158 
between the projection centers of the left and right camera. The left image pixel coordinates i and j are further 
obtained from ( ) 
; y=-AyN | (r—AnN 
=N EA mm + | 
’ Si 9j E t5 2) 
where Az, Ay, and s are numerical constants depending on the camera setup and the image rectification 
process. The parameter N defines the size of the (sparse) disparity map Py which is a matrix of size N x N 
with elements PyG, j) —p(1,j) forizi,..Nandjzl..,N. 
Consider now the case of two disparity maps Py and Qa obtained by the same stereo head from distinct 
positions. Let X, Yi, Zj and X5, Yo, Z3 be the left camera centered coordinates in the two positions, respectively. 
The task is to find the best fitting coordinate transformation from system one to system two. Dividing this 
transformation into rotation R and translation T, we have 
X 2 X 1 T 
eisRIÍSmI-[79ll. (3) 
2 Zi T. 
where the rotation matrix is decomposed into rotations around the coordinate axes, i.e., R = R,(w)R,(p) Rz (0) 
with 
cosw sinw 0 cose 0 —sinp 1 0 0 
R:w)= | —sinw cosw 0), Rıle)= 9.1 0 |; Re()=10 cos? sind (4) 
0 0 1 sing 0 cose 0 —sin0 cos0 
for 9 € (-2,£], e € (—7,7], and v € (-x,x]. The parameters to be estimated are 0, o, w, Tz, T,, and T; 
while the parameters H, B, Az, Ay, and s are assumed to be known. In particular, we assume that there are 
no projective deformations present in the disparity maps or that they have been corrected, see (Niini, 1994) for 
details. 
3. RELATIVE ORIENTATION USING MODELED FEATURES 
In this section, we estimate the relative orientation of two disparity maps using modeled features independently 
determined from each map. Any linear features consisting of lines or points can be used, but we have particularly 
in mind that there are planes, right circular cones and cylinders present in the scene and the modeled features 
include plane normals, axes of cones and cylinders, and vertices of cones. We assume that the disparity maps 
contain observations of the same geometrical objects but the maps need not necessarily overlap in case the 
correspondence between modeled features is otherwise known. This is often the case with planes, cones, and 
cylinders. 
Modeling of cones and cylinders follows our earlier work in (Jokinen, 1994). The vertex of a cone is estimated as 
an intersection point of a set of tangent planes for the cone. These tangent planes are obtained via surface normal 
estimation in the measurement points. Estimates for the cone axis and vertex angle are subsequently obtained 
by using the fact that the scalar product of the axis vector and a vector drawn from the estimated vertex to a 
measurement point should be a constant for every measurement point. In the cylindrical case, the mean value 
of a set of vectors obtained by computing vector cross products of nonparallel surface normals gives an estimate 
for the direction of the cylinder’s axis. The location of the axis can be estimated by determining the intersection 
of a fan of planes obtained by computing in every measurement point a plane defined by the normal vector and 
the axis direction vector. A constraint is added to the regression based axis location estimation in order to fix 
a unique point from the axis. The radius of the cylinder is estimated as a mean distance of the measurement 
points from the estimated axis line. The parameter estimates of cones and cylinders are assured and refined 
by minimizing the merit function F squared and summed over all the measurement points, where F equal to 
zero 1s the equation of the quadric surface. The minimization problem is solved by the Levenberg-Marquardt 
nonlinear least squares algorithm described in more detail in Section 4. 
The unit direction vectors of modeled linear features are used to estimate the relative rotation R in (3). Having 
no fixed location, these vectors can be thought to start from a common origin and the endpoints define two sets 
of points given by s. {=1,...,L, in the first system and by SQ, l.— 1,..., L, in the second system. The 
IAPRS, Vol. 30, Part 5W1, ISPRS intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995