untitled

Published in IET Intelligent Transport Systems
Received on 27th March 2008
Revised on 4th July 2008
doi: 10.1049/iet-its:20080017

ISSN 1751-956X

Wide-angle camera technology for automotive

applications: a review

C. Hughes

M. Glavin

E. Jones

P. Denny

Connaught Automotive Research Group, Department of Electronic Engineering, National University of Ireland, Galway, Ireland

Valeo Vision Systems, I.D.A. Business Park, Dunmore Road, Tuam, County Galway, Ireland

E-mail: ciaran.hughes@nuigalway.ie

Abstract:

The development of electronic vision systems for the automotive market is a strongly growing ﬁeld, driven

in particular by customer demand to increase the safety of vehicles both for drivers and for other road users,
including vulnerable road users (VRUs), such as pedestrians. Customer demand is matched by legislative
developments in a number of key automotive markets; for example Europe, Japan and the US are in the process
of introducing legislation to aid in the prevention of fatalities to VRUs, with emphasis on the use of vision systems.

The authors discuss some of the factors that motivate the use of wide-angle and ﬁsh-eye camera technologies in

vehicles. The authors describe the beneﬁts of using wide-angle lens camera systems to display areas of a vehicle’s
surroundings that the driver would, otherwise, be unaware of (i.e. a vehicle’s blind-zones). However, although wide-
angle optics provide greater ﬁelds of view, they also introduce undesirable effects, such as radial distortion,
tangential distortion and uneven illumination.

These distortions have the potential to make objects difﬁcult for the vehicle driver to recognise and, thus, potentially

have a greater risk of accident. The authors describe some of the methods that can be employed to remove these
unwanted effects, and digitally convert the distorted image to the ideal and intuitive rectilinear pin-hole model.

Introduction

Safety in vehicles is a growing concern for most modernised
countries. For example, the European New Car Assessment
Program (Euro NCAP)

[1]

, which was established in 1997,

provides objective information on the safety of drivers
and passengers in cars in crash situations. In a study
commissioned by the Euro NCAP, 94% of respondents list
safety in vehicles as a major concern

[2]

. There are similar

organisations in both Japan

[3]

and the US

[4]

(known as

JNCAP and USNCAP, respectively).

However, more recently, interest in the protection of

vulnerable road users (VRUs), for example pedestrians and
cyclists, has increased. Back-over collisions caused by the
inability of drivers to detect VRUs when they are present in a
vehicle’s blind-zones is a major cause of VRU fatalities
globally. The term ‘blind-zone’ is used in preference over the
other commonly used term ‘blind-spot’, as this is the term
used in some of the jurisdictions described in this paper, and

more accurately describes the areas that cannot be seen by a
driver of a vehicle. It is somewhat ironic to note that, with
vehicle manufacturers increasing the emphasis on improving
the Euro NCAP ratings of their vehicles by strengthening
and increasing the size of the vehicles’ A-pillars (the vertical
or near vertical shaft of material that supports the vehicle roof
on either side of the wind-screen), they are also increasing the
size of the blind-zones caused by those A-pillars. Increased
A-pillar size is often used to improve a vehicle’s ‘secondary
safety’, which can be deﬁned as all structural and design
features that reduce the consequences of accidents as far as
possible’

[5]

. However, this is at a potential loss of direct

vision, an element of ‘primary safety’, which can be deﬁned as
‘the vehicle engineering aspects which as far as possible
reduce the risk of an accident occurring’

[6]

To match the consumer desire for increased safety in

vehicles, the European Union has introduced legislation that
requires large goods vehicles (LGVs) to have large portions
of their blind-zones made visible to the driver. The term

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‘LGV’ is used in this paper, instead of the term ‘heavy goods
vehicle (HGV)’, as this is the terminology used in most
European Union documentation because of the fact that the
word ‘heavy’ does not have a direct translation in all
European languages. Japan is also in the process of
introducing similar legislation

[7]

. While European and

Japanese legislation focuses primarily on LGV safety,
proposed legislation in the US is targeted at privately owned
vehicles, with the aim of preventing back-over collisions.

In Section 2, we describe radial distortion, which is the

primary distortion introduced by wide-angle cameras to an
image. We review several models of radial distortion and
describe some of the problems in camera calibration and
distortion correction. This is followed, in Section 3, by a
description of some of the other geometric distortion
factors in wide-angle cameras, such as centre of distortion
(COD), tangential distortion and uneven illumination, and
certain methods that may be employed for their mitigation
are discussed.

