BUNDLE ADJUSTMENT IN NO NEED OF APPROXIMATIONS OF PARAMETERS 
Susumu Hattori, Akiyoshi Seki 
Dep.of Information Processing Eng.,Faculty of Eng., 
FUKUYAMA Univ., Fukuyama, 729-02, JAPAN 
ISPRS Commission III 
ABSTRACT 
Bundle adjustment has been widely used in orientation and camera calibration. But since 
the observation equations are non-linear, 
approximations of parameters are necessary at 
the start of adjustment. This paper discusses a method and a procedure to evaluate the 
approximations automatically associated with any model or object space coordinate sys- 
tems. This method can realizes semi-automatic bundle adjustment or photogrammetry with- 
out control points. The method is based on relative orientation by the linear coplanari- 
ty condition and decomposition of rotation matrices to angular elements. This is vali- 
dated by experiments of simple orientation of a pair of photographs and a camera cali- 
bration without control points using a 3-D target field. 
Key Word: Bundle Adjustment, Automatic Adjustment, Camera Calibration, Close Range Photo- 
grammetry, Approximations of Parameters 
1. INTRODUCTION 
Bundle adjustment has been widely used in 
camera calibration and triangulation. But 
since observation equations are non- 
linear, approximations of all parameters 
are required at the beginning of computa- 
tion. 
In close-range photogrammetry the approxi- 
mations of exterior orientation parame- 
ters are usually recorded at exposing 
positions. But it is time consuming and 
sometimes hard, because a convergent or 
parallel imaging configuration rather than 
vertical one is often used. For a digital 
plotter (digital-image-based plotter) 
which is now being developed in many 
organizations (Lohmann, 1989, 
Ohtani,1989), easy manipulation is sub- 
stantially required by operators who are 
not familiar with photogrammetry. Hence an 
automatic or semi-automatic adjustment 
procedure is now strongly called for. 
This paper shows a method to automatically 
calculate approximations of exterior 
orientation parameters and coordinates of 
object points associated with any model or 
object space coordinate system. The method 
is based on relative orientation using the 
linear coplanarity condition and decompo- 
sition of rotation matrices to angular 
elements (Hattori, 1991). 
In practice the purpose of many industrial 
measurements is focused only on shapes of 
objects, not absolute coordinates. And 
camera calibration works also can be 
executed only by the coplanarity condition 
in any coordinate system (Fraser,1982). 
The authors' method solves the problem 
about the selection of a coordinate sys- 
tem, and realizes photogrammetry without 
control points. It is very useful in 
digital plotters, because one can easily 
define any coordinate system on the 
screen, observing a model stereo-optical- 
jv. 
2. OUTLINE OF EVALUATION OF INITIAL 
VALUES OF PARAMETERS 
Fig.1 shows an example of an imaging 
configuration in a camera calibration 
200 
which will be again referred to in 
experiments. A three dimensionally allo- 
cated targets are imaged convergently at 
various positions and with various camera 
rotations. The following is a flow of the 
procedure to obtain approximations of 
parameters. 
(1) Overlapping photographs are separated 
to each independent model. Rotation ma- 
trices of independent models are evaluated 
and decomposed to angular elements (see 
S. X. 
(2) The independent models are linked to 
make a global model(see 4.1). 
(3) If necessary, the global model coordi- 
nate system is transformed to the object 
space coordinate system using more than 
three control points(see 4.2). 
(4) Object space coordinates of target 
points are calculated. Then the rotation 
matrix of each photograph in the object 
space coordinates system ( or in the 
global model coordinate system) is decom- 
posed to angular elements(see 4.3). 
3. —RELATIVE ORIENTATION BY THE 
LINEAR COPLANARITY CONDITION 
3.1 Coplanarity condition 
Let us start with a pair of overlapping 
photographs. The interior orientation is 
assumed complete. Model coordinates of 
two corresponding points are expressed, 
as shown in Fig.2-1, as 
Xp " m44 m45 m43 X4 
Yp4 | "21 "22 "23 E 
Zp4 m31 "32 "33| 1-6 
(15 
Xp» " 044 012 713 Xa lo +18 
Yp2 Nay Nag Nag Ya 0 
Zp; mango ml 0 
where (X4 Yq -a)T, (x, Yo —o)l are, photo- 
graphic coordinates, (Xp4 Yp4 ipi) ; (Xpa 
Ypo Zpa)! are model coordinates, c is a 
camera distance and B is a base length ( 
unity with an unknown sign). The copla- 
narity condition: 
whe 
pa 
P3 
a? 
e 
"3 
It 
igi 
wh: 
re: 
mi. 
th 
mu 
By