Page 267 - Innovations in Intelligent Machines

260 J. Gaspar et al.
where θ is the angle formed by the x axis of the camera and that of the world
coordinate system. This angle will be determined from the vanishing points
[14] of these directions.
A vanishing point is the intersection in the image of the projections of
parallel 3D lines. If one has the images of two or more lines parallel to a given
3D direction, it is possible to determine its vanishing point [79].
In our case, information provided by the operator allows for the determi-
nation of alignments of points along the x and y directions. It is thus possible
to compute the vanishing points of these directions and, from there, the angle
θ between the camera and world coordinate systems.

Reconstruction Algorithm

Having determined the projection matrix R in Eq. (27), we proceed to esti-
mate the position of the 3D points P. This will be done by using the image
points p to linearly constrain the unknown quantities.
From the projection equation, one has p × RP =0 3 , which is equivalently
written
S p RP =0 3 , (29)
where S p is the Rodrigues matrix associated with the cross product with
vector p.
Writing this equation for each of the N unknown 3D points gives the linear
system:
⎡ ⎤ ⎡ ⎤
R
S p 1 P 1
R
⎢ ⎥ ⎢ ⎥
S p 2 P 2
⎢ ⎥ ⎢ . ⎥ = A.P =0 3N . (30)
⎥
⎥ ⎢
.
⎢ .
.
⎣ . ⎦ ⎣ . ⎦
R
S p N P N
where A is block diagonal and P contains the 3N coordinates that we wish
to estimate:
Since only two equations from the set defined by Eq. (29) are independent,
the co-rank of A is equal to the number of points N. The indeterminacy in
this system of equations corresponds to the unknown depth at which each
points lies, relatively to the camera.
This indeterminacy is removed by the planarity and alignment information
given by the operator. For example, when two points belong to a z = constant
plane, their z coordinates are necessarily equal and there is thus a single
unknown quantity, rather than two. Equation (30) is modified to take this
information into account by replacing the columns of A (resp. rows of P)
corresponding to the two unknown z coordinates by a single column (resp.
row) that is the sum of the two. Alignment information likewise states the
equality of two pairs of unknowns.
Each item of geometric information provided by the user is used to trans-
form the linear system in Equation (30) into a smaller system involving only
distinct quantities: