Game Development Reference
In-Depth Information
w
R . In
Zhang (2000), the imaging-distortion model is used and k 1 Im and k 2 Im are
considered. In phase 1, a planar object strategy is employed. Then, in phase 2,
the Levenberg-Marquardt method is used to minimize the following objective
function over all camera parameters p c :
in p are modified for fulfilling the orthonormality of the rotation matrix
(
)
(
)
()
2
2
im
im
Im
im
im
Im
,
(44)
f p
=
x
ˆ
x
f
+
y
ˆ
y
f
c
x
x
y
y
where the summation is done over all available data and all variables were
defined in equation 10. Instead of the linear estimation method in phase 1, a
nonlinear optimization can also be carried out to get a better initial guess of p l with
all distortion coefficients set to zero.
Optimization issue
At phase 2, each iteration of the optimization can also be split up into the following
two steps (Weng et al., 1992):
Step a: The function f ( p c ) in equation 44 is minimized w.r.t. all distortion
coefficients in p d by a simple linear least-squares method, while p l (contain-
ing all linear intrinsic and extrinsic parameters) is fixed.
Step b: The function f ( p c ) is minimized by an iterative optimization method w.r.t.
p l while p d remains unchanged.
However, due to the tight interaction between the linear parameters and the
distortion coefficients, this two-step optimization converges very slowly.
In Chatterjee, Roychowdhury & Chong (1997), the nonlinear optimization part
(phase 2) is further divided into three stages by following the Gauss-Seidel
approach. The first stage is similar to the optimization phase in Sid-Ahmed &
Boraie (1990), except that the optical center
[
]
T
, the aspect ratio s y / s x ,
and all distortion coefficients are fixed. The second stage is the same as Step “a”
in Weng et al. (1992). Finally, in the third stage, the function f ( p c ) in equation
44 is minimized only w.r.t. the optical center
x
y
0
0
[
]
T
x 0 and the aspect ratio s y / s x ,
while all other camera parameters are fixed. Convergence analysis for this new
parameter space partition method has been given in Chatterjee et al. (1997).
However, no convergence speed was provided.
y
 
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