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the linear structures (feature paths) in the EPIs; join together paths that are collinear, although 
broken by occlusion; use the slopes of the linear paths to determine scene depth. Figures 10 
and 11 show the linear paths and the computed 3-space positions of those features found in 
the EPI of Figure 9 (this EPI is 30 lines from the bottom of the imagery). The predominant 
diagonal streaks of Figure 10 are mainly from the shirt in the foreground. The horizontal line 
at the bottom of Figure 11 shows the camera path. 
The EPIs in Figures 8 and 9 were constructed from a simple right-to-left motion with the 
camera oriented at right angles to the path. For what other types of motions can EPIs be 
constructed? In fact they can be constructed for any straight-line motion. As long as the lens 
center of the camera moves in a straight line the epipolar planes remain fixed relative to the 
scene. The camera can even change its orientation! about its lens center as it moves along 
the line without affecting this partitioning of the scene. Orientation changes move the epipolar 
lines around in the image plane, significantly complicating the construction of the EPIs, but 
the epipolar planes remain unchanged since the line joining the lens centers remains fixed. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
| TT IS IS TT ud 
U-T Straight Ledgels for slice Y«30 X-Z Potnts for slice Y«30 
Fig. 9. A second EPI (v -— 30) Fig. 10. Linear feature paths Fig. 11. Depth (z, z) values 
  
If the lens center does not move in a line, the epipolar planes passing through a world point 
differ from one camera position to the next. The points in the scene are grouped one way for 
the first and second camera positions, a different way for the second and third, et cetera. This 
makes it impossible to partition the scene into a fixed set of planes, which in turn means that it 
is not possible to construct EPIs for such a motion. The arrangement of epipolar lines between 
images must be transitive for EPIs to be formed. One further observation about EPIs: since 
an EPI contains all the information about the features in a slice of the world, the analysis 
of a scene can be partitioned into a set of analyses, one for each EPI. In the case of a right- 
to-left motion, there is one analysis for each scanline in the image sequence. This ability to 
partition the processing is one of the key properties of our motion-analysis technique. Slices 
of the spatio-temporal data can be processed independently (and in parallel), with the results 
being combined into a three-dimensional representation of the scene. 
Figure 12 is an EPI formed from a sequence of images taken by a camera moving forward and 
looking straight ahead. Again the image is very structured, except that, instead of lines, it is 
composed of curves. For this type of motion, in fact for any straight-line motion in which the 
camera is at a fixed orientation relative to the direction of motion (see Figure 13), the trajectories 
in the EPIs are hyperbolic. Feature tracking in this more general case would involve hyperbolic, 
and not linear, curve fitting. Viewing at right angles to the direction of travel, as in the motion 
mentioned above, is the one special case where the hyperbolas degenerate into lines. 
Epipolar Plane 
   
     
  
Epipolar 
Line 
Fig. 12. EPI from forward motion 
  
  
! Notice that only in the case of orthogonal viewing will the 
intersections of epipolar planes with image planes be parallel. Fig. 13. Forward motion geometry 
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