International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
have to be carried out to make sure that all points can be 
measured with satisfying quality. Each iteration includes an 
update of the visibility model of object points followed by 
camera placement using the fuzzy inference system based on 
the updated visibility model. 
2. MODELING VISION CONSTRAINTS 
Sensor placement in computer vision (Cowan and Kovesi, 
1988) has to deal with satisfying some vision constraints as well 
as with optimization of accuracy and cost criteria (Fraser, 
1984). To prepare for fuzzy modelling of vision constraints in 
our approach, vision constraints are divided into three classes 
(Figure 2). 
User accessibility N ae 
Iser accessibility constraint 
constraint 
     
    
Resolution 
Field-of-view — constraint 
constraint 
Workspace | x 
constraint Y m 
Visibility 
trees X 2 goal 
u ^ 
7 
Nuniber and m 
distribution of,” 
Depth-of-field image points 
constraint M 
  
     
ba 
Incidence ">. 
angle "———T— 
constraint 
  
Targeted object 
  
  
  
  
Figure 2. Vision constraints for camera placement (motivated 
by the work of Mason (1995)) 
Range Related Constraints are a class of constraints which limit 
the distance between object points and camera stations 
including image resolution, image scale, field of view, and 
workspace upper limits, depth of field and number and 
distribution of image points for the lower limit. 
Visibility Related Constraints: The visibility of an object point 
from a camera station is a complex interrclated matter that 
depends on radiometric and geometric constraints. Radiometric 
constraints with constant “point to image quality” are easily 
satisfied in presence of retro-reflective targets and special 
flashing equipment. Geometric constraints include an incidence 
angle constraint, workspace obstructions, camera field of view, 
and position and situation of the camera station (Figure 3). 
Accessibility Related Constraints depend on camera position 
accessibility, the workspace constraint, and object and 
obstructions inside. In addition to positional accessibility of 
camera, time accessibility is might have to be taken into 
account. 
   
    
   
  
Camera 
z \ 3 FOV 
   
Target“ 
cone 71 / — Visible ray 
Le Invisible rays 
Figure 3. Visibility is a function of the target cone, camera 
FOV, and hidden areas 
As already mentioned, a given or simulated model of the object 
and workspace is not assumed. Based on a high number of 
images of object and workspace in the primary network taken 
from different directions, target visibility is predictable by 
studying the corresponding image point observations and 
camera accessibility is predictable taking closeness to existing 
camera stations into account. 
2.1 Visibility Prediction Modelling (VPM) 
VPM concept is based on two principals: 1) Visibility of each 
direction toward a point is constant and does not depend on 
distance. 2) Visibility of a direction is the same as the visibility 
of its immediate vicinity directions for a point. The first 
principal causes to simplify the modelling by defining a 
visibility prediction sphere (VPS) and second principal is a 
basic for predicting hidden area in unknown directions by using 
known directions. 
As illustrated in Figure 3, modelling of target visibility can be 
done by sequentially. This implies considering the effects of the 
camera field of view, target incidence angle, and hidden area 
constraints on the visibility of all rays between the target and 
related camera stations. In our modelling a fuzzy visibility 
index v between 0 (perfectly invisible) and 1 (perfectly visible) 
is assigned to each ray. A corresponding visualisation is shown 
in Figure 4. 
  
Figure 4. Three examples of visibility prediction on VPS. 
The black and white stars are visible and invisible rays 
correspondingly which predict visible (bright) and 
invisible (dark) areas. 
2.2 Accessibility Prediction Modelling (APM) 
Briefly, APM concept is based on closeness to the positions of 
existing camera stations. In other words, a point closer to these 
positions probably has a higher accessibility. A proper way to 
model this concept is using analytical function like as 
Butterworth function which is a low pass frequency filter in 
signal processing (Gonzalez, 1993). In Equation 1, D, is the 
accessible vicinity radius around existing camera stations. 7 is 
the fuzzy behaviour factor that controls the width of fuzzy 
boundary between accessible and unknown transit areas. 
Usually 5 is less than 4 especially when camera stations are far 
from each other. D, is specified depending on object and 
workspace conditions and is usually about half of the average 
density of camera positions. 
] 
x 2] (1) 
1+ (Puis y» 
w= 
0 
As a general rule, in the open workspaces with low obstructions 
a high value for D, and a low value for n are proper. Notably, a 
    
   
    
    
   
  
  
  
  
  
  
  
  
     
        
    
     
   
     
    
   
  
  
  
    
    
    
   
    
   
    
   
   
   
   
    
    
     
  
    
   
   
   
    
   
   
    
     
   
   
    
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