Information Technology Reference
In-Depth Information
3.2 Context Modelling and Classification of 3D Objects
Once the borders and surface shape of object are determined, next step to analyze
is object context (i.e. voxel labels). Suppose a 3D image object
G
≡
N
x
×
N
y
×
N
z
voxels. Assume that this image contains
K
subvolumes and that each voxel
v
is
decomposed into a voxel object
o
and a context (label)
l
. By ignoring information
regarding the spatial ordering of voxels, we can treat context as random variables
and describe them using a multinomial distribution with unknown parameters
π
k
.
Since this parameter reflects the distribution of the total number of voxels in each
region,
π
k
can be interpreted as a prior probability of voxel labels determined by
the global context information.
The finite mixtures distribution for any voxel object can be obtained by writing
the joint probability density of
o
and
l
and then summing the joint density over all
possible outcomes of
l
, i.e., by computing
()
( )
∑
p
o
=
p
o
,
l
, resulting in a sum
v
v
l
of the following general form [4]:
K
()
()
∑
=
p
o
=
1
π
p
o
,
v
∈
G
(11)
Γ
v
k
k
v
k
where
o
v
is the grey level of voxel
v,
p
k
(
o
v
)s are conditional region probability
K
density functions with the weighting factor
π
k
, if
π
k
> 0, and
∑
=
π
=
1
.
k
k
1
Index
k
denotes one of subvolumes
K
. The whole object can be closely ap-
proximated by an independent and identically distributed random field
O
. The cor-
responding joint probability density function is
N
K
()
( )
∑
= =
P
o
=
∏
1
π
p
o
(12)
k
k
v
i
1
k
[
]
where
.
Disadvantage of this method is that it does not use local neighbourhood infor-
mation in the decision. To improve this, the following should be done. Let
Q
be
the neighbourhood of voxel
v
with an
N
×
N
×
N
template centred at voxel
v
. An in-
dicator function
o
=
o
,
o
,...,
o
, and
o
∈
O
1
2
(
)
I
, is used to represent the local neighbourhood constraints,
where
l
v
and
l
w
are labels of voxels
v
and
w
, respectively with
v
l
w
, . The pairs
of labels are now either compatible or incompatible, and the frequency of
neighbours of voxel
v
, which has the same label values
k
as at voxel
v
,
can be
computed as:
v
w
∈
Q
(
)
1
( )
(
v
)
∑
π
p
l
k
|
l
I
k
,
l
=
=
=
(13)
k
v
Q
w
3
N
−
1
w
∈
Q
,
w
≠
v
(
v
k
)
where
l
Q
denotes the labels of the neighbours of voxel
v
. Since
is a condi-
tional probability of a region, the localized probability density function of grey-
level
o
v
at voxel
v
is given by:
π
