Information Technology Reference
In-Depth Information
N
Av
(
m
)=
{
(
v
j
(
m
)
,φ
j
(
m
)) :
j
=0
,...,q−
1
}.
(1)
where
φ
j
(
m
)is a
naming function
indicating the name of a
j
th neighbor of the
cell (
v
m
,m
). Let's agree that
φ
0
(
m
)=
m
.
Averaging procedure consists of computation for each
m ∈ M
the state values
v
m
=
1
q
v
j
(
m
)
,
(2)
N
Av
(
m
)
resulting in
Ω
Av
=
{
(
v
m
,m
):
v
m
∈
R
,m∈ M}
The inverse procedure which transforms a cellular array
Ω
R
=
{
(
u
m
,m
):
m ∈ M,u
m
∈
R
}
, into a Boolean array
Ω
B
=
{
(
v
m
,m
):
v
m
∈{
0
,
1
},m∈ M}
,
is called a
Boolean discretization
. As distinct to the averaging which is a de-
terministic and precise procedure, the Boolean discretization is an approximate
and probabilistic one. The formal constraint on the resulting Boolean array is
that for any
m ∈ M
u
m
−v
m
<
m
(3)
where
v
m
∈
R
is the averaged value of
Ω
B
, and
q
is the admissible approxi-
mation error.
There is no exact solution of this problem. The approximate one is obtained
by constructing
Ω
B
according to the following rule: the probability of the fact
that
v
m
= 1 is equal to
u
m
, i.e.
P
(
v
m
=1)
=
u
m
.
(4)
The above simple rule is a straightforward from the probability definition
provided
u
m
is constant on the averaging neighborhood. In the general case the
expected value M(
v
m
)is the mean cell state over the averaging neighborhood,
i.e.
M(
v
m
)=
1
q
v
j
(
m
)
P
(
v
j
(
m
)=1)
=
1
q
u
j
(
m
)
.
(5)
N
Av
(
m
)
N
Av
(
m
)
So, the approximation error
vanishes only in a number of trivial cases,
where the following holds.
u
m
=
1
q
u
j
(
m
)
.
(6)
N
Av
(
m
)
The set of such cases includes, for example, all linear functions and parabolas
of odd degrees when considered relative to coordinate system with the origin in
m
. In general case
δ
j
(
m
)=
u
j
(
m
)
−u
m
=0
, j
=1
,...,q−
1, and
M(
v
m
)=
1
(
u
m
+
δ
j
(
m
)) =
u
m
+
1
q
δ
j
(
m
)
.
(7)
q
N
Av
(
m
)
N
Av
(
m
)
