189 
  
  
  
  
Fig. 2: Line segment defined by two vertices 
sector, with its vertices V;', (i=1,2,3), implicitly contains all the depth information we can acquire from the original line 
segment. This is because the plane defined by the planar sector divides the object space into a half space 'beyond' the 
plane that cannot contain any object points, and into another half space that contains object points at yet unknown 
locations. The plane corresponding to the transformed planar sector is defined by the equation 
Nex-NeV4 -0 (1) 
where x denotes a point in space and N is the normal formed by the vector product (V4' - V4) ^ (Ve' - V4). We now have 
to associate the depth information underlying the planar sector with the corresponding pixels of the image frame of the 
frontal view. This image frame will serve as the final depth map. 
To do so, we project the planar sector, i.e. its vertices V;', into the image plane. The resulting vertices vj, (i=1,2,3), form a 
corresponding sector in the image frame (Fig. 5 left). The pixels (i,j) confined inside the sector represent points of the 
planar sector in space projected into the image frame. The distances of these points to the origin, and therefore their 
depth values Z(i,j), can easily be calculated by intersecting viewing rays through every pixel with the plane (Fig. 5 right). 
All transformations that turn the original vertices V; into the projected vertices v;' are calculated in homogeneous 
coordinates. The relevant translation and rotation matrices as well as the perspective projection and the consecutive 
conversion into the discrete pixel coordinate system are described in (Seetharaman, 1994). : 
For the intersection of the viewing rays with the planar sector, the pixels (i,j) are converted into the continuous image 
system which leads to the points x; = (x; , Xj, f). The depth values Z(i,j) of the intersection points are then calculated by 
Ne V3 
Ne Xj 
  
Züj - f (2) 
The resulting values Z(i,j) represent the depth map of the frontal view associated with one single line segment of the 
silhouette of angle x. To receive the complete depth map of this silhouette, we have to repeat the above process for all 
  
Fig. 3: Silhouettes reducing the space where the 
object can possibly be located 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995