The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008
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another and with specific angular relations with one another
need to interact. Using projected LiDAR roof contours and their
error projections, the neighboring set of straight lines associated
with each building roof can be further reduced. This paper is
organized as follows. Section 2 presents the proposed
methodology. The preliminary results are presented and
discussed in Section 3. Finally, the paper is finalized in Section
4 presenting some conclusions and outlook.
2. METHODOLOGY
The proposed methodology comprises preprocessing steps, the
establishment of the energy function (U(x)) based on an MRF
model, the solution of the energy function by applying a
minimization algorithm, and the completion of the detected
straight lines' groupings for reconstructing the refined image-
space roof contours. In the following sub-sections, details on
briefly described steps of the proposed methodology are
described. However, it will be given considerable emphasis on
basic MRF theory and on development of the energy function.
2.1 Preprocessing
The preprocessing steps mainly comprise the projection of the
3D roof contours onto the image-space and the extraction of the
image straight lines that are nearby the projected LiDAR roof
contours. The techniques used in these steps are well-known
and, as such, only more general details are presented.
In order to project the 3D building roof contours onto the
image-space, two basic steps are necessary. First, the
collinearity equations are used, along with the exterior
orientation parameters, to transform the roof contours into the
photogrammetric reference system. Second, an internal camera
model and the associated interior orientation parameters are
used to add systematic errors and to transform the roof contours
from the photogrammetric reference system to the LC-image
coordinate system. The error projections are estimated in order
to construct a registration error model.
The registration error model is a simple bounding box
constructed around each projected LiDAR straight line, which
enables the straight line extraction process to be focused only
on limited regions of the image, avoiding the extraction of
irrelevant information. There is a large amount of research in
the literature in the subject of straight line extraction. Examples
of methods are the Bums line detector (Bums et al., 1984) and
the Hough transform based methods (Balard and Brown, 1982).
The algorithm for straight line extraction is based on standard
image processing algorithms and seems to be effective for the
present application. First, the Canny operator is used to
generate a binary map with thinned edges. Next, an edge
linking algorithm is applied to the edge map for organizing the
pixels that lie along edges into sets of edge contours. In order to
extract the straight lines, the edge contours are approximated by
polylines through the recursive splitting method (Jain et al.,
1995). Very small straight lines (2-3 pixels length) and straight
lines differing too much in orientation (e.g., 20°) from the
projected LiDAR roof contour are removed, since they are
unlikely to be valid candidates for constituting roof contours. In
the last step, simple perceptual grouping rules (i.e., proximity
and collinearity) are used to merge collinear straight lines and
then to further reduce the number of candidates for representing
the roof contours.
2.2 MRF concepts and the energy function
2.2.1 Basic concepts of the MRF theory
MRF theory provides an efficient way to model context-
dependent features such as straight lines forming a roof building
contour. In an MRF, the sites in S= {1, ..., n} are related to one
another through a neighborhood system defined as N= {N i; i G
S}, where Nj is the set of sites neighboring i. A random field X
is said to be an MRF on S with respect to a neighborhood
system N if and only if,
P(x) >0, V x e X
p ( x i I x s-{i}) =p ( x i I x Nj )
(1)
Note that x is a configuration of X and X is the set of all
possible configurations. Also note that Xj e x and x s . {i j (or
x N )cz x - As stated by the Hammersley-Clifford theorem, an
MRF can also be characterized by a Gibbs distribution
(Kopparapu and Desai, 2001), i. e.,
where:
P(x)=
exp(-U(x))
Z
(2)
Z= Z exp(-U(x))
xeX
(3)
is a normalizing constant and U(x) is an energy function, which
can be expressed as:
U(x) = Z V c (x)
csC
(4)
Equation 4 shows that the energy function is a sum of clique
potentials (V c (x)) over all possible cliques c G C. A clique c is a
subset of sites in S in which every pair of distinct sites are
neighbors. The value of V c (x) depends on the local
configuration on clique c. For more detail on MRF and Gibbs
distribution see e.g. Kopparapu and Desai (2001) and
Modestino and Zhang (1992).
2.2.2 The energy function:
Straight lines resulting from the image processing techniques
are used to construct an MRF model expressing the specific
shapes of building roofs, having as reference the polygons
resulted from the photogrammetric projection of LiDAR roof
contours. The associated energy function is defined in such way
that each straight line is associated with a discrete random
variable (x;) assuming binary values according to the following
rule: