2012 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'12, Honolulu, Hawaii, USA. March 4-6, 2012

A Robust Pose Estimation Method for Nearly Coplanar Points
Haiwei Yang1,a, Fei Wang1,b, Jingmin Xin1,c, Xuetao Zhang1,d, Yoshifumi Nishio2,e1Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University, China2

Department of Electrical and Electronic Engineering, Tokushima University, Japanayanghw.2005@stu.xjtu.edu.cn, bwfx@mail.xjtu.edu.cn, cjxin@mail.xjtu.edu.cn, dxuetaozh@gmail.com,
enishio@ee.tokushima-u.ac.jp

Abstract The pose of a calibrated camera cansolutions. However, as the noise increases, allbe estimated from known correspondences withthe two possible solutions will be very close toknown scene structure. There is a number of

each other, therefore, the rate of wrong pose willpose estimation algorithms which solve thisincrease regardless of the selected strategy.

problem, however, all of these algorithms suffer

In this paper, we present a thoroughfrom pose ambiguity. This paper analyses theanalysis of the reason of pose ambiguity. We

reason of pose ambiguity from the aspect of thepoint out that pose ambiguity will also exist for

distribution of object points. Then, we develop anearly coplanar points. In this situation, thenovel robust method to solve pose ambiguity fortraditional pose estimation algorithms will result

nearly coplanar points, through analyzing theto large re-projection error and the coplanar

influence of coplanar points and non-coplanar

algorithms cannot be adopted directly. Then wepoints comprehensively. Simulation experiments

show that pose ambiguity has a compact

and the real image results have demonstrated therelationship with the distribution of object points,

effectiveness of our new algorithm.

not only for coplanar points but also for 3D

Keywords Pose Estimation, Pose Ambiguity,

object point configurations, in Section 2. Section

Coplanar Points

3 proposes a robust novel algorithm to calculate
the pose. Section 4 presents experimental results
thorough testing on both synthetic and real-data

1. Introduction
The aim of pose estimation is to determine
the position and orientation between the camera

andcomparison

withstate-of-the-art

pose

algorithms. Conclusions are drawn in Section 5.

and the object. It has many applications incomputer vision, such as hand-eye robot systems,

2. Pose Ambiguity Description

augmented reality, photogrammetry and so on.

In general, for algorithms which use

In the literatures of pose estimation, several

objective function optimization, there are twoapproaches have also been proposed. Most ofkinds of objective function because of different

them work for 3D points configurations, andprojection methods. One is image space error

some have been extended to coplanar points

function:
NR2  pi  Ty 2R  p TEis ( R, T )  [( xi  1 i x )2  ( yi )]
R3  pi  Tz
R3  pi  Tzi 1

configurations. It is a degenerate configuration
when the object points are coplanar and poseambiguity will arise. Oberkampf et al. have also

And another is object space error function:
2Nv vTEos ( R, T )   ( I  Vi )( Rpi  T ) , Vi  iT ivi vii 1

discussed these pose ambiguities. They give astraightforward interpretation and develop theiralgorithm for planar points. Schweghofer and(1)(2)

Pinz analyse the general case of perspective

Pose ambiguity problem denotes situations
where Eis or Eos have several local minima for aprojection. They choose the correct pose withgiven configuration. We now illustrate pose

minimum re-projection error from two possible

ambiguity for a comprehensive example. For a- 345 - xi P li  yi   RPi  T  R T  i 1 1 

special object scene, which has five points, wechange the distance ||t|| between camera and theobject ceaselessly and keep all the other(3)parameter fixed, and then we observe thevariation trend of Eos around some axis, as

where Pi  X i Yi Zi T and vi  li xi yi 1T are thedepicted in Fig.1.

corresponding object point coordinates andnormalized image coordinates. We can write

5||t||=2||t||=5||t||=10||t||=15||t||=20

4.5

Error function Eos (m2)

43.5

projection function in a matrix form,~~V  [ R T ]P  AP(4)3where V  [v1 , v2 ,, vN ] , P~  [ P~1 , P~2 , P~N ]

2.5
21.5

and

A=[R,T] is the pose to be determined.

1From the equation (4), we can find that the0.5

stability of pose A has a compact relationship

0-100

-80

-60

-40
-20
020
40
Rotation Angle about Y-axis (degree)

6080

100

the theory of linear equations, the more

Fig.1. Error function for varying distances ||t||.

