7A-6-2
are followed in section 3. In section 4, we give
a simple conclusion.
2. SOLUTION ROUTE OF THE
METHOD
The method is set out not only from
application, but also from geometric space.
Different geometric spaces have different
geometric construction and characteristics.
For example, a 3D Euclidean space, the
commonly used geometric space, is
constituted by three axes, which are
orthogonal each other and extend to infinity; a
3D affine space is constitute by three un
coplanarity vectors; a 3D projective space is
built by an enclosed tetrahedron and a unit
point. Different ways to construct a geometric
space bring different description and solution
route to a same problem. And in a certain
geometric space, the problem can be got its
solution with more convenience. From this
point, we select projective space to do the
work of object positioning.
The method can be divided into the following
three steps:
S, : Transforming Euclidean coordinates
of image points into projective
coordinates
For digital camera is widely used in
photography, pixel coordinate (i, j) of image
point is easily got, which is in a 2D Euclidean
coordinate system. The coordinate firstly is
changed into its quasihomogeneous form of (i,
j, c) by adding a nonzero instant number c at
its behind. Then a transformation matrix is
used to transform it into projective
coordinates (u, v, w). The transformation
matrix is formed from four image points'
quasihomogeneous coordinates and these four
points are constituted the 2D projective
coordinate system on image plane.
S 2 : Getting object points' 3D projective
coordinates from image points' 2D
projective coordinates
There are 3 ways to get object point's
projective coordinate, which are different with
image number. These ways are closely
connected with getting the projective
invariants in computer vision [Hartley, 1994]
[Quan,1995] [Xiong, 1995] and they have been
adapted to fit our work.
Based on 2 images: When two images are
used, according to the coplanarity condition,
we use 8 or more pair of homologous points to
calculate the fundamental matrix, then
decomposing the matrix as a product of a
skewsymmetric matrix and a regular matrix,
from which two camera matrices can be easily
obtained. At last using the intersection to
reconstruct the object model and get the object
points' projective coordinates.
Based on 3 images: From the projective
coordinates of object and image point and the
image forming equation in projective space,
we can build another relationship between
object and image points' coordinates. From
one image, we can build one relationship
between them, in which there are 3
independent unknowns about object points'
coordinates. When 3 images are used, it is
enough to get the values of these unknowns.
To get the final solution, certain approximate
values of object point are required to be used
as reference.
Based on 4 images: The coefficient matrix of
the linear equations, built from the
relationship between the object and image
points' projective coordinates with multiple
images, has the rank of 4, even through 6
unknowns are needed to be determined. So
when 4 images are used, two series of basic