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The translational motion is represented as the 3D location of the
object reference frame relative to the current camera reference frame
using the vector
The t_{X} and t_{Y} components correspond to directions parallel to
the image plane, while the t_{Z} component corresponds to the depth of
the object along the optical axis. As such, the sensitivity of image
plane motion to t_{X} and t_{Y} motion will be similar to each other,
while the sensitivity to t_{Z} motion will differ, to a level
dependent upon the focal length of the imaging geometry.
For typical video camera focal lengths, even with ``wide angle''
lenses, there is already much less sensitivity to t_{Z} motion than
there is to (t_{X}, t_{Y}) motion. For longer focal lengths the
sensitivity decreases until in the limiting orthographic case there is
zero image plane sensitivity to t_{Z} motion.
For this reason, t_{Z} cannot be represented explicitly in our
estimation process. Instead, the product
is estimated.
The coordinate frame transformation equation

(15) 
combined with Equation 12 demonstrates that only
is actually required to generate an equation for the image
plane measurements (u,v) as a function of the motion, structure, and
camera parameters (rotation
is discussed below).
Furthermore, the sensitivity of
does not degenerate at long
focal lengths as does t_{Z}. For example, the sensitivities of the
u image coordinate to both t_{Z} and
are
demonstrating that
remains observable from the measurements
and is therefore estimable for long focal lengths, while t_{Z} is
not (
approaches zero for long focal lengths).
Thus we parameterize translation with the vector
True translation
can be recovered postestimation simply by
dividing out the focal parameter from .
This is valid only
if
is nonzero (nonorthographic), which is desirable, because
t_{Z} is not geometrically recoverable in the orthographic case. To
see this mathematically, the error variance on t_{Z} will be the error
variance on
scaled by ,
which gets large for
narrow fields of view.
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19990517