Complexity of the Geological Framework and Use of Mathematics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 23

COVARIANCE

D 2

D 1

1.0

0.6

RADIUS= 1.0

“ PERIMETER ” = 6.6

Fig. 1.11 Three types of measurements performed during image analysis of two-dimensional
objects (Source: Agterberg 1980 , Fig. 2)

1.5.1 Geometrical Covariance, Intercept and Rose Diagram

Two basic concepts of image analysis illustrated in Fig. 1.11 are the intercept
D α for direction α and the geometrical autocovariance (or covariance) K α ( x )for

direction and displacement x . In practice, the intercept of an image which

consists of black and white picture points is measured by counting the number
of times a black picture point is adjacent to a white picture point in a given
direction. The geometrical covariance is obtained by shifting the pattern with
respect to itself and measuring the area of overlap after each displacement. The
area of any original or derived pattern is determined by counting the number of
picture points it contains. For zero displacement, the covariance simply measures
the area A which is independent of direction, or K α (0)

α

A . The change in area
which arises from an infinitely small displacement dx in direction

¼

α yield the

intercept because

dK α ðÞ

dx

x

D α ¼

x¼ 0