2004 
[Let 
Since 
| links 
, and 
ess nll 
c of a; 
to the 
rigid 
It is 
(if not 
s, can 
essary 
iximal 
‘his is 
ferent 
to its 
tion is 
'ssible 
lences 
ion is 
ng the 
their 
be the 
to ma, 
erance 
gy to 
dence, 
etry: 
(aj; bj) 
es of 
step, if 
m can 
ig the 
othesis 
laving 
dex of 
of the 
spatial 
ability 
s have, 
ite Chat 
be put 
tions. 
asured 
le side, 
reneric 
orming 
he first 
ned by 
eyama, 
(4) 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol XXXV, Part B4. Istanbul 2004 
where: 
1 Hn 1 n 2 
m, = —X x. OUT 2523 —m, | 
ju 
"n i 
] 3 LS . 
mg == xy, Os e xs. = m,| 
nj Hj 
| & ^n t i (5) 
Ka => (xy, mi Xx, -m,) -UDV (SVD) 
ri 
I if det (K,, ) 2 0 
if det(K,, ) «0 
ii 11) 
In this formulation, the vertices of the transforming 
configuration correspond to those ones of the reference 
configuration. It happens therefore that x,; corresponds to xp; 
with / = |, .., n. Furthermore, in the Singular Value 
Decomposition (SVD) of Kg, the eigenvalues have positive 
values, and are located in D in decreasing order (D = diag(d;); d| 
> d, > … > d; > 0 where k is the dimension of the reference 
system). 
The term £^ provides an index of the shape difference for the 
two considered configurations (s? = O means that the two 
configurations have the same shape). To limit the shape 
difference to the only presence of measurement random errors, 
it is necessary that the range of the values the term £^ can 
assume must be contained within a given threshold. In the 
proposed procedure this threshold will be represented by a 
shape parameter à? referred to the reference configuration. The 
test of shape is accepted for &? < 5°: this event definitely ends 
the comparison procedure. 
If this does not happen, and previously the presence of doubt 
correspondences had been recognised, it is necessary to identify 
which ones of the correspondences are wrong or are inverted. 
Remembering what already said about the correspondent points 
and the rigid links, the determination is based on the following 
criterion: for each doubt correspondence (a;; bj); if À ME 
then (a; ; bj) is a wrong correspondence. The same reasoning is 
repeated for all the doubt correspondences found out at the 
preceding step. At the end of this phase, three are the cases that 
can be present: 
- there is only one wrong correspondence; 
- the number of the wrong correspondences is equal to two; 
- the number of the wrong correspondences is greater than two. 
The first situation means, for the largest part of the cases, that 
all the proposed correspondences are apparently correct: the 
final decision is then remitted to the shape test. If the number of 
the wrong correspondences is equal to two, a swap is carried 
out: if (a; b; and (a; bj) correspond to the wrong 
correspondences, then (a;; bj) and (aj; bj) represent the necessary 
correction. In the third case, the most complex, the procedure is 
iterated just for the subsets of A and B containing 
correspondences not yet solved, recalling again the 
considerations done for the two preceding cases. 
An alternative solution of the comparison problem can be found 
in Sossai (2003), and in Beinat, Crosilla & Sossai (2003, 2004). 
2.2 The inclusion case 
The situation of inclusion occurs when a group of points is 
entirely contained in a more numerous and topographically 
extended set. This means that the geometrical entities taken into 
consideration can have a different number of points: the 
configuration with less vertices will be called “enclosed, while 
the other will be the “enclosing one. This situation is not 
95 
mandatory: the general method proposed works well also for 
configurations having the same number of points, solving in this 
way, by another approach, the comparison case. 
Other conditions are required instead: all the points of the 
enclosed configuration must find univocal correspondence 
within the enclosing configuration, and every entity must not be 
represented by a degenerate geometrical configuration, like that 
whose vertices are all approximately aligned. As for the 
preceding problem, also for this case, we do not need the 
knowledge of the scale ratio between the configurations, neither 
any structural or topological information. 
According to these conditions, the method described in the 
following allows to identify the correspondences between 
homologous points, also in the presence of some errors present 
in the coordinates of the considered vertices. In Figure 1 and 
Figure 2 two different problems, relative to two distinct 
application fields, are shown. In the following, we will indicate 
with A the enclosed configuration, and with B the enclosing 
one. 
A 
  
  
  
— 
Figure 1. The problem of inclusion for a cadastral 
map, that is to find the point correspondences between 
à parcel À fully contained into a more general map B. 
  
Figure 1. The problem of inclusion for a CAD design, 
that is to locate some predefined structural elements 
fully contained into a more complex drawing. 
2.2.1 The general method using a correspondence kernel: 
The idea leading to the general solution of the problem of 
correspondence is based on the following two considerations: 
- the less is the number of the vertices of the enclosed set 
(considering constant the number of the enclosing points) the 
easier is the solution of the inclusion problem;