i
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8.3. A Brief Look at Some Advanced Ideas in Computer Vision 211
where the matrix M is derived from the translation matrix. Using the camera
calibration matrices P and P
to project the world points X
3
and X
3
onto the
viewing planes, X = PX
3
and X
= P
X
3
, Equation (8.8) becomes
(P
−1
X
)
T
M(P
−1
X) = 0,
or
X
T
((P
−1
)
T
MP
−1
X) = 0.
And finally, by letting F = (P
−
1)
T
MP
−
1, the final result is obtained with
an equation that defines the fundamental matrix:
X
T
FX = 0. (8.9)
Equation (8.9) affords a way to calculate the nine coefficients of F using
the same techniques of estimation that were cov ered in S ection 8.3.4. When
written in full for some point x
i
projecting to X
i
and X
i
, Equation (8.9)
resembles some of the results obtained earlier, e.g., the camera matrix ...