Estimating 3D Polyhedral Building Models by Registering Aerial Images - Advanced Concepts for Intelligent Vision Systems - page 243

where g j is the gradient score for image I j . It is given by the average of the
gradient norm over all pixels coinciding with the projected 3D model segments.
Since we want the dissimilarity measure (3) and the gradient score (5) to help
us determine the best 3D polyhedral model, we must combine them in some way.
One obvious way is to minimize the ratio:

e
g

= arg min
w

w (6)

It is worth noting that during the optimization of (6) there is no feature extrac-
tion nor matching among the images. The use of the image gradient norms in
(6) is not equivalent to a feature-based method. In order to minimize (6) over
w , we use the Differential Evolution algorithm [10]. This is carried out using
generations of solutions—populations. The population of the first generation is
randomly chosen around a rough solution. The rough solution will thus define a
given distribution for the model parameters. The rough solution is simply given
by a zero-order approximation model (flat roof model) which is also obtained by
minimizing the dissimilarity score over one unknown (the average height of the
roof). We use the Differential Evolution optimizer since it has three interesting
properties: (i) it does not need an accurate initialization, (ii) it does not need
the computation of partial derivatives of the cost function, and (iii) theoretically
it can provide the global optimum.
In brief, the proposed approach proceeds as follows. First, the algorithm de-
cides if the building contains one or more facets, that is, it selects either the
model of Figure 2. (a) or the model of Figure 2. (b) . This decision is carried out
by analyzing the 3D normals associated with four virtual triangles forming a
partition of the whole building footprint. Second, once the model is selected,
its parameters are then estimated by minimizing the corresponding dissimilarity
score. Note that in the case of one facet building we only need to estimate the
plane equation using the criterion (4).

4 Experimental Results

4.1 Semi-synthetic Data

We have used a real triangular roof facet in two different aerial images. The
3D shape of this facet is computed using a high resolution Digital Elevation
Model. The rawbrightness of this facet is reconstructed in the second image
by warping its texture in the first image using the relative geometry and the
estimated 3D shape of the facet. The two textures are then perturbed by an
additive uniform noise. For every noise level we run our proposed approach 10
times. For every run, we compute the error as being the difference between
the estimated parameters and their ground truth values. Figure 4. (a) and 4. (b)