In: Stilla U, Rottensteiner F, Paparoditis N (Eds) CMRT09. IAPRS, Vol. XXXVIII, Part 3/W4 — Paris, France, 3-4 September, 2009 
129 
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(d) Horizontal False Positives (HFP) 
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(e) Horizontal False Negatives (HFN) 
Figure 3: Horizontal Qualitative Evaluation: The recognition-driven process efficiently detects, in an unsupervised manner, scene 
buildings and recovers their 3D geometry. 
gable-type one ($1.4) and if all are zero we have a flat one ($1.1). 
The platform and the gambrel roof types can not be modeled but 
can be easily derived in cases where the fit energy metric is as 
sumed on local minima. The platform one (l>i, 2 ), for instance, 
is the case where all angles have been recovered with small val 
ues and a search around their intersection point will estimate the 
dimensions of the rectangular-shape box above main roof plane 
Pm. With the aforementioned formulations, instead of searching 
for the best among ixj (e.g. 5x6 = 30) models, their hierarchical 
grammar and the appropriate defined energy terms (detailed in the 
following section) are able to cut down effectively the solutions 
space. 
from the grouping criteria. The simplest possible approach would 
involve the Mumford-Shah approach that aims at separating the 
means between the two classes. Above equation can be straight 
forwardly extended in order to deal with other optical or radar 
data like for example in cases where multi- or hyper-spectral re 
mote sensing data are available. 
Furthermore, instead of relying only on the results of an uncon 
strained evolving surface, we are forcing our output segments to 
inherit their 2D shape from our prior models. Thus, instead of 
evolving an arbitrary surface we evolve selected geometric shapes 
and the 2D prior-based segmentation energy term takes the fol 
lowing form: 
3 MULTIPLE 3D PRIORS IN COMPETITION 
EXTRACTING MULTIPLE OBJECTS 
Let us consider an image (X) and the corresponding digital eleva 
tion map (Pi). In such a context, one has to separate the desired 
for extraction objects from the background (natural scene) and, 
then, determine their geometry. The first segmentation task is ad 
dressed through the deformation of a initial surface —► 1Z + 
that aims at separating the natural components of the scene from 
the man-made parts. Assuming that one can establish correspon 
dences between the pixels of the image and the ones of the DEM, 
the segmentation can be solved in both spaces through the use 
of regional statistics. In the visible image we would expect that 
buildings are different from the natural components of the scene. 
In the DEM, one would expect that man-made structures will ex 
hibit elevation differences from their surroundings. Following 
the formulations of (Karantzalos and Paragios, 2009), these two 
assumptions can be used to define the following segmentation 
function 
Eseg (0) 
J |Vc/>(x)| dx 
+ 
+ P 
J 
Jq 
H € (4>) r obj (J(x)) 
[ H e {<f>) r obj (7T(x)) 
Jci 
+ [1 - He(</>)] r bg (T(x)) dx 
+ [1 - He(4>)\ r b g (Pi(x)) dx 
(1) 
where H is the Heaviside, r ob j and r bg are object and background 
positive monotonically decreasing data-driven functions driven 
E2d{4>, Ti, L) 
He(<f>{x)) — H e (4>i (Ti(x))) 
Xi(L(x))d,x + 
J \ 2 Xm(L(x))dx + p'^2 J IVL(x)|dx 
i=l 
(2) 
with the two parameters A, p > 0 and the A>dimensional label 
ing formulation able for the dynamic labeling of up to m = 2 k 
regions. 
In this way, during optimization the number of selected regions 
m = 2 k depends on the number of the possible building segments 
according to <j> and thus the (¿-dimensional labeling function L 
obtains incrementally multiple instances. It should be, also, men 
tioned that the initial pose of the priors are not known. Such a 
formulation E seg + E2D allows data with the higher spatial res 
olution to constrain properly the footprint detection in order to 
achieve the optimal spatial accuracy. Furthermore, it solves seg 
mentation simultaneously in both spaces (image and DEM) and 
addresses fusion in a natural manner. 
3.1 Grammar-based Object Reconstruction 
In order to determine the 3D geometry of the buildings, one has 
to estimate the height of the structure with respect to the ground 
and the orientation angles of the roof components i.e. five un 
known parameters: the building’s main height hm which is has 
a constant value for every building and the four angles u> of the