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OBJECT RECONSTRUCTION USING 2-D PROJECTIVE TRANSFORMATIONS 
Ilkka Niini 
The Academy of Finland 
Present: Helsinki University of Technology 
Institute of Photogrammetry and Remote Sensing 
SF-02150 Espoo, Finland 
Abstract 
A method for object reconstruction from arbitrary images is presented. Relative orientation of the images is based 
on the 2-D projective transformations. As a result, the normal case of stereophotogrammetry is obtained. The 
images are resampled along the rows in such a way that the new rows coincide with epipolar lines. Therefore 
the digitization of the object is easily performed with matching along the rows. As an experimental example the 
reconstruction of the Cathedral of Helsinki is presented, where the images are parts of digitized old viewgraphs. 
KEY WORDS: Close-range, 3-D, Photogrammetry 
1. INTRODUCTION 
In conventional  stereophotogrammetry object 
reconstruction is usually performed using nonlinear 
relative orientation, based on the coplanarity 
condition. As a final result an object model is obtained 
which can be digitized. Using digital images, this 
procedure has been tried to be automatized, but these 
algorithms have been more or less restricted to certain 
applications. 
In this paper an object reconstruction method is 
presented which uses 2-D projective transformations 
for relative orientation of images. These transform- 
ations are linear in principle, and in addition, no 
information about the interior orientation of the 
images is needed. As a result of the relative 
orientation, a stereopair of images is obtained which 
satisfies the normal case of stereophotogrammetry. 
After the orientation, the 3-D model coordinates of the 
object can be produced along parallel epipolar lines. 
The measurement stage of this method is a 
combination of manual and computerized measure- 
ments. In future, more automation is included. The 
method is developed at the Helsinki University of 
Technology. 
The images may be anything between digital aerial 
photographs and postcard views digitized with a grey- 
scale scanner. Typically, digitized video images of size 
512 x 512 pixels have been used. 
2. RELATIVE ORIENTATION USING 2-D 
PROJECTIVE TRANSFORMATIONS 
The relative orientation method has its base on the 2- 
D projective transformations instead of the perspective 
collinearity condition. There is a more precise 
mathematical presentation of the method in the paper 
by Haggrén & Niini, 1990. 
677 
The method has three parts: 1) computation of 
singular correlation between the images, 2) 
computation of projective transformation coefficients 
of both images from the correlation parameters, and 
3) rectification of the digital images on a new plane, 
which corresponds to the plane where the images are 
asina normal case of stereophotogrammetry. Finally, 
a 3-D object model can be computed using 
conventional parallax equations. 
For convenience, the main parts of the method are 
presented here. Readers who are already familiar with 
the mathematical concepts may continue reading from 
chapter 3. 
2.1 MATHEMATIC CONCEPTS 
The camera model is assumed to be a pinhole camera, 
ie. no radial distortion exists. The 2-D projective 
transformation equations for the first image are 
aux by;c 
Xe 1514 Dy; C, (1a) 
"o guy hy1 
> > f 
n dx eiy it 1 (1b) 
^o gx hyÿ,+ 1 
and for the second image 
a9x";+ Daÿ";+ C; 
n (2a) 
xe gx'hjy'-1 
E ur dx" e;y',- f, (2b) 
'ogx'i+ hy'+ 1 
where X’, Y;, and X", Y", are the new image 
coordinates of the first and second image at any 
arbitrary plane. The x,y’; and x",,y", are the original 
image coordinates. Index i is the number of the point 
(i=1...n).