ROBUST 3D-OBJECT REPRESENTATION BY LINEAR FEATURES 
Anja Wilkin 
Institute of Photogrammetry and Remote Sensing 
Helsinki University of Technology 
Otakaari 1, SF-02150 Espoo, Finland 
Abstract: 
The use of linear features - instead of points - in photogrammetry allows a 3D-object reconstruction without corresponding 
points in the different images. The goal of this paper is the reconstruction of a straight line in space from contaminated 
image data. A new algorithm, based on the Random Sample method combined with the least squares adjustment, is able 
to perform the robust 3D-reconstruction of the line only by image observations from calibrated cameras. In numerous 
simulation experiments the algorithm is tested. 
Key words: Robust, object reconstruction, linear feature, straight line, random sampling consensus. 
0. INTRODUCTION 
In order to improve our existing photogrammetric station 
for close range applications we aim at an automatic object 
recognition for robot vision and at precise measurements on 
the object (e.g. a car carosse). For these purposes the 3D- 
object is represented by different types of linear features in 
the space like lines, circles or ellipses. The problem 
consists in reconstructing the 3D-structure from several 2D- 
images. The feature-based approach has the advantage that 
no point correspondence is required in the different images. 
So signalization of points on the object can be avoided and 
no image matching is necessary. 
The projection of a three dimensional linear feature into 
image space produces a two dimensional linear feature, 
which can be detected in the image as arbitrarily measured 
feature pixels. After extraction of these pixels from the 
images the determination of the 3D-feature parameters will 
be carried out with robust estimation methods. 
The paper is organized as follows: Chapter 1 introduces a 
model for the straight line representation, chapter 2 deals 
with robust methods for the 3D-reconstruction, chapter 3 
treats the details of the presented algorithm and chapter 4 
finally reports about the simulation results. 
1. MODEL FOR STRAIGHT LINE 
REPRESENTATION 
1.1 Representation of a straight line in space 
A straight line in the 3D-Euclidean space has 4 degrees of 
freedom in its parametric representation (Roberts, 1988). 
Defining the straight line in terms of an arbitrary point C 
= {C,C,,C,} and an orientation B = (B,B,.B,) is a 
representation which is not unique and uses more 
parameters than necessary. For that reason two constraint 
equations are imposed: First, the direction vector B is 
forced to be a unit vector. Second, C is chosen as the line's 
nearest point to the origin, the line center point. 
constraint 1: |B] = B2+ B +B =1 (1-1) 
constraint 2: BeC = B, C,+ B, C,+B,C, = 0 (1-2) 
  
  
X 
  
  
  
Fig. 1: Straight line representation, using two constraints 
The only remaining weakness of this representation is the 
undetermined sign of the vector D. Because this plays no 
role in the calculations later on, no conventions are made 
concerning the sign. 
1.2 Photogrammetric treatment of space lines 
Usually the relationship between object- and image space 
is expressed by the collinearity equation. The weak point of 
this pointwise representation is the need of many nuisance 
parameters in addition to the line parameters. À better way 
is a feature description based on the object space geometry 
(Mulawa, 1988). 
The image ray p is the vector from the perspective center 
L to the observed image point (x,y). Its direction in space 
is calculated from the orientation data of the corresponding 
perspective center. The image coordinates are assumed to 
be corrected from systematic errors. 
X 
p=Riy 
-C 
(1-3) 
R = rotation matrix; ¢ = camera constant