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Inferring from the 2D features in the image to the 3D 
structures of the object is the task of the geometric reasoning 
process which is solved by building up hypothesis about 
relations and geometric properties of the object in order to 
compensate the information lost in central projection. The 
assumptions mentioned above belong to these relationships. 
Especially the regular structure of polyhedral objects supplies 
several relationships, e.g. parallel and perpendicular lines as 
well as horizontal and vertical lines or a trihedral corner. 
The geometric reasoning process, implemented in the system, 
is devided into two parts: 
First relationships between 3D entities are determined by the 
system either automatically by grouping procedures or by an 
operator. Rules of the perspective geometry supply the mutual 
orientation of camera and object. In the second step the inverse 
perspective problem is solved by using the relationships as 
geometrical constraints. The shape of the object is determined 
by the system, calculating the planes of the object in space step 
by step. 
3.2 Relationships between 3D entities 
The following relationships between 3D entities are used in the 
reasoning process: 
given hypothesis 
* Faces of the object are assumed to be planes (polyhedral 
object). 
* Incidence relation: line is in plane, is_in(line, plane) 
determined by grouping 
* Parallel relation 
* Perpendicular relation 
* Collinear relation 
provided by an operator 
* Trihedral corner 
* Features with known length -» scale factor 
* Features with known angle 
The first two relations are introduced as hypothesis to the 
system before the reasoning process starts. The next three 
relations can automatically be determined by grouping 
procedures. Observe, that all the grouping procedures are not 
based on any knowledge about the 3D shape of the scene, any 
how they are able to supply the basis to build up hypothesis. In 
case of not finding enough relationhips, the system asks the 
operator to provide information about geometrical constraints. 
3.2 Rules of the perspective geometry 
The relationships between 3D entities, introduced as 
hypothesis about the shape of the object, are components of 
several rules of the perspective geometry used in the system. If 
a hypothesis is built up, the rule calls routines for calculating 
real values for attributes of spatial objects or relations, e.g. the 
direction vector of parallel lines. 
In the following some rules of the perspective geometry are 
presented: 
1. Vanishing points 
If the grouping procedure has found at least three lines in the 
image assuming them to be parallel in space, the vanishing 
point and the corresponding direction vector is calculated (Fig. 
3). 
Procedures for locating vanishing points are described in 
(BARNARD 1984, MAGEE/AGGARWAL 1984 and BRILL- 
AULT O’MAHONY 1991). 
2. Two parallel lines 
If two lines can be assumed to be parallel, the direction vector 
of the parallel lines can directly be calculated (NEV A- 
TIA/ULUPINAR 1991). 
3. Rectangular corner 
A rectangular corner consists of three lines, being 
perpendicular to each other and meeting in one point. As 
polyhedra or buildings often have a rectangular corner, this 
information can be applied as a starting rule, in case the corner 
is assigned by an operator. A known rectangularity allows to 
determine the mutual orientation of object and camera as the 
rotation matrix just represents the direction vectors of these 
three lines in space. There exist two solutions for the problem 
because the rectangular corner can be considered to be in front 
of the lines or back as well (KANATANI 1990, PAN 1990). 
4. Two parallel lines intersecting one perpendicular line 
If these three lines are located by an operator or a grouping 
procedure, the direction vector of the normal of the plane can 
be determined (Haralick, 1989). 
5. Three known vanishing points 
If three vanishing points are located in the image and their 
direction vectors enclose three angles of 90 degrees, the 
rotation matrix, the focal distance and the position of the 
principle point can be estimated. 
6. Two known vanishing points 
If two vanishing points are located in the image and their 
direction vectors enclose an angle of 90 degrees, the rotation 
matrix and the focal distance can be estimated. 
7. Plane of the object 
If two direction vectors of lines, not being parallel, and one 
point in space are known, the parameters of this plane of the 
object can be determined in space. 
8. Point in a known plane 
  
If a point, given in the image, belongs to a plane known in 
space, the 3D coordinates of this point can uniquely be 
calculated by intersecting the image ray of the point with the 
plane in space. 
9. Known scale factor 
In case of a distance between two object points is given, it is 
possible to transform the 3D model to an object of the real 
world. 
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