Babak Ameri 
  
  
Figure 1: Uncertainty buffer of homologous model edges in corresponding images 
3.1. Linearity: A Local Internal Geometric Constraint. The estimation process is an orthogonal linear least 
squares regression problem. It acts as a functional model with the objective to simultaneously minimize the perpen- 
dicular sum of the Euclidean distances between the candidate edge-pixels and the projected 2D model edges in all the 
images. A set of constraints is also integrated into the estimation model to support the minimization process. In fact, it 
would be sufficient to simply solve a resection equation for modifying the coarse building model if we are able to find the 
corresponding matches between the model points and their homologous points in the respective images. Thus, in order to 
overcome the problem of feature corespondence, the match is actually established between the projected model edge and 
candidate edge-pixels in the image. In other words, since the precise position of the endpoints of image edge is unknown 
or is missing e.g., due to the occlusion, it is necessary to minimize only the perpendicular distances from the representative 
points of an image edge to the projected model edge. 
In order to measure the perpendicular distance between a representative edge-pixel (z ;, y;) and the projected 2D model 
edge ey, it is useful to express the projected edge in image space in the following form: 
€; ! T; Sin; — y: COS; — d; = 0 (1) 
where d; is the distance between the origin and the line, and 0; expresses the angle between the edge and x axis. In 
practice, the initial parameters of the 2D edge (85, d?) are obtained by back projecting the two endpoints of the 3D model 
edge into the corresponding images using the collinearity equation (8). The computed 2D points are then plugged into the 
equations (2), (3), in order to derive the initial 2D edge parameters. 
Vend — Vstart 
0; — arctan —— ————— (2) 
Tend — Estart 
dj = start SiINÔ; — Ystant COSÔ; (3) 
The projected 2D edge e,, ;, in image I, serves as initial guess for finding the exact position of the edge model within the 
corresponding images, An uncertainty buffer with a user specified width is generated around each edge model in image 
space based on the initial position of the edge and is used as the search space to find the representative edge-pixels. Figure 
1 shows the generated buffer around the homologous model edges of a reconstructed coarse building model in four images 
taken from different view points. 
Each pixel within the specified buffer is selected as a representative edge-pixel if it satisfies the following two conditions: 
e its gradient direction is approximately perpendicular to the edge model direction, and 
e the magnitude of its gray value gradient is more than a data-driven adaptive threshold. This threshold is computed 
based on a cumulative histogram of the gradient magnitude of all the candidate pixels within the buffer, which satisfy 
the first criteria. Thc gradient magnitude associated with each selected pixel is used as its weight in the estimation 
model. 
Applving these two conditions for the selection of representative edgc-pixcls which should satisfy equation (1) has the 
following advantages. Firstly, pixels which are laid on the edgc image have stronger cffect during the fitting procedure 
because they have more weight in the estimation process. Secondly, the gradients caused by background objects will not 
interfere with the parameter estimations, as they are not in the approximate direction of the edge model. Figure 2 indicates 
the selected edge-pixels of the homologous 2D model edges in corresponding images within the generated uncertainty 
buffer in the first iteration of the estimation process. 
At this point, after selecting the edge-pixels representative, we are ready to introduce the linearity constraints into the 
estimation model. Let us represent the equation of the projected 2D edge in image 7 ,., passing through the selected edge- 
pixel (z7^?, y; ?) in the following form: 
img 
/ A un Og. „Mg — „Mg 
fir. (0,d) 2 2; sind.) — y; ? costi. —dij - eir; yi ) (4) 
  
26 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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