Uncalibrated Visual Odometry for Ground Plane

Motion Without Auto-calibration

Vincenzo Caglioti

and Simone Gasparini

Politecnico di Milano - Dipartimento di Elettronica e Informazione

Piazza Leonardo da Vinci, 32 - I-20133 Milano (MI), Italy

Abstract. In this paper we present a technique for visual odometry on the ground

plane, based on a single, uncalibrated ﬁxed camera mounted on a mobile robot.

The odometric estimate is based on the observation of features (e.g.,salient points)

on the ﬂoor by means of the camera mounted on the mobile robot.

The presented odometric technique produces an estimate of the transformation

between the ground plane prior to a displacement and the ground plane after

the displacement. In addition, the technique estimates the homographic transfor-

mation between ground plane and image plane: this allows to determine the 2D

structure of the observed features on the ground. A method to estimate both trans-

formations from the extracted points of two images is presented.

Preliminary experimental activities show the effectiveness and the accuracy of the

proposed method which is able to handle both relatively large and small rotational

displacements.

1 Introduction

Robot localization is a fundamental process in mobile robotics application. One way

to determine the displacements and measure the movement of a mobile robot is dead

reckoning systems. However these systems are not reliable since they provide noisy

measurements, due to the slippage of the wheel. Localization methods based only on

dead reckoning have been proved to diverge after few steps [1]. Visual odometry, i.e.

methods based on visual estimation of the motion through images capture by one or

more cameras, is exploited to obtain more reliable estimates. Cameras are mounted on

the robot and the images are processed in order to recover the structure of the sur-

rounding environment and estimate the motion among images captured from different

viewpoints.

Usually, 3D reconstruction from images taken by a moving uncalibrated camera go

through auto-calibration. Autocalibration from planar scenes requires either nonplanar

motion [2], or several planar motions with different attitudes of the camera wrt the

ground plane [3].

In a mobile robotics framework, however, changing camera attitude requires addi-

tional devices as, e.g., pan-tilt heads, not directly connected to the robot functionality.

In particular, mounting a ﬁxed monocular camera on a mobile robot does not allow to

change the camera attitude wrt to the ground plane, making auto-calibration impossible

Caglioti V. and Gasparini S. (2007).

Uncalibrated Visual Odometry for Ground Plane Motion Without Auto-calibration.

In Robot Vision, pages 107-116

DOI: 10.5220/0002067301070116

 SciTePress

without additional information. A similar scenario is that of a ﬁxed camera mounted on

a moving vehicle (such as, e.g., a road car).

However, in this paper we will present a technique for visual odometry on the

ground plane, based on a single, uncalibrated ﬁxed camera mounted on a mobile robot.

The mobile robot is supposed to move on a planar ﬂoor, called ground plane. No map of

the environment is needed. The odometric estimate is based on the observation of fea-

tures (e.g., salient points) on the ﬂoor by means of the camera mounted on the mobile

robot.

The presented odometric technique produces an estimate of the transformation be-

tween the ground plane prior to a displacement and the ground plane after the displace-

ment. In addition, the technique estimates the homographic transformation between

ground plane and image plane: this allows to determine the 2D structure of the observed

features on the ground, as a side effect. The presented technique does not determine the

camera calibration parameters.

However, it is argued that any further step towards auto-calibration is not needed in

the context of mobile robot odometry. In fact, auto-calibration would only allow to de-

termine the spatial transformation between ground plane and camera: auto-calibration

alone does not allow to determine the robot-to-camera transformation. Therefore, de-

termining the transformation between the robot and the ground plane should require

further extrinsic calibration steps: these steps could consist in e.g., acquiring visual data

while the robot is executing self-referred displacements (such as, a self-rotation and a

forward translation).

On the other hand, the presented technique for visual odometry estimates the trans-

formation between ground prior to a displacement and ground after the displacement.

If needed, the robot-to-ground calibration can be accomplished by the same additional

step, namely visual observation of self-referred robot displacements, required when

starting with auto-calibration.

The technique works for generic planar displacements, but it does not work for

translational displacements. However, once the homography between ground plane and

image plane has been determined as a side effect, further displacements, including pure

translations, can be analyzed directly by using the (inverse) homography.

1.1 Related Works

In the last years methods to estimate the robot motion (ego-motion) based on visual

information provided by cameras have gained attention and some approaches have been

presented. Early methods were based on estimation of the optical ﬂow from image

sequence in order to retrieve ego-motion. McCarthy and Barnes [4] presented a review

and a comparison between the most promising methods.

