polygons, but it is also possible to perform a local datum
exchange from one reference frame to another, by way of a
similarity transformation restricted to the typical extent of a
fiducial polygon (of the order of hundreds of metres).
In the actual implemented procedure, both the steps of the
adjustment operate at the same time. It results in fact the same
process, with the same rules, algorithms and goals. It has been
divided, in the explanation, in order to make it more
comprehensible.
3. THE MATHEMATICAL MODEL
We focus first on the problem of establishing the most probable
shape of a given fiducial triangle (or polygon), resulting from a
series of geometric determinations of it, considering their
different accuracy and their datum variability.
Let A; ... Am be a set of m data matrices, containing each one,
by rows, the points coordinates of the same set of p fiducial
points, belonging to a given fiducial polygon A, as determined
by m different field surveys, carried out at different times. For
more generality we can assume tri-dimensional point
coordinates (two planimetric components x and y, and the
height z), consequently every A; has dimension px3 (Figure 2).
At the present, of course, this does not represent the situation of
Italy, but other countries might consider to develop a 3-D
cadastral map in the future.
y ix yz PF2 ;
4 PF1 r777777 o: i z
| i : 5 No WP
I - í umi \ à = X poo
i A i N nz
I T. | X x, Zu
| I +, y, zy)
Y
Figure 2. A generic fiducial polygon A and the corresponding
coordinate matrix A;.
The further assumption is that the coordinates of every fiducial
polygon A; i=1...m, are defined in a local arbitrary datum, that
can be related to the others and to the general map reference
frame by an appropriate similarity transformation.
The most probable shape of mean size A of a given fiducial
polygon A can not be directly computed by simply averaging
the corresponding point coordinate values, since there could be
systematic components among the different fiducial polygon
determinations A; (i=1...m) (different origin, orientation, and
scale) that must be removed at first. Therefore, a proper
unknown similarity transformation {c,T,t}; — where T is an
orthogonal 3-d rotation matrix, t a translation vector, and c a
scaling factor — must be computed for every A; (i=1...m) in
order to contemporaneously fit it to all the others A, (k71...m;
kz i). This leads, unless an individual residual error matrix E;
(/=1...m), to the unknown real configuration matrix A.
À = c, A, T, + jt, + E,
…=CA,T + jt, + E, (1)
04A, T, *E,
m m m Jt
m
32
In Equation (1), j is an auxiliary column vector of size px 1, with
all the components set to 1, vec(E;) is normally distributed with
zero mean and covariance matrix £=c"1.
Among the infinite similarity transformation solutions
satisfying Equation (1) — excluding the trivial ones for which
c;-0 for every, i — we look for that one which fulfils the
following least squares condition:
m
s= tr SL (GALT, vit) - (e, A,T, . it; )] ; (2)
i<k
(CAT, + jt,)-(c, A, T, + jt, )] = min
It represents the simultaneous best fit among Ay... Am, that is
the result of the well known Generalised Procrustes Analysis
problem (GPA), described and solved by Gower (1975), Ten
Berge (1977) and Goodall (1991).
After the minimum condition of Formula (2) is numerically
satisfied, the estimate À of the consensus configuration A is
given by:
À = S e; A.T B it; (3)
i=l
We refer to the above mentioned literature for the theoretical
aspects and for the description of a problem solution method.
Figure 3 aims to provide a graphical interpretation of the GPA.
Four geometric configurations (A; i=1...4) of the same
fiducial polygon A, produced by four separate surveys, are
represented in Figure 3-a, with their differences deliberately
exaggerated. In Figure 3-b the various polygons are centred one
over the others by way of a proper translation, and in Figure 3-c
they are rotated and scaled up to their maximal agreement.
Figure 3-d illustrates the final shape assumed by each of them,
and their relationship with their most probable shape of mean
size A, computed averaging the corresponding point
coordinates of the transformed original matrices A;.
Original configuration [a] Centring [b]
Resulting GPA shape of
GPA best fit [c] mean size (dashed line) [d]
Figure 3. Geometric representation of the GPA process
Once all the most probable shapes of mean size of every
polygon have been determined, taking into account their relative
accuracy, the following problem requires to fit each polygon to
its adjacent ones, by way of the common fiducial points. This
task presents an approach similar to the photogrammetric block
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