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3.2 Image-Object Based Observations 
These types of observations are the essential parts of the estimation model. They are integrated into the estimation process 
in order to establish the required link between the image and object space. They are acting as a bridge to tie the estimated 
corrections of the unknown parameters obtained in image space to their respective model parameters in object space 
during each iteration. 
3.2.1 Collinearity: A Global External Geometric Constraint. The mapping relation between a point in 3D object 
space P;(.X, Y, Z), and its perspective projection in 2D image space 2"? (z, y), can be represented by the classical 
collinearity equations as follows: 
SUMA. FO YZs 
y 7 + EJ)(X.Y,Z) =0 (8) 
Additionally, a mapping relation is needed to relate a point in camera system p §*™(z;, y;) to its corresponding point in 
image coordinates system p;”"? (x;, y;). The transformation parameters are expressed in the terms of an affine transforma- 
tion: 
cam img img 
Xi S: 011 T; T€: y. + C10 
cam. img img 
Vi = C1, +20 +1; * +C20 (9) 
where c11, C12, €21, C29 ANd C10, Cap are the rotation and translation parameters of the affine transformation respectively. 
Assuming the interior and exterior orientation parameters of each image are given, then the unknowns to be determined 
are the image coordinates (x;""?,y;"") of the model point and its corresponding coordinates (X :, Yi, Z:) in the 3D object 
space. Therefore, if the coordinates of a point is given in object space, the corresponding image coordinates of the point 
is simply derived from the equations (8) and (9), or alternatively, if an object is imaged from more than one image and the 
interior and exterior orientation parameters of the images are given, then 3D coordinates of the point can be reconstructed 
by simultaneous intersection of the above collinearity conditions (resection in space). 
Hence, the linearized equations concerning a 2D point in image coordinates pine (z,y) with respect to its 3D position 
P;(X, Y, Z) can be formulated as: 
OR: OF? OF? 
4v ———AXi t z——AYi t —— — AZ + 
OX|x=x° ÖYıy=yo OZiz-zo 
cu Ain? d Ci Ay?" — Ir = ez 
(10) 
OF? OF” OF? 
4v —— — AXi t zZ5L— — AY; t AZ 
OX|x — xo OY y yo OZ|z=7° % 
co Az! T Co2 Ay P9 10 = el 
where 
IF = Nami) pr x0. yo, 20) 
ini’ Mf ATi0} 4 PBX, VY, 2%), 
The equation (11) for all the model points can also be arranged in the standard Gauss-Markov model (see equation 6), 
whereas x is the vector of unknowns consisting the correction of the model points in object (A.X ;, AY;, AZ;), and image 
space (Aæ!"#, Ay:"#), 
3.3 Object Based Observations 
As it has been discussed so far, the main objective of the FBMV method is to integrate the model-driven information into 
the estimation model as supporting constraints. The set of observation equations which is classified in this category are 
directly obtained based upon the discription of the reconstructed coarse building model, before or during the estimation 
process. These constraints are introduced between the model primitives in object space as global or local geometric 
constraints and are linearized and applied as weighted observation equations. In this manner, the integration of the model 
primitives as unknowns into the total system of equations is completely flexible. Introducing the relationship between the 
model primitives as a strict condition by increasing its weight or alternatively reducing its influence into the system by 
decreasing its weight. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 29 
Rs Emm 
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