
−
[
v(l)
−
v(
a
)
]
cylinder centre, and
f
v
(l,
a
)
=
exp
(
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