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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004
angular information of linear objects, location relationships,
shape feature of the objects (i.e. lines length, polygon perimeter
and area, etc.). The geometric method is used in most cases and
requires that two data sets have similarity in geometric location,
thus map alignment or rubber sheeting may be involved in the
processing.
Topological methods use topological information such as
connectivity between lines, adjacency between polygons and
composition relationships to correlate objects: arcs meet at a
node, arcs form a polygon, and so on.
Topological matching is usually used to reduce the search range
or check the results of geometric matching and it can be used
only when topological information is available.
The attribute method is also referred to as the semantic method.
This method can be used to match features if both data have
common attributes with the same known semantics. Otherwise
a relationship table must be established.
Once the correspondence between different data sets are
established, the spatial features need to be put together and
some transformation may be done so that the data describing
the same object coincide.
Theoretical and/or empirical transformations are available in
literature to obtain maps in a unique system with a fixed
tolerance. These transformations are nevertheless insufficient to
completely remove differences and deformations.
1.3 Problems of the most common transformations
Geometric transformation is the process of converting a digital
map from one coordinate system to another by using a set of
control points (also known as homologous pairs) and some
transformation equations. There are several types of
transformations.
Polynomial transformations between two coordinate systems
are typically applied in cases where one or both of the
coordinate systems exhibit lack of homogeneity in orientation
and scale. The small distortions are then approximated by
polynomial functions in latitude and longitude or in easting and
northing. Depending on the degree of variability in the
distortions, approximations are carried out using second, third,
or higher degree polynomials.
Polynomial approximation functions themselves are subject to
variations, as different approximation characteristics may be
achieved by different polynomial functions. The simplest
polynomial is the affine transformation (or first order
polynomial) which is a 6 parameter transformation (rotation,
shift in X and Y, differential scaling in X and Y and skew).
The most important advantages using affine transformation are:
straight lines are transformed in straight lines; parallel lines are
transformed in parallel lines; incident lines are transformed in
incident lines; the ratio between parallel lines is preserved.
Using particular restriction on the polynomial coefficients it is
also possible to preserve the areas of the features (congruence
transformation) or the shapes (similarity transformation).
The higher order polynomials are useful in registering maps
with varying localized distortions, i.e. where the distortion can
not be easily modelled by affine transformation. The greater the
local distortions is, the higher is the polynomial function to be
used.
One property of polynomial interpolation, which is undesirable
in practical applications, is that the interpolating polynomial
can show oscillations which are not present in the data. These
oscillations get worse as the degree of the polynomial increases.
To clarify this concept a one dimensional case can be taken into
account: figure 1 shows the famous example of this
phenomenon due to Runge using 11 equally spaced data points
on the interval [-1,1] and the interpolating polynomials of
different degree (3,7 and 11 respectively).
08 06 04 202 c €2 04 06 08 1 + 8 06 Dé DZ c C2 04 05 T] 1
02 04 28 08 1 1 08106 4 62 © {7 04 06 08 1
08 25 .04 .02 t
Figure 1. Oscillation problem of polynomial interpolation
The second method commonly used applies a variable
transformation to different portions of the unadjusted data. A
possible solution is based on the triangulated data structures
method suggested by Gillman (1985) and Saalfeld (1987) and a
piecewise linear homeomorphic transformation, known also as
rubber sheeting, suggested by White and Griffin (1985),
Saalfeld (1985) and Gabay and Doytsher (1995). This
approach, again based on homologous points of the two maps,
is today the most popular (Lupien and Moreland, 1987;
Doytsher and Hall, 1997; Cobb et al., 1998).
The main disadvantage of the rubber sheeting transformation is
that it holds the control points fixed, that is the control points in
the two maps match precisely, therefore they are treated as
being completely known and with no error. This kind of
approach is purely deterministic and it doesn't consider the fact
that any coordinate in a geographic database has a measurement
error. This second consideration is particularly important: while
rubber sheeting allows for a better solution from the numerical
point of view (the control points coincide and therefore null
residuals are obtained) it can bring about the description of the
phenomenon of transformation far from the physical reality.
Another problem related to the rubber sheeting transformation
is that each error in the selection of control points affects,
without any error filtering, the deformation of its no
homologous neighbouring points.
2. THE INTEGRATION PROCESS
2.1 The first step: the automatic research of homologous
points
The starting point to estimate every transformation is the
homologous points detection.
Control points are points that are in the same location in both
datasets. Usually they are manually chosen interactively in
