oits the 
etween 
lar line 
: object 
'ramids 
values, 
lability 
a very 
n depth 
ines is 
aultiple 
trifocal 
ced by 
25. We 
te, and 
points. 
camera 
on the 
'eneral, 
form a 
local 
ige, for 
) find a 
satisfy 
ties. In 
ed, i.e. 
images 
larities. 
ls to a 
hts. In 
overall 
led to 
and to 
rds, at 
eses. 
| to be 
of the 
an be 
n two 
A in à 
onship 
(3) 
urns to 
hrough 
etween 
(4) 
'R and 
called 
f the 
ulation 
or. the 
(5) 
That is, searching for the corresponding points is performed in 
the vicinity £ of the epipolar lines. Now, we consider three 
cameras with distinct optical centres C;, C; and C; (making the 
trifocal plane) before image planes R;, R; and R;, respectively 
(Fig. 3). Given a physical point M, its image m; on camera i is 
defined as the intersection of line MC; with an image plane i. 
Points m;, m; and m; form a triplet of homologous image 
points. Any triplet (m;,m;m;) is such that m; lie at the 
intersection of the epipolar lines L;; and L; defined by the two 
other image points m; and my. 
Considering the epipolar line, represented by F;3m,, of m; and 
the one, represented by Fm», of m; in the third image, if those 
two vectors are nonzero, i.e. if they do represent lines, and if 
they are not proportional, i.e. if they do not represent the same 
line, we can predict their point of intersection m; which is the 
image of the point of intersection M of the two optical rays of 
m, and m; via the following transfer (Fig. 3): 
m; € Fm, x Fj4m» (6) 
  
  
  
  
  
Figure 3. The Trifocal Arrangement 
Thus, the search for corresponding points can be theoretically 
reduced by a simple check at the intersection of two epipolar 
lines in the third image. 
3.1.2 Ambiguities 
The procedure that we used in MEDPHOS reduces the 
probability of the ambiguities drastically, but not totally. 
Depending on the number of detected points per image, the 
system configuration, the accuracy of the measurements, the 
pattern projected, the quality of the imaging system, and 
complexity and depth extension of the object, a problem of 
ambiguities may occur. The fact that a feature in one image 
may match equally well with a number of features in the other 
images is referred to as the Fake Target Problem (Fig. 4). 
  
1% Candidate Set 2" Candidate Set 
  
  
  
Figure 4. The fake target problem for 2 images. 
Among the matches, we may find two types of outliers due to 
e False Matches. The accuracy in the establishment of 
correspondences depends on the validity of heuristics. 
e Bad Locations. The location error of a point of interest is 
assumed to exhibit Gaussian behaviour. This assumption is 
reasonable as long as the error in localisation for points is 
small, otherwise, accuracy of the estimation will be 
severely degraded. 
Moreover, there might be some missing matches due to 
e photometric differences (illumination, reflectance,...) 
e texture properties of the scene (repetitions) 
e non equal coverage of the cameras 
e occlusions 
e weakness of matching primitive 
e distortions in the feature extraction process 
3.1.3 Disambiguating 
Disambiguating of the correspondences in MEDPHOS is 
accomplished by applying a set of auxiliary constraints that 
make up a cost function, supporting some matches and possibly 
inhibiting others. For finding an optimal solution, 
(dis)similarity of the attributes and the consistency of the 
correspondences have to be weighted properly. Detection of 
false hypotheses is based on using graph theory, allowing a 
rigorous evaluation of the matches under the assumption of the 
geometric and photometric relationships that exist between 
images and by solving a minimum weighted matching problem. 
Only hypotheses consistent with that relationship will be 
accepted. The applicable auxiliary constraints are: 
Compatibility Constraint : Similarity of invariant attributes of 
the points, e.g. information content in the vicinity, local 
contrast of the point, etc. 
If I,(p), I;(p)and I;(p) are the local contrast intensity values at 
position P in the three images, the cost of that matching can be 
defined as: 
A Lip) - D0)D (7) 
  
E(p) = max(\I,(p) - I»(p) 
If the point is affected by specular reflectance or occluded in 
one or more of the images, then the error will be large and 
consequently the point will be null matched by this cost 
function. 
Uniqueness Constraint : The primitive of an image may have at 
most one homologous primitive in the other image(s) provided 
the object is opaque. 
Disparity Gradient Constraint : The disparity of a point is 
supposed to be similar to the disparity of the nearby points. 
Topological Constraint : Assumes the preservation of the order 
of matched points along the corresponding epipolar lines unless 
the scene contains transparent objects. 
Euclidean Distances : Of each potential corresponding point to 
the epipolar line in the 2" image and of each candidate to the 
predicted location in the 3" image. 
Consistency : Checking for the precision of multiple-image 
forward intersection in the object space. 
The quality of a match between a set of elements is given by 
the weighted sum of the magnitude of the feature attribute 
differences: 
—267— 
  
PE RE a diee,