-2r
d 1 1 1 1 1 1 1 1 1 1
4 2 f2 = 0 1 2 3 4 5
Figure 4
Matching procedure can described as follows:
- Compute the least square match under rigid motion
transformation between the curves. We use the
algorithm from (Pirhonen et al.,1994). The algorithm
there is for 2-D affine case but can be replaced by a
rigid transformation.
- For the first curve compute the invariants. These can
be computed also before matching because they
remain unchanged under rigid motions.
- Compute local least squares match between curves.
By local match we mean no predefined global trans-
formation is computed between curves. Just the
coefficients of the first curve are changed so that first
curve fits to the second curve. Local match needs
corresponding points. These are selected for example
by the closest point algorithm (Besl et al., 1992) or by
curve parameter transformation (Pirhonen et al.,-
1994) which one we have selected. The curve para-
meter transformation assumes that curve parametri-
zation is computed using rigid invariant curve
parametrization. Curves are usually approximated
from data points and for this reason the curve
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
parameter transformation can not be very accurate.
So also combination of these methods might be
useful, especially because closest point algorithm
needs an initial value.
- For the first curve compute the invariants once
again.
- Rank the differences between invariants computed
before and after local match. Select some absolute or
relative criterion that cluster the ranked coefficients
to two groups. First group includes the coefficients
where the change in invariant measures have been
below the criterion. For the final motion computation
curve points from the first curve can be chosen from
the support areas of the coefficients that belong to
the first group. Corresponding points from the second
curve are selected same way as in local matching
phase.
The algorithm does not handle curve identification
problem at all. If corresponding curves are not known
that problem must be solved first.
3. RESULTS
Sixteen test cases were generated. In the first basic
group of 8 cases we added random differences to random
places of the second curve. Four groups that have
different number of differences were included, 16%, 25%
‚35% and 50% of the coefficients of the second curve
changed. First degree curve and third degree curve cases
was chosen. The reason for this was that coefficients of
the first degree curve have smaller support area than
coefficients of the third degree curve. The rest 8 cases
random differences were added to constant and consecu-
tive coefficients. Otherwise these cases was generated
same way as in the first basic group. Each of the 16
cases were generated 50 times. The rigid motion under
these test is 2-D transformation, so it includes two shifts
and one rotation.
Results can be seen in figure 5. Each bar graph has 48
bars. First 16 bars are for x-coordinate shift, second 16
bars are for y-coordinate shift and rest 16 bars are for
the rotation angle. Each bar defines how much the mean
value of the 50 computations deviates from zero. If there
is no deviation the original rigid transformation is
recovered exactly. The subgroups that includes four bars
are: First bar defines deviation just after least squares
matching. The rest three bars defines deviation after
final motion computation when features were dot
product, angle, and product of distances respectively. It
is noticed that this number of cases and computations
does not produce purely robust statistical information.
When 50% of coefficients have differences the method
does not have any positive effect. In all 16% and 25%
cases the method have positive effect. From the first
degree curves the method recovered the original motion
better than from the third degree curves. That was
expected because in the first degree curve the invariants
0.35
0.0!
o 1r
0.02
O it
0:05
0.15
0.05
Figur
