International Archives of the Photogrammetry, Remote Sensing and Spatial [Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
both datasets: they are displayed on the screen and the user 
clicks on a location in one map and then the same 
corresponding location in the second map. Typically control 
points are easily identifiable features such as building corners, 
major road intersections, etc. The selection of control points 
must be done carefully; their number and the quality influences 
the types of curve fitting that can be performed (1.e. at least four 
points are needed for an affine transformation estimation). 
Moreover control points must be spatially scattered over the 
datasets and in a number greater than the minimum necessary to 
compute the parameters of the chosen transformation. 
The estimate of the parameters, independently from the kind of 
transformation used, becomes better (more accurate) as the 
number of homologous points increases. 
To avoid the time-consuming manual 
correspondence and the possible human errors, a strategy is 
needed to automate the procedure. 
The idea is to reproduce as much as possible what the operators 
manually do when they try to superimpose two maps: they 
visually search for the same geographic features represented on 
the two different cartographic supports. To detect the feature 
related to a certain entity the operators implicitly makes at the 
same time geometric, semantic and topological analyses. 
search of these 
During the visual analysis, the operators compare the shape of 
the features on the maps. We can summarize this operation by 
considering three steps: an analysis of the coordinates of the 
points that geographically describe the shape of the objects, an 
analysis of the "directional" compatibility of the segments 
starting from the points and finally a semantic analysis. 
Therefore, the basic hypothesis is that, since every cartographic 
entity is essentially defined by points (coordinates) and 
semantic attributes, the simplest way to make the search is to 
focus on them: a point P; on map c, is homologous of a point P5 
on map c; if the geographic feature related to the two points 
corresponds: figure 2 shows the example of homologous points 
that can be manually detected on two corresponding maps. 
  
  
  
  
  
  
Figure 2. Homologous points on two different maps 
2.2 The second step: the choose of the transformation 
Once homologous pairs have been detected a warping 
transformation follows to optimally conflate the different maps. 
To make it more adaptive and localized a combination of finite 
support functions can be used. 
In this way the estimation of each function coefficient will only 
depend on the data within the corresponding finite domain. The 
most common functions used for this estimation approach are 
the splines. 
  
208 
2.2.1 The classic spline interpolation approach 
In general terms, we want to interpolate a field d(t) sampled on 
N spread points ti, t, ..., ty in a plane. 
The main idea is that the observed value d,(t) can be modelled 
by means of opportune spline combinations (deterministic 
model) and residuals v; thought as noises (stochastic model). 
The one-dimensional 0 order spline (see figure 3.a) is defined 
as: 
[1 re[0.1] 
(0) [! 
\ = > A = ! 
QU vsu) lo t €|0.1] e 
The basic function can be shifted and scaled throw: 
gi (1) =" 21 =k) 2) 
where j fixes the scale and k the translation. 
The splines of higher orders can be obtained starting from 
qt) by means of convolution products. 
The expression of the first order mono-dimensional spline (see 
figure 3.b) is therefore: 
9 e on) ds 
and then: 
  
  
  
  
0srs] 
r>1 (4) 
Lc) 
o (t) 
] 
0 ! LE 0| p* 
(a) (b) 
Figure 3. Mono-dimensional 0 (a) and 1 (b) order splines 
Using a linear combination of g order splines with a fixed j 
resolution we obtain the following function: 
+2 
d(t) V, A (21 — k) (5) 
AS 
which represents a piecewise polynomial function on a regular 
grid with basic step [k2*, (k+1)2“]. 
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