TERRAIN ANALYSIS BY KARHUNEN-LOEVE EXPANSION 
Anders Östman 
Department of Photogrammetry 
Royal Institute of Technology 
Stockholm 
Sweden 
ABSTRACT 
The Karhunen-Loeve expansion is a method for the 
transformation Of a set of funetions into a set of 
orthonormal functions. The method has earlier been used in 
different fields such as digital image processing and 
hydrology. In this paper it is applied to terrain analysis, 
with special attention to terrain classification. 
For terrain classification, the autocorrelation functions of 
the elevations of six different areas have been used. The 
autocorrelation functions are transformed into orthogonal 
base functions by the Karhunen-Loeve expansion. Although the 
result presented in this paper is limited, it is concluded 
that the Karhunen-Loeve expansion may be used when designing 
a model for terrain classification. 
1. INTRODUCTION 
During the latter years, interest in terrain classification 
has grown. New mathematical and statistical concepts have 
been introduced in the geoscience field (Agterberg, 1982, 
Frederiksen et ai, °-1984), increasing our theoretical 
framework. These new tools improve the daily use of terrain 
information. One subject of special interest for 
photogrammetrists is terrain classification. This problem is 
here regarded as a problem of data reduction and feature 
extraction. From a continuous terrain surface, a limited set 
of characteristic parameters are extracted, containing the 
information needed for a specific purpose. 
The form of the terrain surface is very complex. It is here 
assumed that it is formed by a number of impellents, such as 
geological processes and human activities. An attempt to 
describe a sedimentation process was made by Jacobi and 
Kubik (1982), but similar descriptions of the remaining 
impellents are still missing. 
Instead of describing the driving processes, the elevations 
of the terrain itself may be described, using mathematical 
and statistical methods. In a review by Frederiksen et.al 
(1984), several such descriptions are discussed, such as 
power spectra, covariance functions, covariograms and 
fractals etc. When establishing a basic model for terrain 
classification, characteristic parameters are derived on the 
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