ct 
11S 
  
159 
task is to find the best fitting rotation which maps the endpoints of vectors s? to the corresponding endpoints 
of vectors s. To do this, we apply the approach presented in (Kanatani, 1994). We seek an optimal rotation 
R such that 
I I NE 
> wi |s$ — rsf | (5) 
ii 
1s minimized. The nonnegative numbers Wi, | — 1,..., L, are possible weights that can be given for the data 
points. The minimization problem is restated as a maximization of tr( RT C), where C is the correlation matrix- 
defined by 
L 
O=Y wish, (6) 
i=1 
To solve this, Kanatani extends three existing methods, namely the method of singular value decomposition, 
the method of polar decomposition, and the method of quaternion representation. We apply here the method 
of singular value decomposition, where C is decomposed into the form C — VAUT with V and U orthogonal 
and A diagonal with singular values in the diagonal. The result is that tr( RT C) is maximized over all rotations 
by 
1 
RzV 1 UT. (7) 
det(VUT) 
The solution is unique if rank(C) > 1 and det(VUT) — 1, or if rank(C) » 1 and the minimum singular value is 
a simple root. The rotation angles 6, ¢, and w can be computed from (4), if necessary. 
Having estimated the relative rotation, we subsequently estimate the translation T in (3) by using modeled 
points such as vertices of cones. In fact, only one point is enough to approximate the translation vector. In the 
case of multiple modeled points, the mean displacement gives an estimate for the translation according to 
iL 
T=) my amb, (8) 
i=1 ; 
where n and RO are the location vectors of the [th modeled point in the systems one and two, respectively. 
4. RELATIVE ORIENTATION BY SURFACE MATCHING 
In this section, we solve the relative orientation of two disparity maps by matching the disparity surfaces in 
the overlap region. This second method thus requires that the maps overlap. The method is intended for cases 
when there are no modeled features available or when the overlap region contains other objects than planes, 
cones, and cylinders yielding more information in addition to the modeled features. The idea is to project the 
first disparity map Py via object space onto the second map Qasr and insist that the disparity surfaces match 
as well as possible. The method is iterative and thus requires initial estimates for the orientation parameters. 
We formulate the method as follows. 
Assume that we have a disparity map Py with coordinates 1, j1,p1. For each pixel (#1, j1), we compute the 
X1, Yi, Z1 coordinates using (2) and (1). An initial orientation given by a rotation matrix Ro and a translation 
vector Tg transforms the X,Yi, Z1 coordinates into X15, Yi», Z1? coordinates of the second system according 
to (3). Applying (1) and (2), we get the corresponding disparity map coordinates i12, j12, P12 in the second 
frame. 
In general, the 419 and ji» coordinates are neither integers nor bounded to the interval [1, N]. On the other 
hand, the pixels with i43 € [1, N] and j12 € [1, N] define the overlap region of Py and Qu. We denote the 
overlap region in Py by Q. For each (i1,j1) € Q, a new disparity value §a(i12, j12) is interpolated by bilinear 
interpolation using the disparities qa of four neighboring pixels in Qa. If we denote by |;| the largest integer 
smaller than 7 and by [i] the smallest integer larger than 7, and interpolate 
(L2], [J121) ^ a2([2]; 121); 
(Fàa2], [312]) ^ ex(P2 b D12])); 
N 
« ss, 
(9) 
N 
d» (|2], J12) 2 as(Là2] (7121) + (ô12 — E121) (9 
1 q2([#12], [j121) + (J12 — [12]) (a 
N 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995