Other approaches exploited stereo vision. Nister et al. [5] proposed a method based

on triangulation between stereo pairs and feature tracking in time-sequence of stereo

pairs, without any prior knowledge or assumption about the motion and the environ-

ment. Takaoka et al. [6] developed a visual odometry system for a humanoid robot

based on feature tracking and depth estimate using stereo pairs. Agrawal and Konolige

[7] proposed an integrated, real-time system involving both stereo estimate in the dis-

parity space and a GPS sensor in a Kalman ﬁlter framework. GPS-based systems can be

108

sufﬁciently accurate for large areas but it can be not used in indoor environments and

require a support framework, which prevent their use, e.g., for planetary exploration.

For such application, Mars Exploration Rover [8] employed a features detection in a

stereo image pair that are tracked from one frame to the next; using maximum like-

lihood estimation the change in position and attitude for two or more pairs of stereo

images is determined.

Davison [9] proposed a real-time framework for ego-motion estimation for a single

camera moving through general unknown environments. The method was based on a

Bayesian framework that detect and track a set of features (usually corner, points or

lines). Assuming the rigidity in the scene, the feature image motion allows to estimate

the motion of the camera; therefore the complete camera trajectory and a 3D map of all

the observed features can be recovered.

Visual odometry system based on catadioptric cameras have been proposed. Bun-

schoten and Krose [10] used a central catadioptric camera to estimated the relative pose

relationship from corresponding points in two panoramic images via the epipolar geom-

etry; the scale of the movement is subsequently estimated via the homography relating

planar perspective images of the ground plane. Corke et al. [11] developed a visual

odometry system for planetary rover based on a catadioptric camera; they proposed a

method based on robust optical ﬂow estimate from salient visual features tracked be-

tween pairs of images, by which they retrieve the displacements of the robot.

Our approach is similar in spirit to the work of Wang et al. [12] who measured

translation and rotation by detecting and tracking features in image sequences; assum-

ing that the robot is moving on a plane, they computed the homography between the

image sequences by which the computed the motion. Similarly, Benhimane and Malis

[13] developed a visual servoing framework based on the estimation of the homogra-

phy between image sequences to retrieve robot motion and close the control loop. Both

methods require camera calibration. Our work differ from these approaches in that we

do not assume camera calibration.

The paper is structured as it follows. Section 2 introduces and describes the ad-

dressed problem. Section 3 shows how the robot displacement can be retrieved by ﬁt-

ting the homography between two images of the ground plane. Section 4 illustrates the

method to estimate the transformation between the ground plane and the image plane.

Section 5 reports and discussed some preliminary experimental activities performed

with a rotating camera. Section 6 concludes the paper.

2 Problem Formulation

A mobile robot moves on the ﬂoor. A ﬁxed, uncalibrated camera is mounted on the

mobile robot: this camera is supposed to be a perspective camera (i.e., distortion is

neglected). The pose of the camera relative to the robot is unknown. The environment

map is unknown, as well as the structure of the observable features (associated to ﬂoor

texture) on the ground. This allows extremely easy set-up: it is sufﬁcient to mount a

perspective camera on the mobile robot in a ﬁxed but unknown position.

For the rigid body consisting of the robot plus the camera, a “ground” reference

frame is deﬁned as follows: the backprojection O of a certain image pixel (say, the pixel

109

′

with cartesian coordinates (0, 0)) on the ground plane is taken as the origin of the

projected reference frame, while vector connecting the origin to the backprojection A

of a second image pixel (say, the pixel A

′

with cartesian coordinates (100, 0)) on the

ground is taken as the unit vector along the x-axis.

As usual within the Robotics and Vision communities, homogeneous coordinates

are used. Let T be the unknown 3× 3 matrix representing the projective transformation,

also called “homography”, between the ground plane and the image plane, as realized

by the uncalibrated camera. The coordinates on the ground plane are referred to the

above deﬁned ground reference frame of the robot+camera system. Therefore, the un-

known homography T does not change with robot motion.

As the robot moves on the ground plane, the robot+camera undergoes a planar mo-

tion consisting of a rotation of an unknown angle θ about an unknown vertical axis. Let

C be the point where this vertical axis crosses the horizontal ground plane. Let R be the

rotation matrix describing the planar displacement. The matrix R is a 3× 3 2D rotation

matrix in homogeneous coordinates, whose third column collects the homogeneous co-

ordinates of the center of rotation C relative to the robot+camera ground reference and

whose upper-left 2× 2 sub-matrix is orthogonal.

Two images are taken: the ﬁrst one is taken before the displacement, while the sec-

ond one is taken after the displacement. The addressed problem is the following: ﬁrst,

given the transformation between the ﬁrst and the second image, determine the center

of rotation, and the rotation angle of the observed displacement; second, determine the

transformation T between the ground plane and the image plane, and use the inverse

transformation T

−1

to measure further displacements. The inverse transformation T

−1

can also be used to determine the shape (i.e., the 2D structure) of the set of the observed

features on the ground.

An interesting problem, which is not addressed in this paper, is that of ﬁnding a

transformation between the ground robot+camera reference frame and a second refer-

ence frame, more signiﬁcant to the robot kinematics. This transformation can be esti-

mated by applying the presented odometric technique to self-referred robot displace-

ment, as e.g., a “self”-rotation and a “forward” translation.

3 Estimation of Robot Displacement

The transformation relating the two images of the ground plane is still a homography,

and it is represented by the matrix H = TRT

−1

, where T is the unknown homography

between ground plane and image plane. (In principle, camera distortion can be compen-

sated by imposing that the transformation between the two images is a homography.)

The homography H between the two images (before and after the displacement) can

be computed from a sufﬁcient number of pairs of corresponding features between the

two images [14].

The eigenvectors of the homography matrix H are given by C

′

= TC, I

′

= TI and

′

= TJ, where the rotation center C, and the circular points I and J are the invariants

under rotation R on the ground plane. In addition, the eigenvalues of H coincide with

the eigenvalues of R (modulo a scale factor).

110

The eigenvectorC

′

is associated to the real eigenvalue of H, while I

′

and J

′

are asso-

ciated to the complex eigenvalues of H. By the eigendecomposition of the homography

matrix H, the parameters of the planar displacement are determined.

In particular, the imageC

′

of the center of rotationC is determined as the eigenvector

correspondingto the real eigenvalue of H. The rotation angle θ is determined as the ratio

between imaginary part and real part of the complex eigenvalue, in fact the eigenvalue

corresponding to I

′

= TI is given by µe

±iθ

, where µ is a real scale factor.

If the displacement is a pure translation, then the images of all the points at the

inﬁnity are eigenvectorsof H. Therefore the translation direction can not be determined.

Therefore, displacements with small rotation angles may generate solutions, that are

numerically unstable.

4 Estimation of the Transformation Between the Ground Plane

and the Image Plane

The shape of the observed features is determined by estimating the transformation

matrix T. This matrix can be estimated by four pairs of corresponding points: these

can be, e.g., the two circular points I = [1, i, 0]

, J = [1, −i, 0]

with their image

projections I

′

, J

′

, plus the two points deﬁning the robot+camera ground reference,

namely O = [0, 0, 1]

and A = [1, 0, 1]

, with their image projections O

′

= [0, 0, 1]

and A

′

= [100, 0, 1]

The homogeneous (world) coordinates of O within the ground reference are [0, 0, 1]

while the homogeneous coordinates of A within the ground plane are [1, 0, 1]. With

these choices, the transformation matrix T between ground plane and image plane is

fully constrained, and it can be determined, imposing that











′

= TI

′

= TJ

′

= TO

′

= TA

Once the transformationmatrix T has been estimated, the shape of any conﬁguration

of observed features can be determined by their images (P

′

, i = 1..n) by P

= T

−1

′

. The

knowledge of T allows to determine the coordinates of the rotation center C = T

−1

′

relative to the (back-projected) robot reference. The estimated motion parameters con-

stitute an odometric estimate of the robot displacement.

Notice that the shape determination requires that the displacement is not purely

translational. However, once the transformation T has been determined by analyzing

a rotational displacement, it can be used also to measure purely translational displace-

ments.

5 Preliminary Experimental Results

In order to validate the proposed method we performed some experimental activities.

111

500 600 700 800 900 1000 1100

0.5

1.5

2.5

3.5

4.5

(a)

400 500 600 700 800 900 1000 1100 1200

(b)

Fig.1. The distribution of the matching score of the features used to estimate θ

5,6

(a) and θ

6,7

(b).

Table 1. The ﬁrst sequence of 7 images taken with relatively large rotational displacements. The

table reports the ground truth references (θ

ref

, in degrees) read on the turntable, the rotational

displacements between two consecutive images (θ

i,i+1

), the estimated rotational angles (

i,i+1

)

and the relevant error (e

i,i+1

). The value of θ

6,7

was obtained with a lower number of features

(N = 10) since many outliers were found.

Step θ

ref

i,i+1

1 107 9 9.37 -0.37

2 116 8.5 8.09 0.41

3 124.5 10.5 10.35 0.15

4 135 8 7.91 0.09

5 143 11 10.97 0.03

6 154 10.5 9.70 0.8

In our experimentations we use a standard perspective camera provided with a very

low distortion optics. The camera was placed on a turntable by which we manually

measured the ground truth rotation with an accuracy of about 0.5

◦

. The camera view-

point was placed in a generic position relative to the rotation axis: therefore the camera

underwent a general planar motion. The camera was pointed towards the ground ﬂoor,

such that the extraction of salient points exploits the ﬂoor texture. Then we took some

images with different rotational displacement and applied the proposed method in order

to estimate the rotation angle between two images.

We tested the method on two sequences of images. The ﬁrst sequence was obtained

considering larger rotational displacements, with a mean angle of about 10

◦

. The second

sequence is characterized by relatively small rotational displacements between images.

The mean rotational displacement of this set is about 5

◦

. As discussed in Section 3,

small rotation angles may lead to numerical instability. On the other hand if the robot

rotates slowly,images with small rotational displacementshave to be taken into account.

Table 1 collects the ground truth values, the estimated values and the relevant errors

for the ﬁrst sequence. For this sequence we employed the following estimation proce-

dure. Given two consecutive images, say I

and I

i+1

, we extracted a number of salient

112

Table 2. The second sequence of 25 images taken with relatively small rotational displacements.

For each step, the table reports the ground truth references (θ

ref

, in degrees) read on the turntable,

the relative rotational displacements among the three consecutive images (θ

i,i+1

and θ

i,i+2

), the

estimated rotational angles (

i,i+1

) and the relevant error (e

i,i+1

Step θ

ref

i,i+2

i,i+1

i+1,i+2

i,i+2

1 321 12 7 5 11.83 0.17

2 314 12 5 7 12.16 -0.16

3 309 8.5 7 1.5 7.83 0.67

4 302 10 1.5 8.5 10.81 -0.81

5 300.5 14 8.5 5.5 7.97 6.03

6 292 10 5.5 4.5 9.25 0.75

7 286.5 7.5 4.5 3 7.03 0.47

8 282 10 3 7 9.36 0.64

9 279 12 7 5 11.58 0.42

10 272 10.5 5 5.5 9.93 0.57

11 267 9.5 5.5 4 8.70 0.80

12 261.5 10.5 4 6.5 9.96 0.54

13 257.5 10.5 6.5 4 10.02 0.48

14 251 9 4 5 8.71 0.29

15 247 7 5 2 7.50 -0.50

16 242 8 2 6 7.01 0.99

17 240 10 6 4 9.03 0.97

18 234 10.5 4 6.5 10.78 -0.28

19 230 10 6.5 3.5 9.52 0.48

20 223.5 10 3.5 6.5 10.91 -0.91

21 220 10 6.5 3.5 10.72 -0.72

22 213.5 7.5 3.5 4 7.21 0.29

23 210 8.5 4 4.5 8.96 -0.46

points from each image using the Harris features extractor [15]. Then we found the

correspondences among these points using the normalized cross correlation and we se-

lected a set of points (usually, N = 20) having the best matching score [16]. We used

this set of points to ﬁt the homography H

i,i+1

using the RANSAC technique [17] pro-

vided by [18]. Once H

i,i+1

was computed, we estimated the rotation angles from the

complex eigenvalues of H

i,i+1

, as explained in Section 3.

As Table 1 shows, the estimates are very accurate and the errors are less than 1

◦

. The

value of θ

6,7

was obtained considering a lower number of salient point, N = 10. Because

of the large rotational displacement (about 10.5

◦

) the matching among features was in

most cases incorrect and the resulting matching score was (on the average) higher with

respect to the other images of the sequence. Figure 1 compares the distributions of the

matching score values of the ﬁrst 50 best matches for image pairs used to compute c

and θ

6,7

respectively: the estimation process of θ

5,6

can rely on many reliable match-

ing features (e.g. at least 20 matches have a matching score less than 800) while for the

estimation process of θ

6,7

there are only few matches under the same threshold. This in-

troduced many outliers that affected the estimate. Decreasing the number of considered

points allowed to discard many outliers, thus obtaining a more reliable estimate.

