The computed epipolar geometry is then used to refine the 
matching process, which is now performed as guided matching 
along the epipolar lines. A maximal distance from the epipolar 
line is set as threshold to accept a point as potential match or as 
outlier. Then the filtering process and the relative orientation 
are performed again to get rid of other possible blunders. 
However, while the computed epipolar geometry can be correct, 
not every correspondence that supports the relative orientation 
is necessarily valid. This because we are considering just the 
epipolar geometry between couple of images and a pair of 
correspondences can support the epipolar geometry by chance 
(e.g. a repeated pattern aligned with the epipolar line). These 
kinds of ambiguities and blunders can be reduced considering 
the epipolar geometry between three consecutive images. A 
linear representation for the relative orientation of three views 
is represented by the trifocal tensor T [Shashua, 1994]; it is 
represented by a set of three 3x3 matrices and is computed only 
with image correspondences without knowledge of the motion 
or calibration of the cameras. For every triplet of views (Figure 
3), 1f pi, p; and p, are corresponding points in the images, then 
for every line b through p; in image 2 and for every line I 
through p, in image 3, the fundamental trifocal constraint 
states: 
I; [Tp J1; =0 [3] 
where [Tp] is a 3x3 matrix whose (i,j) entry is 
[TpiJj = Tx; + Ty; + TV [4] 
If we consider only the corresponding points, each triplet pi, p; 
and p; must satisfy the matrix equation: 
[p;]. [Tp] [ps]. =0 [5] 
with [p], the skew-symmetric matrix of an homogeneous 
vector, built as 
0 a3 a” 
ah=| a; 0 -a [6] 
n4» aj 
where a = (a, a, 23). 
If a triplet of points pi, p; and p, satisfy equation (5), it means 
that the corresponding points support the tensor T;5, (Figure 4). 
L 
P pet 
^ 
*e---l---- 
^ 
ge 
   
e 0, 
Figure 3: Three views geometry: correspondences p; and I; 
corresponding to point P and line L 
Relation (5) can be used to verify whether image points (or 
lines) are correct corresponding features between different 
views. Moreover, with constraint (5), it is possible to transfer 
points, e.g. compute the image coordinates of a point in the 
third view, given the corresponding image positions in the first 
two images. The exterior orientation of the cameras is not 
required (as with collinearity equations) and only image 
measurements are needed. This transfer is very useful when in 
one view are not found many correspondences; calling p, and 
p» the point correspondences in the fist two images, the image 
coordinates of the corresponding point p; in the third view are 
given (up to a non zero scalar factor) by: 
PP3 = [Tp, [. 7 x2[Tp; B» or 
7 
tp z [Tp E. - yo[Tp. D. E 
where: 
[Tp,];« denotes the i" row of [Tp]; 
P1=[X1> Yi, M. 
p.t are non-zero scale factor. 
In case of noise-free image measurement, both equations are 
equivalent. The same transfer can be done with lines. The point 
transfer can be solved also using the fundamental matrix, but 
the trifocal constraint can avoid ambiguities and remove 
blunders. Moreover, from the tensor it is possible to derive the 
fundamental matrices between the first and the third view; e.g., 
given 3 images, M; between image 1 and 3 is given by: 
Ma -lel[n. T. Tı]e, [8] 
where: 
e; is the epipole of image i; 
[ei], is the skew-symmetric matrix (6) formed with e;. 
Therefore, the transfer of p; is expressed as: 
P5 - (Myspi) x (M53p;) [9] 
e.g. the intersection of two epipolar lines in the third view. 
The 27 unknowns of the tensor T, defined up to a scale factor, 
can be computed from at least 7 correspondences: using 
equation (5), each correspondence gives 9 equations, 4 of them 
linearly independent. In our process, for each triplet of images, 
the tensor T is computed with a RANSAC algorithm [Fischler 
and Bolles, 1981] using the correspondences that support two 
adjacent pair of images and their epipolar geometry. The 
RANSAC is a robust estimator, which fits a model (T tensor) to 
a data set (triplet of correspondences) starting from a minimal 
subset of the data. As result, for each triplet of images, a set of 
corresponding points, supporting a trilinear tensor, is available. 
After the computation of a T tensor for every consecutive 
triplet of images, we consider all the overlapping tensors (Ts, 
T234, T345,...) and we look for those correspondences which are 
present in consecutive tensors. That is, given two adjacent 
tensors Typ. and Tyg With supporting points (XarYas: Xb-Yb> Xe Yo) 
and (x'yy'y, X'esY'es x'uy'a), if (xy,ys, XcsYc) in the first tensor is 
equal to (x'y,y'y, X'esY'e) in the successive tensor, this means that 
the point in images a, b, c and d is the same and therefore this 
point must have the same identifier. Each point is tracked as 
long as possible in the sequence. The obtained correspondences 
are used as tie points for the successive bundle-adjustment. 
  
CAUSE: Character Animation and Ünderstanding from SEquance of images 
  
Figure 4: The relative geometry between a triplet of images 
—592— 
  
  
  
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