Our goal is to directly get the unique solution
from a redundant polynomial equation system.
Finding the common roots
is equivalent to the determination
of the zero-dimensional variety
generated by the ideal
.
We can algebraically get a linear polynomial in generic cases
and in one of
the unknowns
by successive application of Ritt method or pseudo division
[1]
of polynomials over 
.
This can effectively be done
with any computer algebra system and
will directly give the unique solution of the problem
for general configurations of the points.
However this algebraic method
is almost useless for practical situations
as the successive elimination will ultimately give complicated
coefficients for the final linear polynomial
which compromise
the numerical stability of the solution.
Instead of doing it algebraically,
our goal is to develope a numerical linear method
which indeed gives the unique solution if it does exist.
For n points,
each pair of points gives a 4th degree polynomial,
we can have
quadratic constraints of type 
on the n unknown distances 
,
and
4th degree polynomials of type 
in one variable 
.
For n=4, we obtain three 4th degree polynomials
Let the 5-vector 
,
this can be rewritten in matrix form:
This can be viewed as a homogeneous linear equation system
in 
for 
.
Since the matrix 
has at most rank 
,
the singular value decomposition
of 
is
The null space of 
is spanned by the right singular vectors
and 
.
Therefore we can construct a
1-dimensional solution space
for 
,
parameterized
by 
and 
:
Now consider the nonlinear constraints
among the components of 
.
It can be easily checked that
Substituting 
from
(4)
in (5)
gives a homogeneous quadratic equation
in 
and 
:
where
We have 7 such equations for the 7 different values of
modulo the interchanges of i and j or k and l:
| (i,j,k,l) |
| (4, 2, 3, 3) |
| (4, 1, 3, 2) |
| (4, 0, 3, 1) |
| (4, 0, 2, 2) |
| (3, 1, 2, 2) |
| (3, 0, 2, 1) |
| (2, 0, 1, 1) |
These 7 quadratic equations can be written in the following matrix form:
This overdetermined system can
be viewed as linear in
, 
and 
,
and solved by SVD
as the right singular vector of the smallest singular value
of 
.
It is clear that these 7 equations are linearly
but not algebraically independent.
The constraints
are the algebraic conditions
for 
,
to be a geometric series.
Given the null vector 
,
we solve for
After obtaining the ratio 
,
and 
can be determined
using one of the scalar equations of the solution (4),
Therefore 
is completely determined.
The final x is taken to be
or the average of all these values.
Since 
,
the final positive depth is 
.
Hence, the pose of the calibrated camera is uniquely determined by 4 point correspondences provided that the 4 reference points together with the perspective center of the camera do not lie in a critical configuration. The unique solution can be estimated by the linear 4-point algorithm.
