Graphics Reference
In-Depth Information
4.12
Reconstruction
Ignoring clipping, which we shall in this section, by using homogeneous coordinates
the mathematics in our discussion of the graphics pipeline basically reduced to an
equation of the form
ab
M =
,
(4.13)
where M was a 4 ¥ 3 matrix,
a
Œ
R
4
, and
b
Œ
R
3
. The given quantities were the matrix
M, computed from the given camera, and a point in the world that determined
a
. We
then used equation (4.13) to compute
b
and the corresponding point in the view plane.
Our goal here is to give a brief look at basic aspects of two types of inverse problems.
For additional details see [PenP86]. For a much more thorough and mathematical
discussion of this section's topic see [FauL01].
The Object Reconstruction Problem.
Can one determine the point in the world
knowing one or more points in the view plane to which it projected with respect to a
given camera or cameras?
The Camera Calibration Problem.
Can one determine the world-to-view-plane
transformation if we know some world points and where they get mapped in the view
plane?
Engineers have long used two-dimensional drawings of orthogonal projections of
three-dimensional objects to describe these objects. The human brain is quite adept
at doing this but the mathematics behind this or the more general problem of recon-
structing objects from two-dimensional projections using arbitrary projective trans-
formations is not at all easy. Lots of work has been done to come up with efficient
solutions, even in what might seem like the simpler case of orthographic views. See,
for example, [ShiS98]. Given three orthographic views of a point (x,y,z), say a front,
side, and top view, one would get six constraints on the three values x, y, and z. Such
overconstrained systems, where the values themselves might not be totally accurate
in practice, are typical in reconstruction problems and the best that one can hope for
is a best approximation to the answer.
Before describing solutions to our two reconstruction problems, we need to
address a complication related to homogeneous coordinates. If we consider projec-
tive space as equivalence classes [
x
] of real tuples
x
, then mathematically we are really
dealing with a map
3
2
T
:
PP
pq p
Æ
Æ=
()
T
(4.14)
Equation (4.13) had simply replaced equation (4.14) with an equation of representa-
tives
a
, M, and
b
for
p
, T, and
q
, respectively. The representatives are only unique up
to scalar multiple. If we are given
p
and T and want to determine
q
, then we are free
to choose any representatives for
p
and T. The problems in this section, however,
involve solving for
p
given T and
b
or solving for T given
p
and
q
. In these cases, we
