International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
length of a data segment is not very meaningful. In our
range images. edges are usually extracted in a fragmented
ray and can be much shorter or much longer than an edge
in the real world.
Because we want to find all possible matches of models in
our data set, we do an exhaustive search of the interpreta-
tion tree and store every consistent solution found.
4.3 Verification of Hypotheses
Photogrammetric methods are used to determine the trans-
formation that transforms our model into data space. Two
transformations in are of particular importance: a Helmert
transformation with 4 free parameters and an affine trans-
formation with 6 free parameters in two-dimensional
space.
Because known approaches usually work for estimating
parameters by transforming points, significant points are
used for each edge instead of transforming lines. There are
several possibilities for this and certain problems associ-
ated with each possibility:
1. Middle points of edges: Because the length of an edge
is of limited significance (real edges are usually frag-
mented into several edges in the edge image), the mid-
dle point of an edge is somewhat arbitrary. Apart from
this, information is lost because a line contains more
information than a single point.
to
Endpoints: Here, the same problem with the length
and exact position of an edge occurs. In fact, the
endpoints of an extracted edge usually don’t coincide
with the endpoints of a real edge. Besides, using end-
points means that edges are given an orientation. To
correctly find all possible solutions, each edge would
have to be considered twice, once in each direction.
This doubles the depth of the interpretation tree.
3. Intersection points: These probably give the most use-
ful information. Intersection points are calculated by
treating edges as lines, so points not directly lying
on an edge are also found. In fact, those intersec-
tion points usually give the best approximation for the
endpoints of the real edges of a building’s facade. The
only problem is that a set of edges can have many in-
tersection points, some of them too far away from the
extracted edge to be meaningful. Solutions containing
such points have to be rejected.
The general form of a 2D affine transformation is as fol-
lows:
X=a+cr—dy 3)
Y=bt+ex+ fy (4)
In a Helmert transformation, it holds that
ecd (5)
1082
and
f=€ (6)
so the equations simplify to
X —a-cx-—dy (7)
Y=b+dz+cy (8)
The Helmert transformation has proven particularly useful
in our example. It provides rotation, transformation and
uniform scaling in both directions.
An approach for estimating parameters based on least
squares adjustment for equally weighted observations is
used (Niemeier, 2001). The coefficients of the transfor-
mation equations for all points are written in matrix form:
19-0 41 —U]
(1 Yi X
1-705 3 3»
A Ty T. (9)
10 m. Up
OL 75
The points in the target co-ordinate system are written as
Xı
Yı
1= | : (10)
x
y
The estimate for the transformation parameters is calcu-
lated as
$ —(ATAy 1411 (11)
5 SEARCH STRATEGY
5.1 Initialization
First, an initial model needs to be fitted. Our model so far
is a rectangle with variable aspect ratio which is automati-
cally estimated. For this, the user of our system is required
to select a structure in the edge image by enclosing it with
a bounding box. A generic rectangle is then fitted into the
edges inside the bounding box. For the initial fitting, an in-
terpretation tree is built for matching four model segments
to the data segments present. No constraints are used. At
leaf level of the tree. an estimate for the aspect ratio of the
rectangle is made. The model rectangle is stretched ac-
cordingly. Then, a Helmert transformation is calculated to
transform the rectangle into data space.
Typically, if there are more than four data segments present
in the bounding box, more than one suitable rectangle is
found. In this case, the rectangle matching most data seg-
ments and providing the best fit is used. Alternaltively,
the user can select one of several solutions. This way, à
custom-made model is found for the structure which we
wish to find in our edge image.
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