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1
Now suppose the camera undergoes a displacement taking it to position 2. Such a general displace-
ment will consist of a translation plus a rotation. Let the translation be given by the vector t such
18
; !
;;
that O1 O2 = t, where O2 is the translation of O1 . A general rotation of about some axis n will
be described by the rotor R, where R = exp (;i n). Thus, the frame f g is rotated to a
1 2 3
2
0 0 0 0
~
frame f g at O2 where = R R for i =1 2 3. At position 2 we have a new image plane
i
1 2 3 i
;;!
, and we let M2 be the projection of P onto . If X = O2P then it is clear that
2 2
X = X ; t: (75)
0 0
Now, our observables in the f g frame are Xi = X . In the f g frame they are Xi0 = X . If
i i
i i
we de ne the vector X0 which lives in the f g frame as X0 = Xi0 then we can write this vector as
i i
0 0
~ ~
X0 = Xi0(R R) = R(Xi0 )R
i i
~
= RXR
~
= R(X ; t)R (76)
Thus, X and X0 are related by
~ ~
X0 = R(X ; t)R or X = RX0R + t: (77)
In what follows we shall take the camera displacement as the unit of distance so that t2 =1.
Suppose we have n point correspondences in the two views and that the coordinates fXig and
fX0 g are known in each of these views. In any realistic case we would expect measurement errors
i
on both sets of coordinates the camera motion will then be best recovered using a least-squares
approach.
For n matched points in the two frames, we want to nd the R and t which minimize
n
h i
X
2
~
S = X ; R(X ; t)R : (78)
i i
i=1
If the fXig and fX0 g are our observed quantities, then we must minimize with respect to R and
i
t. The di erentiation with respect to t is straightforward
n
h i
X
~ ~
@tS =2 X ; R(X ; t)R @t(RtR): (79)
i i
i=1
At the minimum, @tS = 0 and therefore
n
h i
X
~
X ; R(X ; t)R =0: (80)
i i
i=1
Solving this gives us an expression for t in terms of R and the data
n
h i
X
1
~
t = RXiR ; X (81)
i
n
i=1
or
n
h i
X
1
~
t = Xi ; RX R (82)
i
n
i=1
0 0
~ ~
where t = RtR. This is obviously equivalent to saying that t = X ; RX R where X and X are
the centroids of the data points in the two views. It is indeed a well known result that the value of
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t which minimizes the least squares expression is the di erence between the centroid of one set of
points and the rotated centroid of the other set of points [Faugeras and Hebert 1983, Huang 1986,
Horn et al. 1988] { although in the past this result has been arrived at by methods other than direct
di erentiation.
We now turn to the di erentiation of equation (78) with respect to R. As the expression for S
is very similar to that given in equation (50), with vi = X and ui = X ; t, the solution is as
i
i
outlined in Sections 3.1 and 3.3 and can be directly written down as
n
X
~
X ^R(X ; t)R =0: (83)
i
i
i=1
If we substitute our optimal value of t in this equation we have
n
X
~ ~
X ^R(Xi ; X ; RX R)R =0: (84)
i
i=1
P P
n 0 n
~
Noting that the X ^X term vanishes, this reduces to vi ^RuiR =0 with
i=1 i i=1
ui = Xi ; X
vi = X0 : (85)
i
~
The rotor R is then found from an SVD of the matrix F where F is de ned in terms of the ui and
vi as given in equation (73). A comparison of this closed form solution to the SVD technique of
[Arun et al. 1987], the orthonormal matrix approach of [Horn et al. 1988] and the dual quaternion
approach of [Walker et al. 1991], reveals that each method should e ectively produce the same
results.
4.2 Camera motion and structure from two scene projections: range data un-
known
We now look at the more di cult problem of extracting the camera motion when only 2D projections
of the true 3D coordinate sets are available. Sabata and Aggarwal 1991 stress that in this case
the errors in the estimation of the range coordinates will tend to adversely e ect the estimation of
the motion parameters or vice versa. In [Huang and Netravali 1994] this 2D-to-2D correspondence
problem is discussed in some detail for a variety of features. For points, all algorithms considered
are e ectively two stage algorithms. In [Mitchie and Aggarwal 1986] the structure is estimated
rst by the application of rigidity constraints these can then be used to estimate the motion. It
is generally acknowledged that the errors incurred in estimating structure rst are greater than
when one estimates motion rst. Consequently there are many more algorithms which attempt
to solve the 2D-to-2D correspondence problem using a motion rst approach. One such method
is described in [Weng et al. 1989] this is a linear algorithm based on point correspondences which
gives a closed-form solution for estimating rst motion and then structure in the presence of noise. [ Pobierz całość w formacie PDF ]