ANGLES ESTIMATION OF ROTATING CAMERA

Samira Ait Kaci Azzou

Computer Science Department, Setif University, Setif, Algeria

Slimane Larabi, Chabane Djeraba

LIFL Laboratory, Lille1 University, Lille, France

Keywords:

Rotation motion, Matching, Interest points, Camera pose.

Abstract:

We address the problem of camera motion from points and line correspondences across multiple views. We

investigate ﬁrstly the mathematical mathematical formula between slopes of lines in the different images

acquired after rotation motion of camera.

Assuming that lines in successive images are tracked, this relation is used for estimating rotation angles of the

camera.

Experiments are conducted over real images and the obtained results are presented and discussed.

1 INTRODUCTION

There are many applications of computer vision and

pattern recognition where the camera orientation is

controlled by a computer. For example, in order to

track and center the object of attentive focus, the cam-

era is rotating once the object is near from the image

border. The amount of rotation depends on the veloc-

ity of the object and its depth.

When a camera is rotated by a certain angle rela-

tive to a stationary scene, different projected images

are seen on the image plane. Consequently, the ex-

tracted low-level features (interest points, segments

of contour, etc) on images change their attributes so

as position, intensity, etc).

From these low-level features, if a set of invariants

are computed in the sense that their new values are

completely determined by the original values and the

amount of the camera rotation, we can then predict the

values of these invariants which would be obtained if

the camera were rotated by a given amount.

Conversely, if we are given two views of the same

scene obtained from different camera orientations,

and if we know the point-to-point correspondence, we

can reconstruct the amount of camera rotation which

would transform the values of these invariants to pre-

scribed values.

When the camera is ﬁxed, the analysis of different

views permits to understand the structure and the mo-

tion of moving objects. We can consider this case as

equivalent to camera rotation.

Camera motion estimation is important for var-

ious computer vision applications such as: 3D re-

construction, objects tracking and so on. Vari-

ous methods were developed and can be classiﬁed

as optical ﬂow methods and direct methods, which

are global, and features correspondences-based ap-

proaches, which are local. From the interesting

methods we cite (R. Ewerth and Freisleben, 2004),

(C. Jonchery and Koepﬂer, 2008), (A. Yamada and

Miura, 2002), (A. Biswas and Venkatesh, 2006)

for optical ﬂow methods well-adapted to small mo-

tions and (B. Rousso and Pelegz, 1996), (Bartoli

and Sturm, 2003), (Urfalioglu, 2004) for feature

correspondence-based methods that are well-adapted

to high motions of the camera giving well separated

views of the scene. In this paper we address the prob-

lem of camera motion from lines and points corre-

spondences across multiple views. We investigate

ﬁrstly the mathematical formula between slopes of

lines in the different images acquired after rotation of

the camera. Assuming that lines in successive images

are tracked, computed relation is used for estimating

rotation angles of camera. Our contribution in this

works is the extraction from low-level features invari-

ants that permits the estimating of motion rotation of

575

Azzou S., Larabi S. and Djeraba C. (2009).

ANGLES ESTIMATION OF ROTATING CAMERA.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 575-578

DOI: 10.5220/0001753805750578

 SciTePress

the camera knowing the correspondence of the fea-

tures. Experiments are conducted over real images

and the obtained results are presented and discussed.

2 POSITION OF THE PROBLEM

In this work, we assume that the camera performs hor-

izontal rotational movement. The proposed geometri-

cal model is illustrated by ﬁgure 1. After each rotation

of the camera, a new image is acquired and noted IM

where i refers to the position number of the camera.

We assume that in each image some of line segments

or interest points deﬁning segment lines are extracted.

The geometrical model of image formation is deﬁned

by (see ﬁgure 1):

- O

is the impact point of the image IM

deﬁned as the

intersection of the optical axis with the image plane

- L is the center of projection of the camera, O

L rep-

resent the focal length f

−−−−→

is the internal referential associated to IM

- The theoretical external referential

−−−−−→

(OXY Z) is de-

ﬁned so as the origin O is the rotation center of the

camera,

−→

OX is parallel to

−−→

−→

OY is parallel to the

optical axis, the axis of rotation

−→

OZ is parallel to

−−→

V .

- The camera is ﬁxed so as the rotational axis passes

through the optical axis. Due to the uncertainty of the

mechanics, the

−→

OZ axis is somewhere not far from the

impact point whose coordinates relatively to

−−−−−→

(OXY Z)

are (d

, d

- ex, ez deﬁne the dimensions of the pixel

We suppose that none of the deﬁned parameters is

known. Our aim is to develop a mathematical relation

that permit to compute the amount of the angle rota-

tion of the camera. This relation must be independent

of the camera model and will use only the coordinates

of image points in the different views.

We can see in ﬁgure 1 that when the camera is rotating

around the origin O, the projection center L and image

plane IM

are rotating also with the same angle.