Table 2 collects the ground truth values, the estimated values and the relevant errors

for the second sequence. In order to overcome possible numerical instability issues we

used three images to robustly estimate the angle. We employed the following estimation

procedure. Given three consecutive images, say I

, I

i+1

and I

i+2

, we extracted a number

of salient points from each image, say c

, c

i+1

and c

i+2

respectively. We found the

113

(a) Image 1 (b) Image 2 (c) Image 3

Fig.2. An example of tracked features between three images. The ﬁrst two images (a,b) are

compared in order to ﬁnd the best matches (depicted in green and red, respectively). The best

features of (b) are matched with (c) in order to ﬁnd the best matches (depicted in blue in (c)).

Hence the chain of matches between the images are used to computed the homography H

i,i+2

(a) Image 5 (b) Image 6 (c) Image 7

Fig.3. The images (from 5 to 7) of the second sequence for which the estimated rotation angle

was incorrect. The found corresponding features among images are depicted: most of the matches

are false matches, which led to an incorrect estimation of the homography H

5,7

correspondences among the features c

and c

i+1

, selecting only those matches having

the best matching score, say c

′

and c

′

i+1

. Then we tracked these matches in I

i+2

matching c

′

i+1

with the features c

i+2

, selecting the best matches and obtaining c

′′

i+1

and

′′

i+2

(where c

′′

i+1

⊆ c

′

i+1

Exploiting c

′′

i+1

, we also obtained c

′′

, which are features of I

that have a matching

feature both in I

i+1

and I

i+2

. By ﬁtting a homography with c

′′

and c

′′

i+2

, we computed

the 3x3 matrix H

i,i+2

with the RANSAC technique, obtaining the relevant rotation angle

i,i+2

. Figure 2 shows an example of the tracked features among the three images.

The reported results proved the effectiveness and the accuracy of the proposed

method. The estimation errors are less than 1

◦

, except for images 5, 6 and 7. In this

case the errors is greater since the method was not able to ﬁnd a correct rotational

angle. This is due to the large displacements among images: the overall displacement

between image 5 and 7 is about 14

◦

, with partial displacements of 8.5

◦

and 5.5

◦

respec-

tively. Figure 3 shows the matching feature that are used to determine the rotational

displacements: there are many false matching that affected the estimate of θ

5,7

Large displacements may cause such errors since we used the normalized cross cor-

relation to ﬁnd the correspondences. Normalized cross correlation is not rotationally

invariant, hence large rotation can corrupt the matching process. Moreover, large rota-

114

tion angles between images reduce the overlapping region of the images, thus reducing

the number of corresponding features. In order to overcome these issues, rotationally

invariant matching function can be employed such as, e.g., SIFT features extractor [19].

On the other hand, in a real application a proper visual sampling rate during robot move-

ment would avoid large displacements between two poses.

6 Conclusions and Ongoing Activity

In this paper we presented a novel method to estimate the odometry of a mobile robot

through a single uncalibrated ﬁxed camera. Assuming that the robot is moving on a

planar ﬂoor, images of the ﬂoor texture is taken. Salient points are extracted from the

image and are used to estimate the transformation between the ground plane before

a displacement and the ground plane after the displacement. The proposed technique

also estimate the homography between the ground plane and the image plane, which

allows to determine the 2D structure of the observed features. An estimation method

of both transformations was described. Preliminary experimental activities that vali-

date the method for small and large rotational displacements are also presented and

discussed.

Ongoing works are aimed at improving the estimate method in order to provide

reliable estimate in presence of large rotational displacements. Other experimental ac-

tivities will be conducted in order to better stress the method in different situations. We

are also planning to implement a real time version of the proposed method on a real ap-

plication in order to use the odometric estimate for localization tasks in a mobile robots.

Other possible future research direction are the employment of catadioptric cameras in

order to exploit their large ﬁeld; however, using catadioptric cameras the transforma-

tions are not homography, unless central catadioptric cameras are used, which are, on

the other hand, difﬁcult to set up.

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