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the verification process enables FBMV to accurately recover the building corners (see figure 7-d), which were trimmed
off during the segmentation process. Figures (7-a and 7-c) show the initial and modified building model overlaid on the
corresponding aerial image, respectively.
The second example deals with reconstruction of a hipped-gable roof structure (see second row of the figure 7). As it
is shown in figures (7-e) and (7-f), although the major structures of the building, the bounding edges and even small
protruding structure of the roof is reconstructed correctly. Due to the low contrast of the edge segment of the hip tile of
the roof, the reconstructed edge is completely shifted away from its correct position. In addition, the position of the corner
points is not precisely defined. The modification of the model based on a multi-photo estimation process and imposing
the global constraints as discussed previously, enables FBMV to correctly recover this part of the roof and forces the
displaced model edge to located in its true position. Furthermore it also defines the positions of the model points more
accurately (see figures 7-g and 7-h).
The last example shown in third row of the figure 7 represents a more complex roof structure. The generated hypothesis
: coarse building model describes the building completely, but there are still displacements in some of the model primitives,
specially the intersection point between three adjacent plane-roof polygons is shifted significantly from its real position
(see figure 7-1). This is due to the failure in defining the correct orientation (slope) of the respective 3D plane-roof poly-
gons, which is caused by the low quality of the utilized DSM (Ameri and Fritsch, 1999). By applying the FBMV process,
the normal vectors 7i; of every plane-roof polygons are recovered precisely and consequently the intersection point is
moved to its real position. In addition, the real bounding edges of the model are also precisely located and the redundant
one is eliminated from the final model (figure 7-q). To complete the performance evaluation of the FBMV, the following
section is dedicated to numerical analysis and assessment of the quality of the final reconstructed building obtained from
the estimation model.
5 QUALITY ASSESSMENT
One of the key issue of the FBMV method is its ability to provide the essential tools for evaluation of the quality of
the reconstructed model and its geometric primitives. The combined least squares solution provides an estimate for the
variance factor ó& which can be used for the performance evaluation of the estimation process. In other words, it is
used to judge whether or not the estimation model is consistent with the earlier assumption that the noise distribution
follows a normal distribution function with a given standard deviation, which was the motivation to apply a least squares
minimization of the error criterion. In addition, considering sufficient agreement between the estimation model and our
early assumption, the standard and statistically well known covariance matrix Cov ,, of the estimated parameters can be
obtained as follows:
Covyz = 63(Y ATP.A.)! (16)
The estimated variances of the unknown parameters, specifically in our case the coordinates of the model points in 3D
space (6%,0%,0%), are the qualitative measures which indicate the accuracy of the model primitives and act as the
decision criteria in order to reject or accept the estimated model elements based on the simple thresholding process. The
evaluation process can be integrated into the whole chain of reconstruction process as an edition process (traffic light
concept (Forstner, 1996)). In a simple manner, these measures give a hint to the end user to perform a visual check on the
end product and perform the required modifications on the signalized model primitives if necessary. In the following are
the numerical results and the statistical analysis of the complex building (figure 7), where its corner points are numbered
as illustrated in figure (8).
5 1
2
4
: 3
3 14
12 li 9 8
Figure 8: Top view of a single complex building
The test was carried out for the verirication of the model based on utilizing two, and four corresponding images taken
from different views. In addition, the reconstructed building in every test is compared with a reference model digitized
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 33
