3-D Reconstruction of Three Views Based on Manifold Study - Advances in Image and Graphics Technologies - page 184

Symbols in figure 2 are the same with those in figure 1. e ij represent the epipole on
the i th image that is imaged from the j th camera center. Similarly, l ij represent the
epipolar line on the i th image of the projection point on the j th image. x , x' and x'' are
the projection points of 3-space point on the image R , R', R'' . they are correspondence
points. So a conclusion can be drawn from three views constraint: any projection point
on one image of a 3-space point is the same with intersection of epipolar lines on the
very image of its correspondence points.
To distinguish whether a pair of points is matching points or not ,the distance form
point to its matching point's epipolar line is always used as the judgment in traditional
robust algorithms. However, the exact position of matching points couldn't be
determined merely with epipolar geometry constraint. Couldn't it guarantee the
matching is right, even though the matching points have tiny residual(distance form
point to its matching point's epipolar line). Suppose there is a point X in 3-space, as in
figure 3, x is its projection point in image R and x' should be another projection point
in image R . Because of the detection error, x a ' , x b ' and x c ' are the undetermined points
of x' . The distance of dotted line in figure 3 represent the magnitude of residual.
Comparing the residuals of x a ' and x b ' , it's easily to make sure that x a ' is the right
one. Actually, has a smaller residual than x a ' ,and it also probability to be regard as the
correct matching point . But it is a error obviously shown form the figure 3.

X

R ′

x ′

x ′

a

b

x ′

x ′

c

e ′

l ′

C ′

Fig. 3. Correctness judgment of matching points Fig. 4. Flow diagram of fundamental matrix
calculation

Three views has stronger constraints between the matching points. For an example
as shown in figure 3, x and x' are the matching points, l 31 and l 32 are their epipolar lines
on image, whose intersection should be the same point with point x'' in theory. Because
of the noise produced during the extracting of matching points, the shadow region as
shown in figure 3, maybe the area where real point x'' is. But distance between the
point of intersection and the real matching point should be limited below a certain
threshold. In conclusion: distance between matching point and the intersection of
corresponding epipolar lines should be large, when the matching points in three views
are not right. Based on that in this article, a more accurate inliers set is obtaining by
getting rid of mismatches.