Though projective motion and shape can be recovered through
factorization of the measurement matrix 
, we have to know
the values of depth parameters 
in order to construct
. Estimating 
is therefore an essential
part of the factorization methods under perspective projections.
One possible approach is to use fundamental matrices and
epipoles[11] for depth estimation; Let us
consider the projection equation of j-th camera:
. Since
, there exists a 4
3 matrix 
s.t. 
. Then the projection
equation can be solved for the object point as
with
an arbitrary scalar, 
, and a 4-vector, 
s.t.
which represents the projection
center of j-th camera. Now we take i-th camera and substitute
into its projection equation. Then we have
where 
, called the
epipole, is a projection of the j-th camera's center onto the i-th
image. Since this equation implies coplanarity of three vectors,
, 
and
, the scalar triple product of them vanishes and
well-known epipolar constraint results:
where 
is a 3
3 matrix called the fundamental
matrix between i-th and j-th images. Estimating the fundamental
matrices and epipoles from point correspondences between two images has
been well
studied[6,17].
Taking cross product of (14) with
, we have
which can be solved for 
, in the least-squares sense, in
terms of 
as:
It should be noted that we cannot know the ratio between 
and 
from (16) because the fundamental matrix
and the epipole are determined only up to scale. However, considering
another l-th object point and its projections, 
and
, onto i-th and j-th images, the cross ratio of four
depth parameters, 
, 
, 
and
, is independent of the scale:
Equation (17) means that if any three of four depth
parameters are fixed, the remaining one is uniquely determined.
Consequently, if F+P-1 depth parameters are fixed as stated in
(12), the remainders are then recursively computed by
(17) using chains of F-1 fundamental matrices and
epipoles between, for instance, adjacent image pairs, i.e.
and
