Complexity of the Geological Framework and Use of Mathematics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 28

Fig. 1.16 Dilatations of set C for 40 deposit points and intersection of resulting sets with original
pattern: (a) C 4 B ;(b) C 9 B ;(c) C 19 B ;(d) A \ ( C 4 B ); (e) A \ ( C 9 B ); (f) A \ ( C 19 B )

(Source: Agterberg and Fabbri 1978 , Fig. 3)

pattern of Fig. 1.14d is set A with measure mes A . This measure is the area and can be
expressed either as mes A 1,264.01 km 2 , because one pixel ¼ 18,843 pixels or mes A ¼

represents an area of 259 m 259 m. Let B be the operator set of the eight-neighbor

square logic. B has an origin which is located in the center of the square described by the
eight neighboring points. The patterns of Fig. 1.14e, f can be represented as the
Minkowski sums

Α Β

and

Α

2 , respectively. A new set nB canbedefinedby

Β

induction with nB ¼ [( n 1) B ] Bfor n ¼ 2,3, ... . It is seen readily that operating on

A with the set nB is identical to applying the successive operations

Α

. By using the concept of Minkowski subtrac-
tion, the patterns of Fig. 1.14a-c can be written as A

n

Β ¼

[

Α

( n 1) ] B for n ¼ 2,3, ...

Β

, respectively.
If the superscript c denotes complement of a set with respect to the universal
set T which consists of all pixels in use, then the patterns of Fig. 1.15a, b are
A

ʘ Β ,

Α ʘ

2 ;and

Α ʘ

3

Β Β

A c , respectively. A set C was formed by assigning each of the
40 massive sulphide deposits in the area to the pixel closest to it on the grid with
259 m-spacing used for the binary images of Figs. 1.14 and 1.15 . C consists of
40 black pixels which can be subjected to successive dilatations by use of B. The
sets C

B ) c and \ ( A ʘ Α Β \

19 B are shown in Fig. 1.16a-c , respectively. Because
each pixel is representative for a cell of 259 m on a side, the length of a cell
generated by n dilatations is equal to (2 n +1) 259 m. Hence the cells obtained by
4, 9 and 19 dilatations of a single pixel are 2.33, 4.92, and 10.10 km, respectively.
The latter two cell sides can be used to approximate (5 km

4 B , C 9 B and C

5 km) cells and

(10 km 10 km) cells, respectively. The patterns of Fig. 1.16a-c can be intersected

with that of Fig. 1.14d . The resulting sets are shown in Fig. 1.16d-f , respectively.