EVALUATION OF THE BETA-SPLINE INTERPOLATOR FOR DIGITAL 
ELEVATION MODELS WITH DIFFERENT BETA PARAMETERS ON EACH PATCH 
by 
Carlos Eduardo Nery 
Assistant Professor 
Paraiba Valley University - UNIVAP 
12201 Sào José dos Campos, SP, BRAZIL 
ISPRS Commision IV - WG 7 
and 
Luiz Alberto Vieira Dias 
Chairman, Center for Associated Technologies - CTE 
National Institute 
for Space Research - INPE 
12201 Sao José dos Campos, SP, BRAZIL 
ISPRS Commision IV - WG 7 
ABSTRACT 
The use of the Beta-spline interpolator in Digital Elevation Models permits to shape 
the interpolated surface to approximate the terrain shape without changing its control 
points. This interpolator has a user defined set of parameters that enables one to 
control the bias and the tension of the interpolated surface. The results are shown in 
graphical form. 
KEY WORDS: Digital Elevation Models, DEM, Numerical Interpolation, Beta-splines. 
1. INTRODUCTION 
Sometimes certain terrain features are 
known a priori, but when an interpolator 
is used, on a Digital Elevation Model, 
DEM, these features are masked by the 
interpolator properties, like the 
smoothing of break lines. 
The solution would be to use an 
interpolator whose shape could be 
controlled by the user. The beta-spline 
interpolator has such capability. It is 
possible, by means of the two parameters 
betal and beta2, to control, 
respectively, the bias (to "push" or to 
"pull" the interpolated surface toward 
or away to the vertices of the contro 
polygon, as will be seen shortly. 
This work presents a case study about a 
terrain in which some information about 
it is known. The original data is used 
for comparision (considered as the 
"real" terrain). The original data is 
sampled and then interpolated by 
Beta-splines. It is used for different 
patches, different sets of beta 
parameters, thus changing the shape 
acoording to the knowledge available. 
Session 2 presents a brief description 
of the Beta-splines, the following 
session the results obtained, and 
finally a conclusion. The computer 
environment used consisted of 
Workstations for the generation of the 
plots, and IBM PC-like computers for the 
calculations. A simple program, written 
in turbo C provided the files to be 
visualized in the workstations and 
plotted in a laserjet printed. It was 
determined that the Beta-splines were 
935 
fast enough to run efficiently in this 
unsofisticated environment. 
2. BETA-SPLINES 
The Beta-spline interpolator (Barsky, 
1987), is an extension of the Bezier 
family of interpolators (Foley and 
VanDam, 1984). The main property is that 
the interpolated surface lies inside the 
convex hull determined by the polygon 
control vertices, thus the interpolator 
does not pass by the control vertices, as 
do other interpolators. 
The mathematical form is as a series, in 
which the control vertices are weighted 
by base functions. Parametrically it 
can be written as: 
z(u,v) * S 5 (V * 
ij 
* B;;(u,v,betal,beta2) (1) 
i=0, n-1 
j=0, m-1 
where S stands for summation, u and v are 
the variable parameters, i, j, the 
summation indexes, m and n the number of 
lines and columns, V.. the position of 
the polyedron control-vertices, B;.( ) 
the base functions. These base fundtions 
are dependent on i, j, u, v, and the user 
defined bias, betal and tension, beta2, 
but are independent on the position of 
the control vertices. 
The action of betal is the following: for 
betal equal to one the bias is neutral, 
for betal from 0 to plus infinity the 
bias is toward the initial values of v 
and u, from 0 to minus infinity toward 
the opposite side. The tension, as the