With regard to the semantic level, it allows to define objects to 
be modeled and their attributes, in answer to modeling needs 
and in accordance with the available geometric data. 
2. Geometric modeling 
The surface is the fundamental unit in our 3D-modeling 
approach. The modeling process indicates here the use of 
volumes bounded only by their surfaces. Planes entities are 
dominant in the interior parts of buildings (wall faces, door 
faces, windows faces, etc.). Every entity will be modeled by 
means of its boundaries (B-rep representation). Therefore only 
the points that constitute the outline of a given entity will be 
measured in the image. The resulted surface model can be 
completed by means of geometric primitives (cone, cylinder, 
prism and sphere). 
CEILING face WALL face 
WINDOW face, wp | P 
  
Pas | / 
^ Mit 
GROUND face "DOOR face 
Figure 1. Essential entities of a room 
In the indoor parts, the establishment of stereoscopic or 
convergent image rays conditions is difficult. In this 
environment, shots are mainly influenced by the distance 
between the camera and the object but geometric constraints 
(parallelism, perpendicularity, symmetry, etc.) are numerous. 
This favors the adoption of a single image technique as a source 
of 3D geometric data required for modeling. It is, however, 
necessary to clarify that it is almost impossible to extract 3D 
characteristic quantities of a scene from a single image except 
when the scene is plane, at least locally (Burns, 1990). 
The technique of 3D single image modeling adopted in our 
method is partly based on former researches (Criminisi, 1999). 
Data acquisition is done by using an application called Mono 
Image Modeling (MMI). To extract 3D data from one image, 
the following algorithms are proposed (figure 2): 
  
| Single image | 
v 
  
  
| Definition of a local coordinates system | 
: 
Determination of the vanishing points 
associated to axes of coordinates 
system 
  
  
  
Intersection of image 
lines 
1 Homography-based solution 
      
  
  
  
  
  
  
  
  
3D geometry Base line-based solution 
  
Distance-based solution 
  
  
  
  
Figure 2. Steps of the geometric modeling 
2.1.1 Determination of vanishing points 
The first step consists in determining the vanishing points 
associated to the axes of a local coordinate system. Two 
algorithms are proposed to compute these points: 
2.1.1.1. Algorithm based on the intersection of image lines 
Homogeneous coordinates are used to represent the end points 
of measured lines. A series of lines parallel to each axis is 
measured in the image. The corresponding vanishing point is 
the intersection of those lines. If the intrinsic parameters of the 
used camera are known, the vanishing point is directly given in 
metric coordinates and radial distortions’ corrections are 
applied. 
A linear regression is applied to evaluate the accuracy of 
measurements. If the camera is not calibrated, the three 
vanishing points resulting from the previous stage can be used 
to calculate the intrinsic parameters (principal point and focal 
length) of the camera (Caprile, 1990). 
2.1.1.2. Algorithm based on line fitting 
Cartesian coordinates are used to express the measured lines. A 
line fitting approach is applied to compute vanishing points 
associated to the axes of the local coordinate system. 
2.1.2 Calculation of the 3D geometry 
To carry out a complete 3D reconstruction from a single image, 
object space must be described as a combination of three planes. 
The plane, which contains the axes X and Y, is the reference 
plane whereas the direction (Z) is the reference direction 
(Criminisi, 1999). 
Three solutions are proposed to compute the 3D geometry from 
one image: 
2.1.2.1. Homography-based modeling 
Modeling is based on the homography of the reference plane. 
To apply the algorithm, four control points situated in the 
reference plane and a distance along the reference axis, are 
necessary. The control points allow to compute the parameters 
of the homography of the reference plane. If more than four 
points are known, a least square solution is possible. By using 
the vanishing point of the reference axis (Z-axis in our case) and 
the known distance, the scale factor associated with this axis 1s 
calculable by applying the algorithm proposed in (Criminisi, 
1999). To determine the 3D coordinates of the point (P), it is 
necessary to measure its projection (P') in the reference plane. 
The coordinates of (P) are determined in two steps (Figure 3): 
AZ 
  
  
++ 
     
  
/ 
  
Figure 3. Steps of 3D coordinates computing 
—132- 
2.2. 
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