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TRANSFORMATION 
ULTRA-PRECISE MEASUREMENT USING AFFINE 
ATSUSHI OKAMOTO 
KYOTO UNIVERSITY KYOTO JAPAN 
COMMISSION V 
ABSTRACT 
Affine transformation plays a minor role in pho- 
togrammetry. However, this transformation shows 
great effectiveness in the cases where projective 
transformation is very weak. This paper presents a 
general orientation theory of two-dimensional 
affine images and clarifies characteristics of the 
model construction and the one-to-one correspon- 
dence between the model and object spaces. Next, 
a general free network theory for affine images is 
constructed with twelve linearly independent vec- 
tors. Practical characteristics of the proposed the- 
ories are discussed through tests with simulated ex- 
amples. 
INTRODUCTION 
Affine imageries do not exist in reality. For this 
reason the orientation theory of affine images has 
probably not been derived until now. However, 
this orientation theory is of great practical use in 
many photogrammetric fields such as precise 
three-dimensional measurement of small objects 
using a conventional comparator and the analysis 
of satellite CCD camera imageries. In the latter 
case the conventional orientation approach of pho- 
tographs cannot be employed, because the field an- 
gle of the CCD camera is very narrow and height 
differences in the photographed terrain are very 
small for the flying height of the satellite. There- 
fore, this paper derives a general orientation the- 
ory of two-dimensional affine images and clarifies 
the geometrical properties of the relative and abso- 
lute orientation. Then, a general free network the- 
ory for affine images is constructed. The theories 
proposed here are tested with simulated examples 
so as to explore the difficulties when applying them 
to practical cases. 
GENERAL ORIENTATION THEORY OF TWO- 
DIMENSIONAL AFFINE IMAGES 
Let a three-dimensional object space (X,Y,Z) be 
projected into a plane based on affine transforma- 
tion(See Figure-1.). The basic equations relating 
    
   
measured image plan 
(affine image ) 
object space 
Figure-1 : parallel projection of an object space 
into the measured plane of the compa- 
rator coordinate system. 
an object point P and its measured image point 
Pe(Xc.ÿc) are described (Okamoto (1989)) as 
Xe = a1X + a2Ÿ + a3Z + a4 
(1) 
yc = asX + agY + a7Z * ag 
in which ai (i=1, --,8) are independent coefficients. 
Geometrically, the eight orientation parameters of 
the affine image are considered to be three rotation 
parameters (9,9K) of the image, two translation 
elements (Xoc, Yoc) which indicate two of the three- 
dimensional coordinates of the origin of the mea- 
sured image coordinate system (Xc;ÿc) with respect 
to the object space coordinate system (X,Y,Z), the 
image scale s, and two rotation parameters (&,B) 
describing the relationship between projected rays 
and the normal to the image plane. The eight ori- 
entation parameters of a single affine image can 
thus be provided uniquely if four control points 
are available. 
Next, we will consider the orientation problem of a 
stereopair of two-dimensional affine images (See 
Figure-2.). The basic equations are written down 
as