Babak Ameri 
  
The system of (6) is the well known Gauss-Markov estimation model. The least squares estimation in this model gives a 
unique and most probable set of estimates for all the parameters of the 2D model edges. 
To make the selection process robust and impose a self-diagnosis mechanism, the generated buffer is updated in a regular 
interval during the iteration process. As the process is iterated, the initial parameters of the 2D edges are updated based 
on the minimization of the orthogonal distance error between the selected edge-pixels and their respective 2D edges. 
Consequently the updated parameters define a new orientation for the generated buffer. In addition by introducing a 
smaller width, the size of the buffer is reduced. As a consequence, as outliers are excluded from the estimation process, 
this procedure reduces the computational burden and increases the accuracy and speeds up the convergence of the process. 
Figure 3 indicates the selected edge-pixels of the corresponding 2D edges in the figure 2 for the last iteration of the 
estimation process. 
3.1.2 Connectivity: A Global Internal Topological Constraint. The connectivity constraints are integrated into the 
estimation model as a topological constraint based on the intersection point between adjacent model edges. Figure 5 
depicts a corner of the building model when two edges edge ;, and edge», are connected through the intersection point 
Dint- 
edge, 
edge, 
Pint © Te 
Pm 
Figure 5: Intersection of two adjacent edges 
In general, it is possible to introduce the connectivity constraint into the estimation process both in object space or image 
space, because it is invariant under the transformation and it is independent of the embedded space. In this model, it is 
categorized as an image-based observation, in order to overcome the problem of correspondence between the model points 
within the images. In other words, by setting up the following formulation, the location of the corresponding 2D model 
points in different images is introduced implicitly based on the topological information, not the geometrical one. That 
means we do not compute the location of the intersection point explicitly based on the intersection of the two adjacent 
edges. Therefore, the problem of finding homologous points in respective images is not encountered as it is required in the 
feature-based matching techniques. In fact, if we had the correspondence relationships between the homologous model 
points in different images, then the verification of the coarse model would be done simply by obtaining the exact location 
of the 3D model points based on the simple resection technique such as multiphoto geometrically constrained matching 
(MPGC) procedure (Baltsavias, 1991, Gruen and Stallmann, 1991). 
Similar to the derivation of linearity observation equations (see equation 5), the connectivity constraint between adjacent 
edges for every associated edge member of a model point is introduced into the total system of equations. Let us consider 
the equation (4), to represent a 2D edge e ,. ;), in image I,.. Linearization of this equation with respect to its parameters, 
in this case, 2D edge parameters (d,. ;), (rj) and 2D coordinates (1,77, y;,;.) of the intersection point in image space 
results in the following formulation: 
Of. j) Of(r j) 
mouse 1 cout + 
i IM n dct) eS 
Of j i Of j i im im 
Fr Min, Bp e Mint — li = ea Yang") 0 
primam Nu=vine 
where 
a img(0) . 0 img(0) 0 
loni d; ; > qune sin Or.) p yin? COS 0(, i): 
The arrangment of the above equations for all the connected edges in a matrix form result in the Gauss-Markov model 
similar to equation (6), except that x is the vector of unknowns consisting of the corrections of the 2D edge parameters 
(d(ir,;)>O(r,;)) and corrections of the coordinates of the intersection point (Az Im9 Ay. 
  
28 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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