International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B1, 2012 
ıme XXXIX-B1, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
FEATURES Therefore each line correspondence provides two constraints. E- Similarly, for the point position vector RP; +t we have: 
quations (1) and (2) are the well known fundamental equations 
of the pose estimation problem from 2D to 3D line correspon- RP; +t =K;(RP; +t). G) 
d s (Navab and F Ss, 1993). ; 
us ) From the Eqs. (4) and (5), we have the corresponding vector 
I differences as: 
10logy = (I-K;)Rd;, (6) 
.China el = b eR (7) 
arity errors, we formulate 
o find the best rotation and 
attain very accurate results 
ccurate, but also robust to 
eal-time applications. The 
ation problem as minimiz- 
ty in object space, and it- 
n matrices in a global con- 
an accurate and efficient 
features. By introducing 
jective functions in terms 
id use different optimiza- 
and translation. We show 
ich fully exploits the line 
rate and robust results ef- 
utliers. 
)DEL 
epicted in Fig. 1. Let c — 
n with the origin fixed at 
ling with the optical axis 
I denotes the normalized 
coordinate system. L; is a 
'e projection on the image 
enter, the 2D image line 
plane, which is called the 
). In the object coordinate 
, where d; = (dj, d diy 
(xi. yi. zi)^ is an UE 
e 2D image line /; in the 
sed as: aix +bjy+c; — 0. 
T. which represents /; as 
the normal vector of the 
i can be expressed in the 
RP; +t, where the 3 x 3 
ector t describe the rigid 
inate system and the cam- 
>ctors are all in the inter- 
(1) 
. Q) 
  
Figure 1: The geometry of the camera model 
3 POSE FROM LINE CORRESPONDENCES 
We developed a new algorithm of pose estimation from line cor- 
respondences. Our algorithm is inspired by Lu et al.’s method (Lu 
et al., 2000) which addresses pose estimation problem from points. 
In comparison with other algorithms, the iterative method of (Lu 
et al., 2000) can attain very accurate results in a fast and globally 
convergent manner. It is regarded as one of the most accurate and 
efficient pose estimation methods (Moreno-Noguer et al., 2007). 
For pose estimation from line features, the existing linear algo- 
rithms (Ansar and Daniilidis, 2003, Liu et al., 1990) often lacks 
sufficient accuracy while the iterative algorithms (Phong et al., 
1995, Kumar, 1994), which generally uses classical optimization 
techniques such as Newton and Levenberg-Marquardt method, 
may not fully exploit the specific structure of the pose estimation 
problem (Lu et al., 2000), and are usually sensitive to the initial 
noise or outliers. Our new iterative algorithm is an extension of 
the algorithm (Lu et al., 2000) to lines, which is named as Line 
based Orthogonal Iteration algorithm (LOI). Experiments in Sec- 
t. 4 demonstrate that our method is very efficient and can attain 
very good performances in terms of both accuracy and robust- 
ness. 
3.4 Object-Space Objective Function 
The fundamental equations indicate that each 3D-to-2D line cor- 
respondence provides 2 constraints. If N-line correspondences 
are available, the pose problem becomes the problem of mini- 
mizing the following objective function (Phong et al., 1995, Lee 
and Haralick, 1996, Kumar, 1994): 
N N 
E(R.t) — Y (n Rd; 4 ) (n; (RP;4-t)) 3) 
i=1 = 
— 
Instead of using the objective function of Eq. (3), we present and 
use an objective metric based on the coplanarity error in the ob- 
ject space. In fact, just as the collinearity of point correspondence 
in the way of orthogonal projection (Liu et al., 1990), the copla- 
narity of a 3D-to-2D line correspondence can be understood in 
the way that the projection of 3D line in the interpretation plane 
should be coincide with its self. In the camera coordinate system, 
the line direction vector is Rd; and its projection in the interpre- 
tation plane can be obtained by (I— nin? JRd; where I is a 3 x 3 
identity matrix. Let K; = I — nn’, we have: 
Rd; = K;Rd;. 4) 
81 
We refer ed and eP as coplanarity errors. We then achieve the 
following objective functions: 
N 
R) = Y ied? — > Ia-K K;)Rd;|?, (8) 
i=1 
N 
O= - ZI0-K (RB;-0l^. © 
i=1 
Compared with Eq. (3), the two objective functions (8) and (9) 
have clear geometry meaning that the optimal solution of pose 
should achieve the minimum value of the sum of squares of copla- 
narity errors (See Fig. 2). 
Note that like the line-of-sight projection matrix V; defined in (Li- 
u et al., 1990), K; is an interpretation plane projection matrix that, 
when applied to a object vector, projects the vector orthogonally 
to the interpretation plane. It owns the following properties: 
IKill > |Kixll,  xeR*, (10) 
Kl =k, (11) 
K? =K;K/ =K;. (12) 
  
Figure 2: Object-Space error of line correspondence 
Since our new pose estimation method involves mainly the least- 
squares estimation, we show a lemma given by the work of (Umeya- 
ma, 1991) here before proceeding to the details of our method. 
Lemma 1: Let A and B be m x n matrices, and R a m x m rota- 
tion matrix, and UDV" a SVD of AB” (UUT = VV" =1, D = 
diag(À;), M > A2 > +--+ > Ay > 0). To minimize the objective 
function 
f(R)=||A-RB]?, (13) 
the optimal rotation matrix R such that 
R =USV/. (14) 
When rank(ABT ) > m — 1, S must be chosen as 
f if det(ABT) > 0, 
s- diag(1, ,1,—1) else. as 
When det (ABT ) — 0 (rank(ABT ) — m — 1), S must be chosen as 
ry if det(U)det(V)=1, 
sel dial. 1° 1-1) df del e(v)--r 9