160
then bilinear interpolation gives
Qa(tı2,J12) = Gelltı2),Ji2) + (2 — [h2]) (42 (P3121, 712) ^ 42(132], 712). (10)
Consider the triples (41, j1, p12) and (à, 71, q2) for (4,71) € €. The disparities pi and go describe the distance
from the second stereo head position to the object: p19 as measured from position one and d as measured from
position two. If the orientation were correct, then the disparities would be equal excluding the measuring error .
and the error resulting from interpolation. So we intend to minimize the difference between the two disparities
and obtain corrections to the orientation parameters. The merit function to be minimized is given by
X’ (a) =: S^ {W(i1, j1)[P12(i1, J1, P13 à) — d2(5, 3 01,05; 3)]"/ K, (11)
(41,71)€0
where K is the number of pixels in 2, the vector a — (0,9, w, Tz, Ty, T. ;)7 contains the orientation parameters,
and the possible nonnegative weights are included in the function W. We thus minimize the mean squared
distance between pi» and dj». In particular, note that considering the mean distance makes the merit function
insensitive to changes in the size of (2. The functions p12 and ÿ2 depend on the orientation parameters as follows:
p12 = p12[Z12(a)],
d2 = da{i2[y12(Y12(a), Z12(a))], j12[712(X12(a), Z12(a))1}- us
The dependence in (12) being nonlinear, the minimization problem is solved by the Levenberg-Marquardt
nonlinear least squares algorithm (Press et al, 1986, pp. 521-525). The algorithm combines the inverse-Hessian
method and the steepest descent method. In the inverse-Hessian method, the increments in the parameters are
obtained as a product of the inverse of the Hessian matrix of x? and the negative gradient vector of x“. In the
steepest descent method, the increments in the parameters are taken into the direction of the negative gradient
of x2. The terms in the Hessian matrix containing second partial derivatives 0*p19/0a;0a; and 0?$5/0aj 0a,
are dropped so that we have to evaluate only the first partial derivatives Op15/0a, and 045/0ay for k — 1,...,6.
According to (12), we have
Opi Opi 0212
day t 04749 Oa,
0d) Od» Oi» (Su: 0Y12 | Oya oe) Od» 0j12 E 0X12 | Oz13 Er
Oak 7 042 Öyı2 ÖYı2 day, 07219 day 0J12 O12 0X19 day 0719 day :
(13)
All the partial derivatives can be evaluated analytically using Formulas (1), (2), (3), (4), and the interpolated
values, e.g., 0ÿ2/0t12(t12, j12) = G2([d12], 12) — 42([t12|, J12)-
To summarize, note that one iteration step in the minimization of (11) consists of the following three interme-
diate steps. First, the disparity map Pw is projected onto Q according to the current orientation in order to
obtain the disparities p12. Second, the overlap region €) is determined and the interpolation for the values q» is
performed. Third, corrections to the orientation parameters a are computed following the Levenberg-Marquardt
procedure. The iteration is stopped when x? essentially stops decreasing, say, when it decreases relatively less
than one per mille.
5. TEST RESULTS
The methods for the estimation of relative orientation were tested with two synthetically generated data sets.
In the first case, the object consisted of a plane and cone (seen from above), while the second data set contained
patches from a plane, cone (seen from the side), and hyperbolic paraboloid (saddle surface). The generated
disparity maps P3» and Qa; for the second data set as seen from two distinct stereo head positions are shown
in Fig. 1. The cone is in the upper left corner, the saddle surface on the right, and the plane in the lower left
corner. Normally distributed noise with zero mean and standard deviation 0.1 were added to the disparities.
The true and estimated parameter values are shown in Table 1. The estimated ones are mean values of hundred
simulations and the sample deviations of the parameter estimates are of the order 0.01 in the case of data 2
solved by modeled features and of the order 0.001 in all the other cases.
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995
rn »-— — — AA
——
M. Li MA. As
