‘size px1, with
istributed with
on solutions
nes for which
ich fulfils the
(2)
Ar. that is
istes Analysis
1-(1975), Ten
s numerically
iguration A is
he theoretical
on method.
n of the GPA.
of the same
surveys, are
s deliberately
re centred one
| in Figure 3-c
al agreement.
each of them,
hape of mean
nding point
i*
ng [b]
E
A shape of
hed line) [d]
»
=
Bd
\ process
mul
ize of every
their relative
th polygon to
| points. This
imetric block
adjustment by independent models, for which a Procrustes
solution has been recently developed (Crosilla, Beinat, 2002).
The mathematical formulation of the global adjustment begins
with the GPA model, extended to consider partially weighted
configurations. The general form of the Procrustes problem
assumes therefore the following expression (Commander,
1991):
zi
S - V tr(c AT, + jt, —c, A,T,—jt,) D,D, p »,| Wie
i<k
(c, AT, * jt, — c, A, T, — jt, ) = min
where D; and D, are general diagonal weighting matrices of size
pxp. It can be demonstrated (e.g. Borg, Groenen, 1997) that the
condition expressed by Formula (4) is equivalent to:
m mA ^
S-Y t(cAT, it, -À) D(cAT »it,- Ájzmm 0)
izl
where À , called the geometric centroid, is given by:
A= 3 D, Jl Sp, (cA,T,+jt,) (6)
and represents the least squares estimation of the real
configuration matrix A (Crosilla, Beinat, 2002).
This second formulation can be managed and computed more
easily than the previous expressed by Formula (4). Equations
(5) and(6) in fact, are computed iteratively, and matrices A; are
continuously updated, until a predefined convergence threshold
is reached. This event represents the GPA best fit solution.
The algorithm that makes possible to update, at each iteration,
every data matrix A; (i=1...m) with respect to matrix À. is
due to Schoenemann and Carroll (1970). Its description, by the
same formalism adopted in this paper, can be found in Crosilla
and Beinat (2002).
One fundamental advantage of the generalised formulation for
the weighted case is the capability to account for situations of
missing corresponding points between matrices. To this aim, an
efficient solution is due to Commander (1991). Every D; can be
considered as the product of a proper weight matrix P; by a
Boolean diagonal matrix M;:
D, -M,P, - PM, (7)
M; is automatically defined, and associated to every matrix A,
Its diagonal components are 1 where the corresponding
elements (rows) of A; are effectively defined, and zero in all the
other cases. Referring to Figure 4, the diagonals of the M;
matrices associated to the corresponding A; are:
diag(M,)=[1 11 0 0 0 0 0]
das(M.)-[o 7111 09 0 0
dez({M,}=[0 0.1 1 11 0 0]
33
Ay p | Polygarnı
5 j d 1 d$
f 8 0
ibid ] Polygon
0 06 0 3
0 9
;
Figure 4. Fiducial polygons, matrix description, and fiducial
point correspondences.
For better understanding how the GPA problem with missing
points can be set out, we have represented all the matrices A,
with the same size, that is the same coordinate dimension and an
equal number of points, although not all are actually defined.
Ideally, every A; contains as many rows as the total number of
fiducial points to be adjusted.
For the algorithm implementation this assumption does not
represent a drawback. In fact, the apparent waste of computer
memory resources can be avoided introducing and managing
variable size arrays by way of usual programming techniques.
The software implementation deserve some additional remarks.
Practically, the two stages in which the network adjustment has
been divided, that is the initial identification of the most
probable size and shape of every fiducial polygon configuration,
and the following reciprocal best fit of the different ones, are
performed simultaneously. Every measured polygon assumes a
proper global weight, as a function of the accuracy and of the
survey techniques adopted for its determination.
4. APPLICATION OF THE PROPOSED METHOD TO A
SIMULATED EXAMPLE
A rigorous evaluation of the Procrustes adjustment model
capabilities, to solve problems relative to the cadastral mapping
recomposition in the presence of different kinds of errors, can
be done if the true measurement values are available. To this
purpose, a real situation has been artificially created,
considering a network of fiducial points with fixed coordinate
values. Afterwards, various kinds of errors have been inserted
into the original data, simulating in this way incorrect
measurements surveyed in the field. The solution of the
classical least squares adjustment of the individual polygon
sides, and of the entire polygons by conformal Procrustes
algorithm, leads to adjusted values that can be directly
compared to the originally fixed values, permitting, in this way,
a significant analysis of the precision and of the reliability of the
two methods.
4.1 Description of the simulated network
A two dimensional network of 30 vertexes has been considered.
In order to simulate a real spatial distribution of the fiducial
points, the vertexes have been located with a reciprocal distance
varying between 227 m and 1153 m; the total area covered by
the network corresponds to 5.7 km”. The network is composed
of a series of triangles and polygons whose vertexes represent,
in reality, the fiducial points used by the technicians to
reference the various surveys. In this way 70 polygons have
been artificially generated, and some of them are characterised
by common fiducial points.
Furthermore, some fixed points have been identified within the
network, that is points of high importance for the Cadastral
Administration, or points whose coordinates have been
determined with a high precision (for instance by GPS). Their
choice depends on:
