IMAGE RECTIFICATION

Evaluation of Various Projections for Omnidirectional Vision Sensors using the

Pixel Density

Christian Scharfenberger, Georg Faerber and Florian Boehm

Institute for Realtime-Computersystems, Technische Universitaet Muenchen, Arcisstrasse 21, 80333 Munich, Germany

Keywords:

Omnidirectional vision, Image rectiﬁcation, Image quality, Calibration, Pixel density.

Abstract:

Omnidirectional vision sensors provide a large ﬁeld of view for numerous technical applications. But the

original images of these sensors are distorted, not simply interpretable and not easy to apply for normal image

processing routines. So image transformation of original into panoramic images is necessary using various

projections like cylindrical, spherical and conical projection, but which projection is best for a speciﬁc appli-

cation?

In this paper, we present a novel method to evaluate different projections regarding their applicability in a

speciﬁc application using a novel variable, the pixel density. The pixel density allows to determine the resolu-

tion of a panoramic image depending on the chosen projection. To achieve the pixel density, ﬁrst the camera

model is determined based on the gathered calibration data. Secondly, a projection matrix is calculated to map

each pixel of the original image into the chosen projection area for image transformation. The pixel density is

calculated based on this projection matrix in a ﬁnal step.

Theory is veriﬁed and discussed in experiments with simulated and real image data. We also demonstrate that

the common cylindrical projection is not always the best projection to rectify images from omnidirectional

vision sensors.

1 INTRODUCTION

Many technical tasks require perspective cameras to

provide useful information of the surrounding envi-

ronment. Examples of common application ﬁelds

are obstacle detection, people tracking and detec-

tion. Therefore, cameras are used for driver assis-

tance systems in cars and robotics. There are a cou-

ple of reasons to use perspective cameras for differ-

ent tasks. However, it is desirable in some applica-

tions of computational vision to use vision sensors

with a large ﬁeld of view. Teleconferencing, remote

surveillance and texture for model acquisition for vir-

tual reality beneﬁt from camera systems providing

a large ﬁeld of view. But normal perspective cam-

era systems are limited in their ﬁelds of view. To

overcome this limitation, researchers and practition-

ers developed camera systems containing either ro-

tating (Krishnan and Ahuja, 1996) or multiple cam-

eras (Utsumi et al., 1998). Other, quite effective ways

to enhance the ﬁeld of view are camera systems us-

ing wide angle lenses like ﬁsheye lenses or mirrors

in conjunction with lenses (Ishiguro, 1998; Daniilidis

and Geyer, 2000; Baker and Nayar, 1999). Omnidi-

rectional vision systems are catadioptric systems and

combine a curved mirror with a perspective camera to

obtain a large ﬁeld of view. In the past, several con-

cepts for optical characteristics such as single view-

point (Baker and Nayar, 1999), (Yamazawa et al.,

1993), equi-resolution (Gaspar et al., 2002) and equi-

areal (Hicks and Perline, 2002) have been developed

to improve omnidirectional camera systems.

1.1 Related Work

Original images of omnidirectional camera systems

are distorted, not simply interpretable and not easy ap-

plicable to normal image processing routines. Meth-

ods and procedures exist which process original im-

ages even of uncalibrated omnidirectional vision sen-

sors, but no useful correlation between real object

properties like size and width can be found in origi-

nal images. So there is a need for conventional image

processing routines requiring transformed panoramic

images as well as calibration information. Yamazawa

et al. (Yamazawa et al., 1993) propose methods to

82

Scharfenberger C., Faerber G. and Boehm F. (2009).

IMAGE RECTIFICATION - Evaluation of Various Projections for Omnidirectional Vision Sensors using the Pixel Density.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 82-89

DOI: 10.5220/0001791700820089

Copyright

c

SciTePress

transform warped images into cylindric panoramic

images using cylindric coordinates. In (Yamazawa

et al., 1998), images of omnidirectional vision sen-

sors (ODVS) are transformed into images as if taken

directly with an ordinary camera. These images are

suitable for further data analysis, so that conventional

image processing routines can be utilised. Gandhi and

Trivedi (Gandhi and Trivedi, 2004) use ODVS’s with

single viewpoints. Knowing the exact parameters of

the hyperbolic mirror and the camera, they present a

plane transformation method to transform an image

to a perspective view looking downwards. Geyer and

Daniilidis propose in (Geyer and Daniilidis, 2001) a

uniﬁying model for the projective geometry and stu-

died its properties as well as its practical application.

1.2 Motivation

As shown above, various projections are developed

to transform original images into unwarped, rectiﬁed

images for a better use of conventionalimage process-

ing routines. However, there has been no evaluation in

literature that studies and evaluates different projec-

tions types to obtain best utilisation of sensor pixels

in rectiﬁed images. In this paper, we propose a novel

value, the pixel density, which characterises the con-

sistence of rectiﬁed image pixels with sensor pixels

at a certain position. Using the characteristics of the

pixel density, the projection for a speciﬁc camera sys-

tem can be chosen for best utilisation of sensor pixels

in rectiﬁed images. But the pixel density does not

predict the distortion of rectiﬁed images for a chosen

projection. We assume, that the pixel density depends

on the chosen projection, the distance between mirror

and camera, the ﬁeld of view of the camera and on the

chosen mirror surface. So we conduct experiments on

synthesised and real images of ODVS’s with different

hyperbolic mirrors, different ﬁeld of views, different

distances between mirror and camera as well as var-

ious projections. This paper is organised as follows:

The camera model and the calibration of omnidirec-

tional vision sensors is introduced in section 2.1 and

in section 2.2. We present the rectiﬁcation methods

and the pixel density in section 3 and illustrate our re-

sults in section 4. The paper ends with a discussion

and conclusion in section 4.2 and section 5.

2 CAMERA CALIBRATION

2.1 Camera Model

We designed ODVS’s based on the well illustrated

SPOV theorem of Nayar and Baker (Baker and Na-

yar, 1999).

z−

c

2

2

− r

2

k

2

− 1

=

c

4

4

k− 2

k

(1)

z−

c

2

2

+ r

2

k

2

+ 1

=

2k+ c

2

4

(2)

Equation (1) for k ≤ 2 and equation (2) for k ≥

2 dimension ODVS’s with single points of view

(SPOV), whereas k speciﬁes the mirror surface and

c the distance between SPOV and pinhole of the per-

spective camera. Furthermore, equation (3) fulﬁls the

SPOV theorem for many mirror types like hyperbolic

mirrors.

1

a

h

2

z−

c

2

2

−

1

b

h

2

= 1 with

a

h

=

c

2

r

k− 2

k

and b

h

=

c

2

r

2

k

(3)

In this paper, we use equation (3) to design some

real and synthesised mirrors, but we use distances dif-

ferent than c for technical reasons as well as to iden-

tify the properties of the pixel density. Using hyper-

bolic mirros, the SPOV constraint is only valid for

an accurate alignment of mirror and camera. But it is

difﬁcult to achievean exact alignment between mirror

and camera, so that the camera system must be cali-

brated to compensate misalignments and to get a pre-

cise relation between world and camera coordinates.

2.2 Calibration

To determine the position of object points, the func-

tion f(~p) (4) which describes a relation between ~p in

world coordinates x

P

, y

P

and z

p

and the camera coor-

dinates u

P

and v

P

has to be found (see Figure 1(a)).

~

P =

u

P

v

P

= f(~p) with ~p = λ ·

x

p

y

p

z

p

, λ > 0 (4)

There exists many methods (Mei and Rives, 2007;

Colombo et al., 2007; Baker, S. and Nayar, S., 2001;

Micusik and Pajdla, 2004; Scaramuzza et al., 2006a)

proposing calibration to determine the function f(~p).

We use the calibration method developed by Scara-

muzza et al. (Scaramuzza et al., 2006b). All points

on a vector ~p in world coordinates (Figure 1(a)) are

mapped to the corresponding point P

′′

on the virtual

plane E

′′

(Figure 1(b)). Scaramuzza et al. propose the

IMAGE RECTIFICATION - Evaluation of Various Projections for Omnidirectional Vision Sensors using the Pixel Density

83

x

y

z

0

P

u’’

v’’

0

P’’

(a) World to virtual

sensor plane E

′′

0

u’’

v’’

P’’

(b) Point P

′′

on the virtual

sensor plane

0

u

v

P

(c) Real sensor plane

Figure 1: The camera model (a) used in this paper. The

world point P is mapped on a virtual sensor plane E

′′

(b)

and the projection transformed to the real sensor plane (c)

using afﬁne transformations.

relation between world and virtual sensor plane as

~p =

x

p

y

p

z

p

= λ·

u

P

′′

v

P

′′

f(ρ)

=

x

p

y

p

a

0

+ a

2

ρ

2

+ ... + a

N

ρ

N

(5)

with λ = 1 and ρ =

q

x

2

~p

+ y

2

~p

. Furthermore, they ap-

proximate the component z of f(~p) depending on the

curvature as a n-polynomial function. The relation

between the real sensor plane and the virtual sensor

plane is realised as an afﬁne transformation (see equa-

tion 6).

~

P

′′

= A·

~

P+

~

t with A =

c d

d 1

~

P =

u

P

v

P

and

~

t =

u

center

v

center

(6)

The parameters a

0

... a

n

, A and

~

t are the calibration

parameters.

3 IMAGE RECTIFICATION

First the projection area in world coordinates based on

individual projection parameters like width M, height

),,( ZYXf

v

u

=

ú

û

ù

ê

ë

é

Interpolation

UndistortedImage

Projection-

parameters

CalibrationData

Source

Cameramodel

0

u

v

0

n

m

Worldcoordinates

ofprojectionsurface

X, Y,Z

Calculationof

projectionsurface

Figure 2: This ﬁgure illustrates the rectiﬁcation of non

panoramic images. We calculate the projection area and

transform original images to unwarped image using the

gathered calibration data. A bicubic interpolation method

is used for interpolation.

X

Y

α

offset

α

α/2

d

P

α

P

(a) Top view

Z

Y

P

Z

top

Z

bottom

Z

P

(b) Side view

Figure 3: This ﬁgure shows the cylindrical projection pa-

rameters in top and side view.

N and region of interest (ROI) is determined. Each

pixel [m,n]

T

of the projection area is stored in a M x

N x 3 dimensional matrix F containing its world co-

ordinates X(m,n), Y(m,n) and Z(m,n). Furthermore,

the sensor coordinates of the projection area are then

calculated using the camera model as well as the cal-

ibration data and stored in a matrix (LUT). Based on

the calibrated camera model, images are rectiﬁed us-

ing the LUT and a bicubic interpolation method. Fig-

ure 2 illustrates this workﬂow. We also postulate the

view of the perspective camera in z - direction for all

projections.

3.1 Cylindrical Projection

A common projection is the cylindrical projection.

Parameters for the cylindrical projections are distance

to the projection area d, rotation width α, rotation off-

set α

of f set

, and the height of the cylinder given by Z

top

and Z

bottom

(see Figure 3(a) and 3(b)). Based on these

parameters, ﬁrst the cylindrical coordinates (α

P

,Z

P

)

T

at position [m, n]

T

in the target image with width M

and height N are calculated (see equation (7)).

~

P

F m,n

=

d

α

P

Z

P

=

d

α

of f set

−

α

2

+

α

M

· m

Z

top

−

Z

top

−Z

bottom

N

· n

(7)

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

84

X

Y

α

offset

α

α/2

P

α

P

d

bottom

d

P

d

top

(a) Top View

Y

Z

P

Z

top

Z

bottom

Z

P

(b) Side View

Figure 4: The parameters of conic projection are shown in

this ﬁgure and deﬁne the projecton area.

3.2 Conic Projection

The projection area of the conic projection relates to

a cone cut-out. Parameters for the conic projections

are distance to the projection area d

top

and d

bottom

,

rotation width α, rotation offset α

of f set

, and the height

of the conic given by Z

top

and Z

bottom

(see Figure 4(a)

and 4(b)). The cylindrical coordinates (α

P

,Z

P

)

T

of

a point P

F

at position [m, n]

T

in the target image are

then determined using equation (8)

~

P

F m,n

=

d

α

P

Z

P

with

d

α

P

Z

P

=

d

top

−

d

top

−d

bottom

N

· n

α

of f set

−

α

2

+

α

M

· m

Z

top

−

Z

top

−Z

bottom

N

· n

(8)

The carthesian coordinates (both for cylindrical and

conic projection) of every point are computed using a

standard coordinate transformation.

3.3 Spherical Projection

The projection area is a cut-out of a shperical surface.

Parameters for the spherical projections are the eleva-

tion width β, the elevation offset β

of f set

, the rotation

width α and rotation offset α

of f set

(see Figure 5(a)

and 5(b)). The spherical coordinates (α

P

,β

P

)

T

at po-

sition [m,n]

T

in the target image are calculated using

equation (9) (r = 1):

~

P

F m,n

=

r

α

P

β

P

=

r

α

of fset

−

α

2

+

α

M

· m

β

of fset

+

β

2

−

β

N

· n

(9)

The spherical coordinates are transformed into

carthesian coordinates using the rotation α

P

clock-

wise beginning at the y - axis and the elevation β

P

anti-clockwise beginning at the x axis.

X

Y

α

offset

α

α/2

P

α

P

(a) Top View

Y

Z

P

-β

P

β/2

β

-β

offset

(b) Side View

Figure 5: This ﬁgure shows the spherical projection param-

eters in top and side view.

3.4 Pixel Density

To evaluate different projections and to compare dif-

ferent conﬁgurations of omnidirectional vision sen-

sors, the position dependent resolution of unwarped

images has to be calculated. Therefore, we propose

a novel value, the pixel density σ. To determine the

pixel density, the distances between pixels in rectiﬁed

images on the image sensor are determined. Figure 6

demonstrates the basic idea of the pixel density.

0

u

v

521 522 523 524 525

353

354

355

356

357

m,n

m-1,n

m+1,n

m,n+1

m,n-1

d

h

d

v

Figure 6: This ﬁgure shows the position of pixels in rectiﬁed

images on the image sensor (left) and its distances (right)

on the sensor plane, which are used to calculate the pixel

density.

The horizontal pixel density σ

h(m,n)

at position

[m,n]

T

is calculated using equation (10)

σ

h (m,n)

=

1

2

· d

h (m,n)

with (10)

d

h (m,n)

=

q

(u

(m+1,n)

− u

(m−1,n)

)

2

+ (v

(m+1,n)

− v

(m−1,n)

)

2

Analogical, the vertical pixel density σ

v(m,n)

at posi-

tion [m,n]

T

is deﬁned as

σ

v (m,n)

=

1

2

· d

v (m,n)

with (11)

d

v (m,n)

=

q

(u

(m,n+1)

− u

(m,n−1)

)

2

+ (v

(m,n+1)

− v

(m,n−1)

)

2

The pixel density σ

g(m,n)

at position [m,n]

T

is deﬁned

as the geometric mean (see equation (12)) of the hor-

izontal and vertical pixel densities

σ

g (m,n)

=

p

σ

h (m,n)

· σ

v (m,n)

(12)

IMAGE RECTIFICATION - Evaluation of Various Projections for Omnidirectional Vision Sensors using the Pixel Density

85

The pixel density σ characterises the consistence

of rectiﬁed image pixels with sensor pixels, which are

used for intensity calculation. Consequently, the pixel

density contains the resolution of rectiﬁed panoramic

images depending on the chosen projection and cam-

era setup. We assume, that best resolution reaches

rectiﬁed images achieving a pixel density nearly 1. A

pixel density less than 1 denotes poor resolution due

to less sensor pixels for one pixel in the rectiﬁed im-

age, while values larger than 1 denote good resolution

but wasting sensor pixels. The pixel density depends

on the camera, the choosen projection, the mirror con-

ﬁguration including the curvature as well as the dis-

tance between projection center and pinhole.

4 EXPERIMENTAL RESULTS

To evaluate the projections proposed in section 3.1 to

3.3 as well as to analyse the properties of the pixel

density, images taken in a synthesised environment

and taken in our laboratory are used. The difference

between the characteristics of densities of real and

synthesised images was less than 0.3 pixels, so that

corresponding curves overlap. Therefore and for bet-

ter understanding, only synthesised images are pre-

sented and discussed. For both synthesised and real

Table 1: Overview of the tested parameters.

d (mm) k α (degree)

50 5,50 43,00

50 7,55 39,50

50 10,00 36,75

50 7,55 39,50

75 7,55 39,50

100 7,55 39,50

50 7,55 39,50

75 7,55 28,50

100 7,55 22,25

images, we designed various ODVS with different

curvatures (k), varying distances (d) between projec-

tion center and pinhole as well as diverse ﬁeld of view

(FOV) (α) of the pinhole camera. Secondly, we deﬁne

a region of interest (ROI) for rectiﬁcation, unwarp the

images using different projections and calculate the

pixel density. Table 1 gives an overwiev of the tested

parameters with constant ROI.

4.1 Rotational Symmetric Projections

using Synthetic Environments

First, we present the results of our measurements for

different distances d between the mirror projection

center and the pinhole (see section 4.1.1). Thereafter,

we present the characteristics of the pixel density de-

pending on the chosen projection using constant dis-

tances d (see section 4.1.2) and discuss the result in

section 4.2. The conic, cylindric and spheric projec-

tions are rotationally symmetric, so that the pixel den-

sity σ remains constant for all number of columns in

the rectiﬁed image.

(a) d=50mm, k=7.55 (b) d=75mm, k=7,55

Figure 7: This ﬁgure illustrates synthesised images ODVS

with constant k=7.55, d=50mm and d=75mm using the

same FOV.

4.1.1 Constant Mirror Type using Varying

Distances d as well as FOV

Figure 7 illustrates synthesised images from ODVS

with constant mirror type k=7.55 and different dis-

tances d=50mm and d=75mm using the same FOV.

Figure 8 shows also the same camera conﬁguration

and the same mirror type, but the FOV is adapated to

get best utilisation of sensor pixels for image rectiﬁ-

cation. The marked ROI (red) is used for rectiﬁcation

as well as for calculation the pixel density.

(a) d=50mm, k=7.55 (b) d=100mm, k=7,55

Figure 8: An ODVS with the mirror constant k=7.55, dif-

ferent distances and an adapted FOV is used.

Figure 9 demonstrates the result using cylindrical

projection, while the characteristics of the pixel den-

sity depending on different distances, projections and

the chosen ﬁeld of view (FOV) is presented in ﬁg-

ure 10. The charts in the left column (10(a), 10(c),

10(e)) present the results using a constant FOV, while

the FOV in (10(b), 10(d), 10(f)) is adapted to get best

utilisation of sensor pixels.

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

86

(a) d=50mm, k=7.55 (b) d=100mm, k=7,55

Figure 9: This ﬁgure shows unwarped images using cylin-

dric projection, the same mirror type k but different dis-

tances d.

(a) Const. α, k=7.55, cyl. (b) Var. α, k=7.55, cyl.

(c) Const. α, k=7.55, con. (d) Var. α, k=7.55, con.

(e) Const. α, k=7.55, sph. (f) Var. α, k=7.55, sph.

Figure 10: Pixel densities for ODVS’s with different dis-

tances d=50mm, d=75mm and d=100mm as well as ﬁxed

FOV’s (left) and adapted FOV’s (right) using cylindrical,

conical and spherical projections.

4.1.2 Results from ODVS with SPOV using

Different k’s

To study the inﬂuence of the projection, images from

omnidirectional vision sensor with single point of

view and different mirror types k = 5,50, k = 7,55

and k = 10, 00 are generated. Figure 11 shows the

original images. We calculated the cylindric, the

conic and the spheric projection and rectiﬁed the im-

ages using LUT (see section 3).

The rectiﬁcation result is illustrated in ﬁgure 12

for chosen mirror constants. Based on the rectiﬁed

images, the pixel density σ for each mirror and each

projection is determined. The characteristics of the

pixel density for the ODVS’s with SPOV for different

k’s illustrates ﬁgure 13.

(a) k=5.50 (b) k=7.55

(c) k=10.00

Figure 11: This ﬁgure illustrates synthesised images from

Single Point of View ODVS with different mirror types.

The marked ROI (red) is used for all rectiﬁcations.

(a) k=5.50, cylin-

dric

(b) k=5.50, conic (c) k=5.50, spheric

(d) k=7.55, cylin-

dric

(e) k=7.55, conic (f) k=7.55, spheric

(g) k=10.00, cylin-

dric

(h) k=10.00, conic (i) k=10.00,

spheric

Figure 12: The rectiﬁed images for a mirror constant k =

5.50, k = 7.55 k = 10.00 using cylindric, conic and spheric

projection.

4.2 Discussion

Figure 10 and ﬁgure 13 present the pixel densities

both for synthesised and real images. Measurements

show that differencesin the pixel density between real

and synthesised images were less than 0.3 pixels and

not signiﬁcant. For a better understanding, only the

synthesised images are presented and discussed.

The pixel density depends on the ROI, the mirror

and camera conﬁguration and the chosen projection.

For our studies, we use a ROI including regions under

the camera, with short and with large distances to the

ODVS. The best resolution reaches a rectiﬁed image

achieving a pixel density nearly 1. A pixel density

less than 1 denotes poor resolution in case of less sen-

sor pixels for one pixel in the rectiﬁed image, while a

IMAGE RECTIFICATION - Evaluation of Various Projections for Omnidirectional Vision Sensors using the Pixel Density

87

(a) k=5.50, SPOV (b) k=7.55, SPOV

(c) k=10.00, SPOV

Figure 13: The pixel densities for the different mirrors both

for the synthesised and the real rectiﬁed images are shown

in this diagram.

value larger than 1 denotes good resolution but wast-

ing sensor pixels.

In a ﬁrst setup, the FOV of the camera remains con-

stant while the distance d between the mirro projec-

tion center and camera pinhole increases. As shown

in ﬁgure 10, the characteristics of pixel density (see

10(a), 10(c), 10(e)) descrease with larger distances d

by constant FOV’s because less sensor pixels can be

used for the light reﬂected by the mirror.

In a second setup, the FOV is adapted during increas-

ing of the distances d to get the best utilisation of sen-

sor pixels. If the FOV is adapted to the distances to

achieve good utilisation of sensor pixels for light re-

ﬂected by the mirror, pixel density is nearly identical

for the same projection type. The small differences

as shown in ﬁgure (10(b), 10(d), 10(f)) result due to

the distance based variance of the vertical FOV. The

small differences also result of the increasing vision

ﬁeld of the ODVS by increasing distances d. Figure

9(b) illustrates the larger ﬁeld of view by the larger

distance d compared to 9(a).

In a third setup, we conduct experiments with differ-

ent projections as well as mirror conﬁgurations us-

ing cameras with a single point of view. As shown

in ﬁgure 13, the pixel density with constant sensor

resolution varies depending on the chosen projection.

The range of the pixel density for cylindrical projec-

tion varies in a large range between the upper and the

lower image area. The pixel density varies less for

the conical projection, therefore the sensor pixel are

mapped homogenous to the rectiﬁed image. The rec-

tiﬁed images using conic projection are the strongest

distorted images, but this need not to be a drawback.

A good compromis is the spherical projection with

less distortion and a nearly homogenous use of sensor

pixels in rectiﬁed images.

Figure 13 also demonstrates that the common cylin-

drical projection is not always the best projection for

image rectiﬁcation due to the large range of the pixel

density characteristic. Furthermore, the variation of

pixel density for cylindric projection as shown in ﬁg-

ure 10 and ﬁgure 10(b) is larger than the variation of

other projections for different distances between the

pinhole of camera and the mirror projection center.

To get best image quality of rectiﬁed images, the pro-

jection with least variance in the pixel density for

different mirrors as well as for various distances be-

tween camera pinhole and projection center is recom-

mended. If the distortion of rectiﬁed images is not a

problem for image processing routines, the conic pro-

jection seems to be the best projection due to less vari-

ances in the pixel density. Otherwise, the spherical

projection can be a good alternative. If only a small

region of interest in the projection area is needed for

image rectiﬁcation (for example the area between 50

and 100 pixel (column)), the cylindrical projection

can also be a good choice for image rectiﬁcation. So

the pixel density helps to ﬁnd the projection with op-

timum use of sensor pixels in rectiﬁed images.

Using the LUT as proposed in section 3, online image

rectiﬁcation is possible. Experiments show that only

6.5ms are necessary to transform images with a reso-

lution of 480 · 480 pixels to unwarped images with a

resolution of 540· 204 pixels on a 2.2 GHz AMD 64

X2 4200+ processor using bicubic interpolation.

5 CONCLUSIONS AND FUTURE

WORK

In this paper, we present methods to rectify and to un-

warp distorted images generated by omnidirectional

vision sensors (ODVS). Furthermore, we propose a

novel value, the pixel density, to evaluate different

projections.

The ﬁrst step is to calibrate the camera providing a

relation between world and camera coordinates. Sec-

ondly, we present several projectionsand calculate the

projection area to transform original into panoramic

images using the calibrated camera model and a LUT

containing the world coordinates of every pixel in the

transformed region of interest. For our research, we

propose a novel value, the pixel density, to evaluate

the chosen projection to ﬁnd best utilisation of sensor

pixels in rectiﬁed images.

To get best image quality of rectiﬁed images, the

projection type with least variance in the pixel density

both for different mirror types as well as for various

distances between the camera pinhole and the projec-

tion center is recommend. So the pixel density helps

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

88

to ﬁnd the projection with optimum use of sensor pix-

els in rectiﬁed images.

Further studies of the pixel density and its relation to

the projections with different regions of interest (ROI)

are necessary. In the next step, the curvature of the

pixel density has to be parametrised to ﬁnd a mathe-

matical relation between the pixel density and the pre-

sented projections.

ACKNOWLEDGEMENTS

The authors would like to thank Michelle Karg

and the reviewers for their comments which have

greatly improved the paper. Support for this re-

search was provided by BMW AG in the framework

of CAR@TUM.

REFERENCES

Baker, S. and Nayar, S. K. (1999). A theory of single-

viewpoint catadioptric image formation. International

Journal of Computer Vision, 35(2):175–196.

Baker, S. and Nayar, S. (2001). Single viewpoint catadiop-

tric cameras. Monographs in Computer Science.

Colombo, A., Matteucci, M., and Sorrenti, D. G. (2007).

On the calibration of non single viewpoint catadiop-

tric sensors. Lecture Notes in Computer Science,

Springer, pages 194–205.

Daniilidis, K. and Geyer, C. (2000). Omnidirectional vi-

sion: Theory and algorithms. In Proceedings of the

XV International Conference on Pattern Recognition.

Gandhi, T. and Trivedi, M. M. (2004). Motion analysis for

event detection and tracking with a mobile omnidirec-

tional camera. Multimedia Systems, 10(2):1432–1882.

Gaspar, J., Decco, C., Okamoto, J., and Santos-Victor, J.

(2002). Constant resolution omnidirectional cameras.

In Proceedings of the Third Workshop on Omnidirec-

tional Vision.

Geyer, C. and Daniilidis, K. (2001). Catadioptric projective

geometry. International Journal of Computervision,

45(3):223–243.

Hicks, R. and Perline, R. (2002). Equi-areal catadioptric

sensors. In Proceedings of the Third Workshop on

Omnidirectional Vision.

Ishiguro, H. (1998). Development of low-cost compact om-

nidirectional vision sensors and their applications. In

Proceedings of the International Conference on Infor-

mation Systems, Analysis and Synthesis.

Krishnan, A. and Ahuja, N. (1996). Panoramic image ac-

quisition. In Proceedings of the IEEE Conference on

Computervision and Pattern Recognition.

Mei, C. and Rives, P. (2007). Single view point omni-

directional camera calibration from planar grids. In

Proceedings of the IEEE International Conference on

Robotics and Automation.

Micusik, B. and Pajdla, T. (2004). Para catadioptric camera

auto-calibration from epipolar geometry. In Proceed-

ings of the 6th Asian Conference on Computer Vision

System.

Scaramuzza, D., Martinelli, A., and Siegwart, R. (2006a). A

ﬂexible technique for accurate omnidirectional omni-

directional camera calibration and structure from mo-

tion. In Proceedings of the Fourth IEEE Conference

on Computer Vision Systems.

Scaramuzza, D., Martinelli, A., and Siegwart, R. (2006b).

A toolbox for easily calibration omnidirectional cam-

eras. In Proceedings of the IEEE/RSJ International

Conference on Intelligent Robots and Systems.

Utsumi, A., Mori, H., Ohya, J., and Yachida, M. (1998).

Multiple-view-based tracking of multiple humans. In

Proceedings of the IEEE Conference on Computervi-

sion and Pattern Recognition.

Yamazawa, K., Yagi, Y., and Yachida, M. (1993). Om-

nidirectional imaging with hyperboloidal projection.

In Proceedings of the IEEE/RSJ International Confer-

ence on Intelligent Robots and Systems.

Yamazawa, K., Yagi, Y., and Yachida, M. (1998). Hy-

peromni vision: Visual navigation with an omnidirec-

tional image sensor. Systems and Computers in Japan,

28:36–47.

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