Digital Signal Processing Reference
In-Depth Information
2
Ideas and Methods
2.1
Basic Concept of MPS
Kriging and stochastic simulation, both based on the variogram which only describes
the relations between two points in space and cannot reconstruct complex patterns
such as curvilinear shapes, are called two-point geostatistics. However, multiple-point
geostatistics describes the relations of multiple points
around a node to be simulated,
so the disadvantages of two-point geostatistics are overcome. By reproducing high
order statistics, MPS allows capturing structures from a training image, and then
anchors them to specific model data. A training image is a numerical prior model
which contains the structures and relationship existing in realistic models [2].
The training image is scanned using a data template
τ
n
that comprises
n
locations
u
α
and a central location
u
. The
u
α
is defined as:
u
α
=
u
+
h
α
(
α
=1,2,…,
n
), where the
h
α
are
the vectors describing the data template. For example, in Fig. 1(a),
h
α
are the 80
vectors of the square 9×9 template. In Fig. 1(b),
h
α
are the 26 vectors of the cubic
3×3×3 template with a blue center
u
.
Consider an attribute
S
that has
K
possible states {
s
k
;
k
=1,2,…,
K
}. A data event
d
n
of size
n
,
centered at location
u
,
constituted by
n
vectors
u
α
in
τ
n
is defined as:
d
n
=
{
S
(
u
α
)
=
s
α
;
α=
1,2,…,
n
}. (1)
where
S
(
u
α
) is the state at the location of
u
α
within the template.
d
n
actually means that
n
values
S
(
u
1
)…
S
(
u
n
) are jointly in the respective states
k
s
. Fig. 2 illustrates
the procedure of capturing a data event with a 5 × 5 template. And Fig. 3 illustrates
two data events captured by the data templates displayed in Fig.1 (a) and (b)
respectively. The different colors in the Fig. 3 mean different states of an attribute.
Scanning a training image using a data template is to get the probabilities of
occurrences of the data events
d
n
, i.e., probabilities of the
n
vectors
u
1
,…,
u
n
within
the
τ
n
jointly in the respective states
s
…
n
s
s
, …,
[6, 7]:
k
n
s
α
Prob{
d
n
}=Prob{
S
(
u
α
) =
;
α=
1,2,…,
n
}. (2)
In the process of scanning a training image using a given data template, it is a
replicate when a data event in the training image has the same geometric
configuration and the same data values as
d
n
associated with
τ
n
. Under the hypothesis
of stationarity, i.e., the statistics are location-independent, the probability of
occurrences of the data events
d
n
is the proportion of replicate number
c
(
d
n
) found in
the training image and the size of effective training image denoted by
N
n
[3, 8]:
cd
N
()
n
s
α
≈
Prob{
S
(
u
α
) =
;
α=
1,2,…,
n
}
. (3)
n
At any unsampled node
u
, we need to evaluate the cpdf (conditional probability
distribution function) that the unknown attribute value
S
(
u
) takes anyone of
K
possible states
s
k
given
n
nearest data denoted by
S
(
u
α
)=
s
α
(
α=
1,2,…,
n
). According
to the Bayesian relation, the above cpdf is defined as:
