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and stitching them along their line of intersection. The positional accuracy of the reconstructed roof elements such as
ridgelines of the roof structure is highly related to the quality of the extracted 3D plane-roof polygons. Failure in correctly
estimating the orientation of the 3D plane-roof polygons in object space causes displacement and rotation of the ridgelines
with respect to their exact positions during the reconstruction process. In addition, due to the nature of region growing
type segmentation algorithm used in the lower level process of this work (Fritsch and Ameri, 1998), the quality of the
roof outline is poor. In fact, in real-world images, object boundaries cannot be detected solely on the basis of their
photometry because of the presence of noise, occlusion and various photometric anomalies. Therefore, methods for finding
boundaries based on purely local statistical criteria are tied to error, finding either too many or too few edges based on
arbitrary thresholds (Fua and Leclerc, 1990). To supplement the weak and noisy local information of the images and
probable misinterpretation of the orientation of the 3D plane-roof polygon, the geometric and topological information
that the coarse object model can provide is incorporated into the chain of the reconstruction process. This information is
introduced into the process of the object model verification based on a weighted least squares minimization process. A
fine building model is obtained in an iterative, top-down, model-driven estimation process by simultaneously fitting the
3D model into the corresponding images where the geometrical and topological model information are integrated into
the process as external and/or internal constraints during the estimation. The ability to apply such constraints is essential
for the accurate modeling of complex objects. In particular, when dealing with a generic object model, it is crucial that
the model elements are both accurate and consistent with each other. For example, individual components of a building
can be modeled independently, but to ensure realism, one must guarantee that they touch each other in an architectural
way. The estimation procedure yields a description of the building that simultaneously satisfies all the constraints within
all the images. As a result, it allows us to perform a consistency check, and refinement of the model across all the
images. Moreover the ability of the estimation method to fuse the information and impose the geometrical and topological
constraints over all the images increases the accuracy and reliability of the reconstruction.
In the same line of the major image matching techniques, i.e. feature-based (Forstner, 1986), and area-based least squares
matching (Fórstner, 1982, Ackermann, 1984), the proposed verification process is called Feature Based Model Verification
(FBMV). Similar to feature-based image matching techniques where a set of image-driven geometric features such as
points, or edges are utilized in one image to be matched to the homologous features in corresponding images in order
to, e.g. describe the surface geometry of the viewed scene. The FBMV uses model-driven geometric primitives to be
matched to the respective homologous features in corresponding images taken from different viewpoints in order to verify
the geometric description of the object model. In recent years, there has been a considerable increase in the number of
publications on parameters solving for model-based vision (Lowe, 1991, Haala, 1995, Fua, 1996, Gülch et al., 1998,
Brenner and Haala, 1998). An interesting similar work is reported by (Gruen and Li, 1997). Their method is a semi-
automatic approach for 3D extraction of linear features. In fact, this is an extension of a point-wise least squares template
matching method (Gruen and Stallmann, 1991, Baltsavias, 1991), where a deformable contour model is used as a template
instead of a square or rectangle which is generally used in conventional least squares matching techniques. In our study,
their work is categorized as an area-based object extraction or alternatively Area Based Model Verification (ABMV ).
3 FBMV-MATHEMATICAL FOUNDATION
The objective of this section is to formulate the verification of the hypothesis building model. This is carried out by back
projecting the 3D coarse model into the corresponding 2D images. Although this transformation is a non-linear operation
it is a smooth and well-behaved transformation, and it is a promising candidate for the application of the well known
Gauss-Markov estimation model based upon an iterative least squares minimization error criterion. This method requires
the appropriate initial guess for the unknown parameters. These values are provided by the geometric and topological
information derived from the reconstructed coarse model itself (Ameri and Fritsch, 1999). In practice, the whole spectrum
of the observations derived from the building model's description are divided into three major categories as 7) image
based, 2) object based and 3) image-object based observation equations which are discussed in the next subsections.
At the starting point of the estimation process a dense internal data structure is built from the model description. The
structure is used to define identical 3D points, edges, and planar surfaces, as well as their topological relationships. In this
manner, the model primitives may move independently while being attached to their adjacent primitives. In this way, an
edge element connecting two model points can stretch under the influence of shifting one of its endpoint from its initial
location and rotate under the influence of the movement of the another endpoint.
3.1 Image Based Observations
The observations concerned in this class are introduced into the estimation process for solving the unknown parameters
of 2D primitives such as the parameters of the 2D image edges or the coordinates of the homologous 2D model points in
image space. Two types of observations 1) linearity which serves as functional model of the estimation process, and 2)
connectivity which is applied as topological constraint are integrated into the system as image based observations and are
discussed next.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 25