3 ESTIMATING OF THE

ROTATION OF THE CAMERA

3.1 Basic Principle

In general case where the camera is rotating by

an angle α, any point M

, y

, z

) of the 3D space

is projected into m

where its projective coordi-

nates on the image plane IM

are (Duda and Hart,

1988)(O. Faugeras and Papadopoulo, 2000):

Figure 1: Geometrical models for image formation and

camera rotation.

u(m

) = f .ex.

−x

.cos(α) + y

. sin(α)

. sin(α)+ y

. cos(α)+ D

(1)

v(m

) = f .ez.

(−z

)

. sin(α)+ y

. cos(α)+ D

(2)

where: x

= x

− d

, y

= y

− d

, z

= z

− d

and

D = d

− f .

As the referential

−−−−−→

(OXY Z) is attached to the initial

position of the camera, the angle α may be considered

as equal to zero. The equations 1 and 2 will serve us

for the writing of the new coordinates of points after

two rotations of the camera. Let S

be a line segment

in the 3D scene and let S

be the image of S

on the

image IM

whose equation relatively to (O

) is

v = a

i,1

.u + b

i,1

The coordinates of m

image on IM

of any point

, y

, z

) of the segment S

are:

u(m

) =

f .ex.X

i,1

, v(m

) =

f .ez.(−Z

i,1

)

i,1

where:

i,1

= −x

. cos(α) + y

. sin(α)

i,1

= x

. sin(α) + y

. cos(α)

i,1

= z

As v(m

) = a

i,1

.u(m

) + b

i,1

, we can write:

i,1

= a

i,1

−

i,1

f .ez

i,1

+ D) (3)

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

576

After a second rotation of the camera with an an-

gle β, the segment S

will be projected on IM

as S

Let v = a

i,2

.u + b

i,2

be the equation of S

Following the same steps described above, we ob-

tain:

i,2

= a

i,2

−

i,2

f .ez

i,2

+ D) (4)

where:

i,2

= −x

. cos(α + β) + y

. sin(α + β)

i,2

= x

. sin(α + β) + y

. cos(α + β)

i,2

= Z

i,1

A set of transformations will give us:

v(m

) = a.u(m

) + b (5)

where: a = (a

i,1

. cos(β)+

i,1

. sin(β)

f .ex

) and b = C

i,2

Knowing that the equation (5) is valid for any

point m

of the segment S

whose equation is v(m

) =

u(m

).a

i,2

+ b

i,2

, we obtain:

i,2

− a

i,1

. cos(β)

sin(β)

i,1

f .ex

(6)

The equation 6 is also valid for any segment S

The use of two equations written for S

and S

gives

us:

i,2

− a

i,1

. cos(β)

j,2

− a

j,1

. cos(β)

i,1

j,1

(7)

For another rotation movement of the camera with

angle θ, a new equation is obtained for segments S

and S

where v = a

i,3

.u + b

i,3

and v = a

j,3

.u + b

j,3

are

the equations of S

and S

on the image IM

i,3

− a

i,1

. cos(θ)

j,3

− a

j,1

. cos(θ)

i,1

j,1

(8)

From 7 and 8, we obtain:

. cos(β) + k

. cos(θ) + k

= 0 (9)

where:

= a

i,3

j,1

− a

i,1

j,3

= a

i,1

j,2

− a

i,2

j,1

= a

i,2

j,3

− a

i,3

j,2

The use of a third segment S

with the segment S

gives us the relation:

. cos(β) + k

. cos(θ) + k

= 0 (10)

where:

= a

i,3

k,1

− a

i,1

k,3

= a

i,1

k,2

− a

i,2

k,1

= a

i,2

k,3

− a

i,3

k,2

The linear resolution of the equations 9 and 10

gives us the values of the angles β and θ.

3.2 Algorithm

The following algorithm gives the steps to be per-

formed in order to compute the angles of rotation of

the camera knowing the correspondence between the

set of lines extracted in the three images. The theoret-

ical study presented above allows estimating the two

rotation angles of camera using only images of three

lines.

In order to increase the accuracy in the computation

of the values of (β, θ), we will use all combinations

of all triplets of lines in the three images.

Let: IM

= {S

, S

, . . . , S

}, IM

, S

, . . . , S

}, IM

= {S

, S

, . . . , S

} be the

set of located straight lines respectively in the ﬁrst,

second and third image. We assume that each triplet

, S

) deﬁnes three matched segment lines in

the three images.

The steps of the Algorithm consist to select

((S

, S

), (S

, S

), (S

, S

)) from

(IM

× IM

)

and to compute the corre-

spondent slopes. For each one of these triplets, we

compute the values of β and θ by resolving the linear

equations 9 and 10

This step is repeated until all triplets should be

selected. At the end, the average values of (β, θ) is

computed.

The number of possible triplets of (S

, S

) in

(IM

×IM

) is equal to C

×n×(n−1)×

(n − 2), and it is identical to the number of triplets

((S

, S

), (S

, S

), (S

, S

)). To reduce the

complexity of this algorithm, we will use a restricted

number corresponding to triplets of segments giving

better results.

