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It is clear that in this case we also have many possible
combinations and sequences. Results show that the
uniqueness problem exists here as it does with the 2D
case.
The original method by Barrett utilizes linear equations,
avoiding the need for initial approximations, and uses
the minimum (non-redundant) number of 5 control
(Basis) points. As in the case of 2D invariance,
discussed in our previous work (Barakat, 1995), the
uniqueness problem exists. Different
combinations/sequences of control (Basis) points lead
to different results.
In our modification, a refined least squares technique,
which allows for iteration on the observables as well as
on the unknown parameters, was applied to alleviate the
sequence problem. In addition, the use of redundant
number of control points is introduced which
significantly improves the results. Also, in view of our
previous work the constrained least squares technique
for the estimation of the fundamental matrix was
implemented to get more accurate results than the
original linear estimation of F.
The improvements in the results due to the modification
of the original method, are presented and discussed in
the following section. Because object coordinates are
involved together with image coordinates, the
photogrammetric equivalent to this invariance task is in
general two-photo block triangulation. — Since in
invariance no information is assumed with regard to the
sensor, all 18 L.O. and E.O. acquisition parameters must
be assumed to be unknown. Five control points yield 20
collinearity equations, and 8 pass points yield 8
coplanarity equations, thus a redundancy of 10 will exist
for the equivalent invariance unique case. If the 5
control points are taken as a subset of the 8 pass points
a redundancy of 5 still remains. The following sections
presents comparative results of both approaches.
3.5 Experimentation and Results
Extensive experimentation has been performed
employing the procedure described above for object
reconstruction using invariance, and comparisons were
made with the equivalent photogrammetric technique.
The results of this experimentation are summarized in
the following cases.
Case A - Simulated Data Two pairs of photographs,
one with convergent geometry and the other with
normal vertical geometry, were simulated such that the
set of perfect image coordinates were perturbed with
errors having 0,=0,=0.01mm. Six well distributed
control points and 16 object check points were used for
object reconstruction. Table 5 summarizes the rms of
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
dX,dY,dZ for the original and the modified invariance
and the 2 photo block. It is very clear that the modified
method results are superior to those of the original
method especially for the convergent (C) case. In the
convergent geometry case, the 11 extended relative
orientation parameters for the two photos (18-7=11)
are distinct and have significant values. Therefore,
lumping those 11 parameters into 7 recoverable
elements of the fundamental matrix affects the solution
and requires more accurate estimation of the
fundamental matrix as implemented in the modified
method. For the normal vertical geometry case, the
number of well defined camera parameters is smaller
than that of the convergent case. The 7 independent
elements of the fundamental matrix can more easily
recover those camera parameters for this geometry, as
can be seen in the small amount of improvement
between the original and the modified methods. It is
important to note that, as in the case of 2D invariance,
the control points configuration and the location of the
check points have significant influence on the quality of
the results. All subsets of 4 points, out of the total 7
points (6 control + 1 check), should be checked not
being close to falling in a plane. The main advantage of
the invariance technique, besides that no knowledge is
required for the image acquisition parameters, is that no
approximations for the ground coordinates of the check
points are required.
Case B - Real Data (Purdue Campus Imagery): The
modified method was applied on a pair of real vertical
images flown over the Purdue campus, at a scale of
1:4000. The equivalent photogrammetric technique was
performed using the same data set. Table 6 lists the
rms of dX,dY,dZ for 20 check points inside and around
the border of the control points frame, wheredX, dY,dZ
are the differences between the estimated coordinates
and the known measured coordinates. Both invariance
and photogrammetry worked equally well because of the
well distributed control points and the location of the
check points. The most significant conclusion from this
experiment is the importance and sensitivity of the
estimation of the fundamental, F, matrix and its effect
on the success of the invariance method. All subsets of
7 points out of the total number of points used to
estimate the F matrix should have different Z values so
that they are not close to being on a plane. This is even
more important than having different Z values for the
control points on the quality of the obtained results.
Case C - Real Data (Bangor Imagery): The data set is
described in Case B in section 2.5. Table 7 lists the rms
of dX,dY,dZ of check points using both modified
invariance and photogrammetric methods. Six well
distributed control points were selected along the model
perimeter (the overlap area of the two photos) with the
11 check points both inside and on the border defined
