Jorge Brito 
  
problem of automatic detection of occlusions in digital images. This can be implemented through a simple Bayesian 
network, which is described in the next section. 
2 IMPLEMENTATION 
2.1 Preliminary Steps 
The first step to solve the problem of automatic detection of 
occlusions in digital images through a probabilistic approach is to 
take another look to figure 1, and changing it slightly. This 
modification is shown in figure 2. 
. One can easily see from figure 2 that, if vectors ab and AB have 
opposite orientations, then "B" is occluded by “A.” In that 
situation, the analysis of occlusion will consist in determining the 
probability of agreement in orientation between vectors ab and AB. 
Therefore, the problem of automatic detection of occlusions in 
digital images can be summarized in the answer to a “simple” 
question: Do vectors “ab” and “AB” agree in orientation? The Figure 2. Lateral view of an occlusion along a 
answer to this question will be provided by the computation of the generic direction of sight “0” in a vertical image. 
probability of agreement between ab and AB and, which comes 
from a simple Bayesian network. The differences between figure 2 above and its predecessor is that the vectors ab and 
AB determined from points “A” and “B”, and their projections onto the image space are now being considered, instead 
of their respective radial distances, Ra , Rg, r,, and ry. 
The approach described above is thought to be more efficient because it avoids some additional steps, necessary to 
compute the radial distances and the linearization of its formula, when propagating the error. Instead of computing 
vectors ab and AB only logical comparisons between the possible projections of “a” and “b” onto the * x' ^ axis will 
suffice for the occlusion analysis. This situation is illustrated in figure 3: 
The geometric condition for checking for occlusions to be A , 
implemented in this research can then be re-stated as follows: given y 4 
that Xg is greater than (or equal to) X4 in the object space / 
coordinate system, if its projection (x’,) onto the line of sight “6” ‘ 
is less than that of “A” (x’,), then "B" is invisible in the image > ; 
space, because it is occluded by “A.” This situation can be 
visualized through figure 2 above. 
The implementation of the boolean statement described above 
requires the computation of the image coordinates of points “A” and 
“B” along the “6” direction, x’, and x', respectively. This task 
requires an additional rotation, R(0). This rotation is necessary to 
set the x’ axis parallel to the line of sight, along which the existence 
of an occlusion will be investigated. This rotation determines a 
radial line parallel to the line of sight, as shown in figure 3. The 
following condition must be satisfied after that rotation: 
  
  
  
  
  
  
  
  
  
Ya=¥'s (1) eis 
The direction “0” is computed from the image coordinates of “a” 
and "b," by equation 2: Figure 3. The image coordinate system (x, y) and 
; its rotation according to the line of sight (ab) along 
T Ya | (2) which the occlusion will be investigated. 
0 - atan 
Ixp- xal 
The coordinates of points "a" and "b," along the x’ axis, x’, and x’,, respectively, and their standard deviations 
constitute the input to the Bayesian network. These parameters are computed a mathematical model for assessing the 
precision of digital orthoimages (Brito, 1997). The preliminary steps described above can be summarized in the 
following algorithm: 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 103