-* AS HO
0g 2D HP 0 Hy
00200 0 DHDMDUHDH-+ o3
o5 m0utrm.u
basis of these descriptions of the terrain. For instance, if
the terrain is described by its power spectrum, this
spectrum may be approximated by an exponential function
(Frederiksen et.al, 1978). The parameters of this
exponential function are then used as characteristic
parameters for describing different terrain types. When at
last the characteristic parameters are selected, they are to
be used for terrain classification in the daily work.
This paper addresses the problem of selecting the
characteristic parameters as described above. That is, given
a certain description of the terrain, for instance its power
spectrum or its autocorrelation function, which parameters
are needed to describe the terrain? The proper selection of
these parameters is very essential for its practical use.
The number of characteristic parameters should be small,
they should be easy to estimate and they should be well
suited to the purpose.
Unless the underlying impellents of the elevations of the
terrain are known, our selection of characteristic
parameters has to be based on empirical studies. The
Karhunen-Loeve expansion is an objective method for this
purpose. The object of this paper is to describe the basic
principles of Karhunen-Loeve expansion when used for terrain
classification. The aim is not to establish a general model
for terrain classification, but rather to pay attention to
the possibilities and limitations of an empirical approach
like the Karhunen-Loeve expansion.
2. BASIC PRINCIPLE OF THE KARHUNEN-LOEVE EXPANSION
The Karhunen-Loeve (K-L) expansion is a method for the
transformation Of a set of functions into a set of
orthonormal functions. The method is well known from
different sciences. In Digital Image Processing, the method
is used for data compression (Rosenfeld and Kak, 1982),
while in hydrology it is used for the study of similarities
and dissimilarities among hydrological data sets (Stokes,
1974). In Rao (1965), the method is described theoretically
in the section covering principal component analysis of
random variables.
For the completeness of this paper, a brief review of K-L
expansion is given in this chapter. A more detailed
description can be found in several textbooks, for instance
in Rosenfeld and Kak (1982) or Rao (1965).
Assume that we in one way or another can observe a variable
t and its outcome e). Assume also that each function
ey (t) can be expressed as a linear combination of a set of
base functions a,(t), such that
- 683 -