International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
value greater than the significant threshold. This corresponds to
find out in I, the number of lengths /; significantly different
among all of them, without the uncertainty due to the coordinate
errors. The vertex with the maximal asymmetry will correspond
to that particular point of A having the largest index of
significance. Formally, m, is such that:
i = max(i,_, . (2)
a-nm, TA
In case of ambiguity, due to the simultaneous presence of more
vertices having the same maximal value of reliability, the vertex
that, for the same i, is characterised by the minimal component
of D, with the largest value, will be chosen as the point of
maximal asymmetry. The reason and the convenience of finding
out a point of maximal asymmetry, will be clear in the
following.
At the next step, the corresponding point of m in the
configuration B is researched; this vertex is indicated with the
symbol mp. To this purpose, the necessary but not sufficient
condition to state that two arbitrary vertices (u and v) are
correspondent, for configurations with the same scale rate, is
that, for each rigid link associated to u, there must exist another
one of the same length, a part for random errors associated to v.
Referring to this property and considering the fact that the
symmetrical configurations have been excluded, the most
probable correspondent of ma is, among all the vertices of B,
that one with the less discrepancy to homologous rigid links
computed with respect to the point of maximal asymmetry in A.
Formally, mp is such that:
| 7 Im] milis. - M) il
PNEU
[n the case of exactly comparable configurations, that is without
errors in the coordinate values, the discrepancy to homologous
rigid links between two corresponding points is necessarily
equal to zero. All this leads to an important consequence: the
comparison between À and a proper threshold (L), proportional
to the admitted tolerances, provides a criterion to evaluate the
correspondences. In fact, given two points z and v, if à > L, the
two considered vertices are not correspondent. Given the
characteristic — function assumed, L is defined as
"correspondence threshold".
Keeping in mind what already exposed, let us consider again the
points of maximal asymmetry and their correspondence ratio.
The verification that m4 e mg are not correspondent vertices,
leads to consequences much more significant than a wrong
correspondence. Referring to the way in which my has been
determined, a negative exit of the mentioned test would imply
the conclusion of the procedure: which other point in B could
overcome the comparison?
This impossibility to proceed is due to the fact that the
comparability hypothesis between the configuration considered
it is not verified. A positive exit of the test about the
correspondence between m4 and mp allows, on the contrary, to
pass to the next phase.
Now, a fast and tentative problem solution of the residual
correspondences will be looked for. The final solution should
try to satisfy the following aspects: to be, as much as possible,
near to the real result, and to require a light computational work.
Now let 4 be a generic vertex of A, and v be the correspondent
one in B: the method can solve the residual correspondences
between A and B by the comparison of the rigid link lengths of
u and v assumed as reference points.
Let a; be the vertex of A joined to u by the rigid link /;, of L Let
b; be the vertex of B joined to v by the rigid link An di, 1... Since
the correspondent vertices define, between A and B, rigid links
of the same length, a part for some random errors, and
remembering the definition of “ordered set of rigid links“, it
seems correct the hypothesis that the correspondent vertex of a;
is b;, with i = 1, ..., n-1. This assumption is as much close to the
real situation as much the correspondence between the rigid
links of the reference points is univocally identifiable. It is
evident that the presence of links with very similar length (if not
equal), combined to the distortions caused by the errors, can
lead to imprecise results in the proposed solution.
With the aim to optimise the procedure, it is therefore necessary
to identify the reference vertex of A providing the maximal
reliability and correctness to the hypothesised solution. This is
the vertex whose rigid links have as much as possible different
lengths with respect to the other points. According to its
definition, the reference point for the correspondence solution is
the point of maximal asymmetry. If nm, is known, it is possible
to find its correspondent mp, and the residual correspondences
can be immediately identified at the first tentative.
In the following, the quality of the obtained solution is
evaluated. First of all it is necessary to define which, among the
proposed correspondences, can originate doubts about their
effective correctness. As mentioned before, these will be the
correspondences characterised by rigid links, connected to ma,
having similar length, that is contained within the tolerance
range.
According to what already explained, the methodology to
search for couples of vertices with a doubt correspondence,
assume the following form: fixed the i" distance of asymmetry:
d, With d,, € Da. + iA dim, ; then (aj; 554) and (a; b;)
im, im,
are doubt correspondences.
The same reasoning is repeated for all the distances of
asymmetry referred to the vertex my. At the end of this step, if
the test has not found doubt correspondences, the problem can
formally be considered solved.
2.1.2 Validation test: To make this method satisfying the
maximal reliability, the true final step is to verify the hypothesis
of comparability of the configurations A and B. Having
identified all the correspondences is not, by itself, index of
correctness; the method of research, based on the lengths of the
rigid links, leaves out of consideration from the effective spatial
disposal of the vertices. To confirm the supposed comparability
it is necessary to verify that the considered configurations have,
a part for some random errors, the same shape. We can state that
two configurations have the same shape if they can be put
coincident by rotations, translations and isotropic deformations.
Let us indicate with e* the square mean of the measured
distances among correspondent points belonging, from one side,
to A (reference configuration) and, from the other, to a generic
configuration having the same shape of B (transforming
configuration). To perform the so called "test of shape“, the first
step is to compute the minimum value that can be assumed by
€; this value is reported with the symbol gh
Given:
A 2 fXAi, Xa» XAí1 7 Reference configuration
B = {Xp1, XB2, -, XB,} = Transforming configuration;
the term & is provided by the following equation (Umeyama,
1991):
94
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