ıme XXXIX-B1, 2012
ion
| Ej (R) is minimized firstly
s then used to minimize the
ine the translation t. This
ne direction to compute ro-
istraints effectively. In the
anslation separately, the s-
are amplified into large er-
1994). To fully exploit the
orithm 1 to optimize alter-
nslation vector.
stimation of Ry, t; is com-
otation estimation iterative
/ rotation value R’;, 1, and
kx1 Via £(R'j,4) from Eq.
e the method of (Lu et al.,
imation by minimizing the
ep is described as follows.
2 and t are iteratively opti-
objective function defined
RP; +t)[[”. (19)
when applied to a point,
line of sight defined by the
btained, the next estimate
owing absolute orientation
Ht— V;ak|?, (20)
rientation problem is then
88).
q. (9) compared with Eq.
nstead of V;. Both projec-
operties (Sect. 3.1). Hence
! c1 i1), an estimate of
rectly using the algorithm
bjective function (9). The
ithm 2.
for both R and t, purely
nethod of (Lu et al., 2000)
t as LOI-3). Since the only
projection vector, we will
ore details, the readers are
NTS
experiments on both syn-
nly within a cube defined
5] (Fig. ??) in the object
then created by linear fit-
ing points in the 3D lines.
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B1, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
Algorithm 2: Alternative Optimization
1. Given N (N > 3) 3D-to-2D line correspondences and initial
rotation Ro, compute: K; for i — l,....N,
B= (di, d»... dy). Set k z: Q.
2. Perform the following steps:
(a) Compute A — (Kj Rid;,:-- , Ky Ridy).
(b) Compute M — AB” and perform SVD: UDV" = M.
(c) Compute R, 4107 USVT, where S is set according to
Eqs. (15) and (16).
(d) Compute t, , , = (R4)
(e) Given R, +; and i 4.1: compute Ry, ; using the
algorithm of (Lu et al., 2000).
(f) Compute tt, = 1(Re+1)-
(g) Terminate the iteration if convergence is attained;
otherwise, k = k+ 1, go to step (a).
We add Gaussian noise to the projections of points and also con-
sider a percentage po,; of outliers, for which a set of 3D lines are
randomly selected and replaced by another line generated within
the cube [—0.5,0.5] x [-0.5,0.5] x [-0.5,0.5]. For each setting
of the control parameters in every plot, the result is obtained by
running 1000 trials and the mean value is recorded. To facilitate
the description, we denote the Algorithm 1 ^ 3 as LOI-1, LOI-2
and LOI-3 separately.
In Fig. 3, we plot the rotation and translation relative errors pro-
duced by the three algorithms as a function of Gaussian noise
with its standard deviation varies from 1 to 10 pixels. The num-
ber of sampling points that are used for creating the 2D lines is
set as 100. The line number is fixed to be 8 and the percentage
of outliers pou = 0. The plots show that "LOI-2" is consistently
more accurate than the other two algorithms.
Rotation relative error
1 2 3 4 5 6 7 8
Gaussian image noise (pixels)
Translation relative error
1 2 3 4 5 6 7 8 9 10
Gaussian image noise (pixels)
Figure 3: Relative rotation and translation error as a function of
image noise when the number of lines is fixed to be 8.
Fig. 4 plots the errors as the function of the number of 3D object
83
lines when the image noise is fixed (0 = 3 pixels). We compare
the performances of our algorithms with the OI method (Lu et
al., 2000), for which the corresponding 2N endpoints of the 3D
lines are used. It can be seen that all these algorithms can achieve
higher accuracy when the number of feature correspondence in-
creases. LOI-2 and OI algorithms show more accurate and stable
performances.
1
& —E— O02
0.9F —Ééc— LOI-1H
\ -7€-— LOI-3
\ —>- I
o
œ
Rotation relative error
=
o
E
o
o
=
=
S
D
=
=
.2 \
a
E S
o N
c :
© i
= =
To.
ne > A ees
dE $e A
ET Lg
0 1 :
5 10 15 20
Number of 3D object lines
Figure 4: Relative rotation and translation error as a function of
the number of object lines when the standard deviation of image
noise is fixed to be 3 pixels.
In Fig. 5(a), we give the percentage of convergence when the
initial poses are generated from a multinormal distribution with
mean as the true pose and the diagonal covariance dX, where
the standard deviation element of X is about 1.5 degree for the
rotation angles, 0.2 for the x and y components of the translation
t, and 0.5 for the z component. ó varies from 1 to 20. The plot
indicates that LOI-2 shows very robust performance and slightly
outperforms the OI algorithm which is proved to be global con-
vergent. In contrast, the LOI-3 algorithm produces very poor per-
formance. We conclude that, without exploiting the direction in-
formation of the lines, LOI-3 is very sensitive to the image noise
as well as to the initial pose. Fig. 5(b) plots the number of it-
erations as the function of the number of object lines. With the
increase of the line number, the number of iterations needed de-
creases. Since in LOI-2 there are two updates of rotation, the
computation time taken by 1 iteration in LOI-2 is about double
of that in LOI-1 and LOI-3. This can be seen from Fig. 5(c),
which gives the computation times. The LOI-1 and LOI-2 use
almost the same running times. LOI-3 is faster with the increas-
ing of the number of lines. We compare our methods with the
iterative weak perspective (IWP) method (Christy and Horaud,
1999), which estimates a pose with a weak perspective camera
model and improves the estimation iteratively by solving an ap-
proximate system of linear equations. Our orthogonal iteration
methods are very efficient and comparable to the IWP method.
4.2 Real Data
We also validated our pose estimation approach for line corre-
spondences, by using the algorithm for 3D line object tracking.
