3-D Reconstruction of Three Views Based on Manifold Study - Advances in Image and Graphics Technologies - page 183

computation of trifocal tensor. Aim to get precise result of fundamental matrix is came
true.
(2) Remove outliers in 3-D point cloud based on the theory of minimum spanning
tree.
(3) Combined implicit surface fitting method with manifold study method to
complete reestablish of 3-D TIN, taking the characteristics of point cloud into
account, witch is steady, sparse and uneven distributed.

2

Calculation of Fundamental Matrix Based on Three Views

2.1 Three Views Constraint

Three views constraint-the trifocal tensor, has analogous properties to the fundamental
matrix of two view geometry: it is independent of scene structure depending only on the
relations between the views. The view matrices may be retrieved from the trifocal
tensor up to a common projective transformation of 3-space, and the fundamental
matrices for view-pairs may be retrieved uniquely.
The essence of the epipolar constraint over two views is that rays back-projected
form corresponding matching points are coplanar, a weak geometric constraint, as in
figure 1

X

C c

e

e

l

l

S

x c

l

R c

e

e

R

R

x

R c

l

x c

l R cc

e

x

l x cc C cc C

l

e

l c

e

e c

C

C c

X

Fig. 1. Epipolar constraint

Fig. 2. Geometric constraints of Three-View

C and C' are the centers of the two cameras when they are getting the views R and
R' . The camera baseline C C' intersects each image plane at the epipoles e and e' , x and
x' are the points which imaged by a 3-space point X in two views R and R' . Any plane π
containing the baseline C C' and point X is an epipolar plane, and intersects the
image planes in corresponding epipolar lines l and l' . l' is the epipolar line of point x ,
and it passes through epipoles e' and point x' ,the corresponding point of x in another
image. So, given a pair of image ,it was seen in figure 1, the epipolar geometry
constraint can only guarantee that any point x' in the second image matching the point x
must lie on the epipolar line l' . But exact position where the point is could not be
determined. Compared with epipolar geometry, the three views geometry is the ability
to transfer from two views to a third: given a point correspondence over two views the
position of the point in the third view is determined as in figure 2.