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3.3.1  Coplanarity: A Global External Geometric Constraint. Due to the pragmatic assumption that building roof 
structures are geometrically described by the aggregation of k > 1 planar surface(s), all the bounding points P ;(.X, Y, Z), 
of the 3D plane-roof polygon 7, should satisfy the coplanarity condition defined as follows: 
fk) (7, D) = Ax Xs + BLY; + Ck Zi 4- Dj — ej (11) 
where (Az, Br, Cr), are the components of the surface normal 73 1, Dy is the distance from origin to the plane-roof polygon 
FX, and ej is an added error parameter. The partial derivatives of this equation with respect to the unknown parameters, 
that is the 3D coordinates of the model points, is obtained by: 
Of(i,k) Of (i,k) Of i,k) 
TVC—AXid4 uM .AY9 AZ;-li-e; 2 
OX xon C ay, uM Das 92 L=e (12) 
where 
L=—(A2X? + BY? + 0070 1 DY, 
In fact, the corrections (A X ;, AY;, AZ;), in each iteration represent changes to the initial location of the model points, 
while the best planar fit to the updated model points is obtained. Introducing an equation of the type (12) for every point of 
the plane-roof polygons FY, in the estimation model and arranging all the equations in the matrix form result in a Gauss- 
Markov model similar to equation (6), whereas z is the vector of unknowns consisting the corrections of the coordinates 
of the model points (AX ;, AY;, AZ;). 
3.3.2 Conditional Constraints. The FBMV is an iterative procedure based on Newton-Raphson method, thus it con- 
verges to the minimum and becomes closer to the correct solution in every iteration, unless the system is degenerated, the 
initial values are so far away from the true solution, or the estimation model is incorrectly established. This property en- 
ables us to integrate additional constraints between the model primitives during the iteration process, if certain conditions 
are satisfied. As we have mentioned previously the strength of our method is a data-driven generic data model. That means 
instead of imposing certain regularities or conditions into the model in the earlier stages of the reconstruction process, 
these regularities and constraints are introduced into the model in the higher level process of reconstruction. Such con- 
straints are the orthogonality, or parallelity between the adjacent model edges, symmetricalness or semi-symmetricalness 
between the adjacent faces, and so on. The decision to impose these constraints into the estimation model is made during 
model verification process when the required criteria are met. The triggered constraints are integrated into the model, 
simply by adding a new row to the total system of equations. For the sake of completeness the orthogonality constraints 
are elaborated in details next, the other constraints can be dealt with in the same manner. 
  
Figure 6: Two orthogonal adjacent edges 
Orthogonality: A Local External Geometric Constraint. Figure 6 represent the angle a, between two 3D model edges 
Ej, and E». The conditional geometric constraint of the orthogonality is applied for every model point P;, when a satisfies 
the following condition during each iteration: 
90—t<a<90+t (13) 
where f is a threshold (e.g., 5°) indicating the small deviation of à from its expected value i.e. 90°. In fact, when two 
adjacent edges are considered orthogonal then the following constraints should be met: 
filX,Y, 4) = Q102 + b1b9 + C1C9 =0= € (14) 
where Ej (01,54, c1), and É,(a,, ba, C2), are the directions of the F, and E» respectively and e; is an added noise pa- 
rameter. Linearization of the equation (14) with respect to the position of the model points P;(X,Y, Z) result in the 
form: 
Of 
OX|x- xo 
Ofi 
Ofi 
Sik 
AX quain. 
OZ|z zo 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 31