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FEATURE-BASED PHOTOGRAMMETRIC AND INVARIANCE TECHNIQUES 
FOR OBJECT RECONSTRUCTION 
H. F. Barakat, K. Weerawong, and Edward M. Mikhail, Purdue University 
Commission III - IWG II/III 
KEY WORDS: Invariance, Linear Features, Object Reconstruction, Least Squares Adjustment 
ABSTRACT 
Linear features, with independent descriptors in the object space, are affectively used in photogrammetric restitution 
and object reconstruction. 
Point-based invariance is discussed and its applications in Image 
Understanding/Computer Vision are contrasted with photogrammetry. Object reconstruction, being a common 
invariance and photogrammetric task is evaluated by both techniques using synthetic and real image data. Research 
is continuing on multi-image invariance, multi-feature construction, and combined invariance/photogrammetry 
techniques. 
1. INTRODUCTION 
Imagery used to reconstruct the objects recorded is in 
general two-dimensional representation of usually three- 
dimensional objects. Image features are of three types: 
points, lines and areas. Until recently, photogrammetric 
methodology has been based primarily on point 
features, particularly because of extensive use of hard- 
copy image input. The increased use of digital imagery 
has opened up opportunities for exploiting linear 
features since they are both abundant in imagery of 
human infrastructure, and amenable to extraction by 
automated algorithms. The inclusion of linear features, 
alone or in combination with point features, into 
photogrammetric reduction algorithms requires careful 
development and analysis. 
Image invariance refers to the existence of properties, 
derived from images, which are invariant under specific 
imaging geometry, the most common of which is central 
or perspective projection. One very early property used 
in graphical rectification is the anharmonic or cross- 
ratio. In recent years, activities in Image Understanding 
(IU) and computer Vision (CV) has resulted in 
significant development in image invariance. As in 
photogrammetric research, point-based development 
preceded line-based invariance. Although IU/CV 
applications of invariance encompass different tasks, 
object reconstruction from overlapping imagery is an 
application which is also common to photogrammetry. 
Feature-based, particularly linear features, 
photogrammetric techniques for the reconstruction of 
imaged three-dimensional objects is discussed in section 
2. A brief introduction to the invariance concept and its 
uses is given in section 3. Results from experiments 
using both simulated and real imagery are provided for 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
both techniques. Conclusions and continuing research 
are in the fourth and last section. 
2. LINE-BASED PHOTOGRAMMETRIC 
RESTITUTION 
2.1 Linear Feature Description 
A point feature is represented by two coordinates in 2D 
space and three coordinates in 3D space. Linear 
features can be similarly described. Considering 
straight lines, they are defined by two parameters in 2D 
space, and four independent parameters in 3D space, 
expressed by equations (2.1) and (2.2), where p (the 
distance from the origin to the line) and a (its angle 
with the x-axis) are the 2 parameters in 2D; q (the 
distance from the origin to the line), and ,. p, p, 
(angles effecting rotation such that the line is along one 
coordinate axis) are the 4 parameters in 3D. A circle 
is defined by 3 parameters in 2D, and 6 independent 
parameters in 3D, and represented by equations (2.3) 
and (2.4) in which x, y, are coordinates of its center in 
the plane, and 7 its radius; X,Y ,Z_ are coordinates of 
the center; R its radius, and a, a, are the angles 
defining the unit vector P perpendicular to the plane of 
the circle in 3D. 
X,cos 0. +y sina =p (2.1) 
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