4 EXPERIMENTAL RESULTS

In the ﬁrst we generated randomly (x, y, z) coordi-

nates of six points, and we computed their projec-

tions on the three images corresponding to three posi-

tions of the camera (initial, ﬁrst and second rotation).

Nine (09) values of rotation angles were used for the

new positions of the camera (10

◦

, 15

◦

, 20

◦

, . . . , 60

◦

Knowing the points correspondence, the application

of the algorithm 3.2 computes the values of (β, θ) for

each group of six generated 3D points. We repeated

this process (100) times and the computed values of

(β, θ) are grouped relatively of the orientation of seg-

ment lines in the image. We distinguish seven cat-

egories of absolute slopes (C

, i = 1..7) representing

the line segments whose absolute slopes are respec-

tively in the intervals: C

= [0, 0.1], C

=]0.1, 0.5],

ANGLES ESTIMATION OF ROTATING CAMERA

577

=]0.5, 1], C

=]1, 5], C

=]5, 20], C

=]20, 200],

=]200, ∞[. From the obtained results, we can con-

clude that the high value of rotation angles are better

estimated than the low values, the better estimation

are obtained respectively by the line segments of the

categories C

, C

and C

. However, It is necessary

to avoid the line segments of the ﬁrst and seventh cat-

egories.

We studied also the inﬂuence of the noise on the

uncertainty estimation of rotation angles. The great

noise decrease the accuracy in the estimation of ro-

tation angles. However, we can conclude that the

slopes of categories C

, C

are more robust to noise.

A set of images of interior 3D scene are taken by the

camera after two rotations. The extraction of inter-

est points is done using Harris detector (Harris and

Stephens, 1988). Some of these interest points are

chosen to deﬁne three line segments (S

, S

). We

used many combinations of interest points in order to

deﬁne the three segments. We applied our algorithm

for these images. Many combinations of the six inter-

est points are used but eliminating the combinations

for which the slopes are near from zero (category C

)

or having high values (category C

). We selected only

the combination of interest points deﬁning segment

lines of categories C

, C

and C

. The average

of calculated values of β and θ by this algorithm are

considered as the estimated values. In our case, the

error in estimated values from the three images are

(1.59

◦

, 0.93

◦

5 CONCLUSIONS

In this paper we addressed the problem of camera

motion from lines and points correspondences across

multiple views. We investigated in the ﬁrst the mathe-

matical formula between slopes of lines in the various

images acquired after the movement of rotation of the

camera.

Assuming that lines in successive images are tracked,

we used the found relation for estimating rotation an-

gles of camera.

The advantage of the proposed method is that

does not require any knowledge about the geometrical

models of the camera; they use only the slope of line

segment as 2D primitive.

The obtained results from experiment conducted over

synthetic and real images are promising and will en-

courage us for their use in different applications so as

head pose estimation where the interest points of the

head are moving around the ﬁxed camera.

REFERENCES

A. Biswas, P. Guha, A. M. and Venkatesh, K. (2006). In-

trusion detection and tracking with pan-tilt cameras.

In Proceedings of the Third IET International Confer-

ence on Visual Information Engineering, pp. 565-571,

Bangalore (India).

A. Yamada, Y. S. and Miura, J. (2002). Tracking players and

a ball in video image sequence and estimating camera

parameters for 3d interpretation of soccer games. In

IEEE, ICPR’02.

B. Rousso, S. Avidan, A. S. and Pelegz, S. (1996). Ro-

bust recovery of camera rotation from three frames.

In IEEE Computer Society Conference on Computer

Vision and Pattern Recognition.

Bartoli, A. and Sturm, P. (2003). Multiple-view struc-

ture and motion from line correspondences. In Ninth

IEEE International Conference on Computer Vision

(ICCV’03).

C. Jonchery, F. D. and Koepﬂer, G. (2008). Camera motion

estimation through planar deformation determination.

In Journal of Mathematical Imaging Vision, Vol.32,

pp.7387.

Duda, R. and Hart, P. (1988). Pattern Classiﬁcation and

Scene Analysis. Wiley New York, USA, 1nd edition.

Harris, C. and Stephens, M. (1988). A combined corner

and edge detector. In In Alvey Vision Conference, pp.

147-152.

O. Faugeras, Q. L. and Papadopoulo, T. (2000). The Ge-

ometry of Multiple Images. MIT Press, Cumberland,

USA, 1nd edition.

R. Ewerth, M. Schwalb, P. T. and Freisleben, B. (2004). Es-

timation of arbitrary camera motion in mpeg videos.

In Proceedings of the 17th International Conference

on Pattern Recognition (ICPR04).

Urfalioglu, O. (2004). Robust estimation of camera rota-

tion, translation and focal length at high outlier rates.

In Proceedings of the First Canadian Conference on

Computer and Robot Vision.

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

578