As we know, when all the object points are
coplanar, there will be two possible poses at
most, and this situation is called coplanar poseambiguity. In Fig.1 we can see that, with theincrease of ||t||, the second minimum becomesincreasing significance. Hence, for the specificconfiguration of object points, pose ambiguity
will come up, especially when the image orobject noise level is heavy.

thorough analysis of pose ambiguities. Then weshow that pose ambiguities have a compact
relationship with the distribution of object points,
not only for coplanar points but also forthree-dimensional points. Finally, we propose arobust novel algorithm to calculate the positionand orientation between the object and thethenewalgorithm,

poseparameter can be uniquely determined and thepose ambiguity can be avoided.
We adopt perspective projection camera
model to interpreter the pose ambiguity, with theprojection equation as follows:In other words, the robustness of pose estimation
has a direct relationship with the configurationof object points. When the object points havedifferent distribution, the stability of solution
will also have some difference, and this is theroot cause of pose ambiguity.
According to the above interpretation, weproposed a novel robust method to solve pose
analyzing the influence of coplanar points andIn this section, we firstly present aUsing

condition number, the less stability of equations.

ambiguities for nearly coplanar points, through3. Interpretation and Robust Algorithm

camera.

with the condition number of P~ . According to

non-coplanar points comprehensively. Firstly,
the singular value decomposition in the object
point set is done, and then a part of the object
points, corresponding to the largest singularvalue and the second largest singular value, isselected as the main plane points. Secondly,using the scale orthogonal projection model, thepose can be estimated. As the presence of poseambiguity, there are, in general, two local
minima in most configurations of object points.
Finally, a verification step will be adopted, using
the remaining set of object points. The
re-projection error is calculated according tothese two local minima, in image space andobject space respectively. Then the unique pose,according to the minimum re-projection error,- 346 -can be chosen.

coplanar, the result is extremely unstable.
Though analyzing the condition number of thematrix which is constructed by object points, we

4. Results
We compare the effect of our approachfind that it has a order of magnitude about 10^4,

against that of state-of-the-art ones, both on

thus the correct rate is extremely low. Fiore +simulated and real image data.

LM algorithm is Fiore algorithm followed with

4.1 Synthetic Experiments

LM optimization, In Fig.2 we find that the rate

For all synthetic experiments, we use thehas a greatly improvement. The slightly lessfollowing setup:robust results (about 10 percent) can be1) There are 9 object points, where 8 points are

explained: in a larger range, the result can becoplanar. The distance ||t||=10.

optimized to the correct pose, however, in some2) For each test, we generate there random angle

range it will reach to the other pose, which willof rotation R, the interval is [-180,180].

lead to pose ambiguity. The rate of our new3)

Thecorresponding

imagepoints

arealgorithm is always around 100 percent, thesecalculated though rotation R and translation T,

results demonstrate the effectiveness of our newand Gaussian noise are added.

algorithm, even if the object points have nearly

For each noise level, we run 1000
independent simulations for each algorithm,

coplanar configuration.

4.2. Real Image

such as Lu's algorithm, Fiore's algorithm and ournovel algorithm. Fig.2 shows the rate of correct

We select a manual model to validate actual
results of each algorithm, as depicted in Fig.3.

pose.
1.5

rate of correct pose (*100%)

Our algorithm
Fiore+LMLu.
Fiore
1

0.5

00

0.5

1

1.5

2
2.5
3
noise level (pixel)

3.5

4

4.5

5

Fig.2. Rate of correct pose for different algorithms.

In Fig.2, we can see that for different noiselevel, the Lu’s algorithm has a correct rate about
50 percent, it is because the Lu’s algorithm getsthe pose through the minimums of the object
space error function Eos . According to theabove analysis, when object points have a nearly
coplanar configuration, there are two minimumsof Eos . Therefore, for the case of nearly
coplanar configuration, Lu’s algorithm chooses
the correct pose for two solutions, and the rate offinding the correct pose is just about 50 percent.
The Fiore algorithm is a linear pose estimation
method, when the object points are nearly- 347 -Fig.3. Experiment results: top: our algorithm; middle: Lu'salgorithm; bottom: Fiore's algorithm.

We select six points from them to estimatepose using the algorithm of Lu, Fiore and ourRefernce

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