Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

So far we discussed spatial relationship between two points by fuzzy
measure. Now, we shall discuss the spatial relationships between two objects.
Let A and B be two objects and {ai , 1 ≤ i≤n }, { bj , 1≤ j≤m } be the set of
points on the boundary A and B respectively. We first compute the angle θij
between each two points aI and bj . Since there are n aI points and m bj points,
the total occurrence of θij will be (m x n). Now, for each type spatial relation
like bj below aIi, we estimate µbelow(θij). Since θij has a large range of value [0,
∏ ], we may find equal value of µbelow(θij) for different values of θij. A
frequency count of µbelow(θij) versus θij is thus feasible. We give a generic
name f (θ) to the frequency count. Since f (θ) can have the theoretical largest
value (n. m), we divide f (θ) by (m. n) to normalize it. We call that
normalized frequency f(θ) = f (θ) /(m .n). We now plot f(θ) versus θ and
find where it has the largest value. Now to find the spatial relationship
between A and B, put the values of θ in µbelow(θ) where f(θ) is the highest.
In fig. 11.7 we illustrate the method of measurement of the possible
θijs. Since abcd is a rectangle and pqr is a triangle, considering only the
vertices, m .n =3. 4 =12. We thus have 12 possible values of θij. So f(θ) =
f(θ)/12. It is appearing clear that f (θ) will have the largest value at around
45 degrees (fig. 11.8); consequently µbelow(θ=45 ° ) gives the membership of
pqr being below abcd.
1.00
F(θ)
0 ° 45 ° 90 ° 135 ° 180 °
Fig. 11.8: Theoretical f ( θ) versus θ for example cited in fig. 11.7.
θ