Chapter 2 Principles of Stereoscopic Depth Perception and ...

d h p(Z) ¢
Z conv ¢
¤
¤
2. Principles of Stereoscopic Depth Perception and Reproduction
and right cameras. Stated differently, when properly aligned, no vertical
image displacement occurs with a parallel camera configuration. The
horizontal disparity, d h p(Z), can be obtained by subtracting x l , the X coordinate
of the left camera projection, from x r , the X coordinate of the right
camera projection:
d h p(Z) ¢ x r (X¡ Z) ¤ x l (X¡ Z) (2.5)
By substituting (2.1) and (2.3) into (2.5) we obtain the following equation:
B
Z
(2.6)
From this equation we can deduce that the left-right camera plate disparity
will increase as the camera base separation, B, increases, and as the focal
length, , increases. The horizontal disparity will decrease as the distance
to the real-world point, Z, increases. For points infinitely far away along
the Z-axis disparity becomes equal to zero.
Converging cameras
For a converging (or toed-in) camera set-up the optical axes of both cameras
intersect at a convergence point, scaled by the camera base separation
(B) and the convergence angle ( ¡ ) of the camera imaging planes, such that:
B
2 tan( ¡ )
(2.7)
The converging camera set-up is illustrated in Figure 2.12. The projection
onto the camera imaging plane of a real world point (X,Y,Z) is slightly
more complex than in the parallel set-up (which actually is a special case of
the converging set-up where equals 0). As the camera axes are no longer
parallel to the Z-axis, both a horizontal translation (along the X-axis) and
¡
a rotation (around the Y-axis) are required, followed by a perspective projection
(Franich 1996). For the left camera, the horizontal translation is as
follows:
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