esamt für 
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ally Con- 
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ongress, 
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grammet- 
No. 3, pp. 
rocessing 
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93 - 71. 
  
COMPARISON BETWEEN AKIMA AND BETA-SPLINE INTERPOLATORS 
FOR DIGITAL ELEVATION MODELS 
by 
Luiz Alberto Vieira Dias 
Chairman, Center for Associated Technologies, CTE 
National Institute for Space Research - INPE 
12201 Sào José dos Campos, SP, BRAZIL 
ISPRS Commission IV - WG 7 
and 
Carlos Eduardo Nery 
Assistant Professor 
Paraiba Valley University - UNIVAP 
12201 Sao José dos Campos, SP, BRAZIL 
ISPRS Commission IV - WG 7 
ABSTRACT 
When interpolators are used in Digital Elevation Models, it is known that certain 
geographic features are not well represented. This work compares the performances of two 
interpolators for different geographic features: the Akima and the Beta-spline. The 
Akima interpolator is widely used in DEM's, while the Beta-Splines have user defined 
parameters that can control the shape of the surface without changing the control 
points. The results are presented in graphical form. 
KEY WORDS: Digital Elevation Models, DEM, DTM, 
Beta-Splines. 
1. INTRODUCTION 
A problem that frequently arises when it 
is necessary to select an interpolator 
for use in Digital Elevation Models, DEM, 
is the choice of the best, or at least 
good, interpolator for a given condition. 
This work compares the widely used Akima 
interpolator with the Beta-Spline 
interpolator. Their characteristics are 
very different, thus it is expected that 
they complement each other, according to 
the situation. 
A characteristic they share is the 
computer run time, that is fast for both. 
The environment used in this work was: 
the files were prepared on IBM PC-like 
computers, the visualization was done on 
workstations, and the graphs made in 
laser jet printers. 
Session 2 presents a brief description of 
both interpolators, in Session 3 the 
results obtained are shown, and the 
conclusions are discussed in Session 4. 
2. AKIMA AND BETA-SPLINE INTERPOLATORS 
2.1 - Akima Interpolator 
  
The Akima surface interpolator (Akima, 
1978) is a very interesting method, for 
it runs very fast on computers, passes 
for all vertices of the control 
polyhedron, has continuity of zeroth 
(passes by points) and first order 
(tangent continuity), at the patches 
borders. It has not second order 
continuity, thus the curvature is not 
925 
Numerical Interpolation, Akima, 
equal at the border of patches. However, 
for terrain description, this property is 
not essential, since the terrains vary in 
an abrupt way in certain cases. 
Mathematically it is represented as a 
cubic polynomial in two variables (Akima, 
1979): 
z(x,y] 9:8 (A,, + x} x yJ) (2) 
17 
i=0,3 
i= 0,3 
The determination of the coefficients 
Ass is made by means of a method devised 
byJAkima (1978). The method uses the 
Hermite interpolator, but instead of 
using the, generally unknown, derivatives 
at the control vertices, Akima devised a 
method to determine the derivatives based 
on the values of the neighbouring 
vertices. It uses, for each patch, with 4 
points, the 32 surrounding points (Akima, 
1978). For each patch, there is a 
different cubic polynomial in two 
variables, to represent the 
interpolated surface. 
2.2 - Beta-Spline Interpolator 
  
The Beta-Spline Interpolator has the very 
interesting property of permiting the 
user to modify the shape of the 
interpolated surface without changing the 
vertices of the control polyhedron. This 
enables onethe shape the surface in order 
to comply with some features already 
known (Barsky, 1987).