1.1 Vehicle blind-zones

A vehicle’s blind-zones are the areas around the vehicle that
cannot be seen directly by the driver by looking forwards or
by using any of the vehicle’s standard rear-view mirrors
(internal and external) from the normal sitting position.

Fig. 1

shows the potential blind-zones around a vehicle.

The sizes of these areas are determined by the size and
design of the vehicle and mirrors, and will vary according
to car model and manufacturer.

Consumers Union

[8]

has examined the rearward blind-

zone on many non-commercial light-duty passenger vehicles
[from small passenger cars to large sports utility vehicles

(SUVs)]. The zone was measured by determining how far
behind the vehicle a 28 inch (0.71 m) trafﬁc cone needed to
be before a person, seated in the driver’s seat, could see its
top while looking through the rear window. For a 5 foot
8 inch (1.73 m) tall driver, the distance measured was up to
44 feet (13.4 m) for a commercially available four-wheel
drive vehicle registered in 2006. In the same vehicle, the
blind-zone distance for a 5 foot 1 inch (1.55 m) driver
extends to 69 feet (21 m).

The blind-zone for LGVs is naturally much larger than

that of light-duty vehicles. Ehglen and Paidla

[9]

calculated

the forward blind-zones of a given LGV as shown in

Fig. 2

. Furthermore, the rearward blind-zones of LGVs

tend to be very large; we have calculated that several HGVs
have a rearward blind-zone that can extend up to 65 m
behind the vehicle on the ground plane.

1.2 Blind-zone collision statistics

1.2.1 Europe:

In Europe, ofﬁcial statistics for VRU

deaths because of the victims not being visible to the driver
of a vehicle are not readily available, as there is no single
repository for such information. However, the European
Commission’s CARE road accident database

[10]

claims

that there were 3961 VRU pedestrian fatalities within
urban areas in 2005. It is reasonable to assume that a
signiﬁcant portion of these deaths were caused by the VRU
not being visible to the driver of the vehicle.

This assumption is supported by several statistics. The

European Commission Directorate-General for Energy and
Transport estimates that the lack of visibility in the blind-
zone towards the rear of a vehicle directly causes 500 deaths
a year in the EU

[11]

. The Commission of the European

Communities estimates that every year, approximately 400
European road users lose their lives in collisions with LGVs,
because the driver did not see them when turning right

[12]

The UK Health and Safety Executive reports that in the 12
months between 2005 and 2006, 18 people were killed and
620 sustained major injuries because of the intentional

Figure 1

Five blind-zone areas (shown in grey) in a

standard left-hand drive car

Sizes of the blind-zones are dependent on the design of the car
and the viewing angle of the mirrors

Figure 2

Blind zones around the front of a given (left-hand

drive) LGV [9]

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motion of the vehicle (either forwards or backwards) in the
workplace

[13]

1.2.2 US:

Statistics for the US are equally disjointed, with

no ofﬁcial statistics directly available for VRU injuries
because of vehicle blind-zones. However, the Kids and
Cars Organisation in the US

[14]

claims that 941 children

were killed in non-trafﬁc collisions in the US between 2002
and 2006. Non-trafﬁc collisions are collisions involving
vehicles while they are not in a trafﬁc situation (for
example, while on private residential property). They
further claim that 49.5% (or 466 children) of the fatalities
were because of the vehicle reversing while children were
present in the vehicle’s rearward blind-zone.

In a study between July 2000 and June 2001, the Centers for

Disease Control and Prevention (CDC)

[15]

reported that

there were an estimated 9160 non-fatal injuries to children in
non-trafﬁc automotive collisions, with

20% (or 1832

children) of these injuries caused by the vehicle moving
backwards. Between 2001 and 2003, the CDC reported that
an estimated 7475 children (2492 per year) were treated for
vehicle back-over injuries

[16]

. Again, although the blind-

zone is not directly implicated in these injuries, it is
reasonable to assume that a signiﬁcant proportion of these
injuries were because of the children being present in the
vehicle’s blind-zone.

Wang and Knipling

[17]

estimated that lane change

merge crashes in 1991 accounted for approximately
244 000 police-reported crashes with 224 associated
fatalities. Furthermore, the authors reported that the
principal causal factor in such crashes is that the driver ‘did
not see the other vehicle’.

1.3 Legislation relating to vehicle
blind-zones

Because of the increasing awareness of the vulnerability of
pedestrians, legislation has been introduced, or is in the
process of being introduced, in several jurisdictions around
the world. In this section, the legislative or potential
requirements for the visibility of blind-zones in the EU,
Japan and the US are described.

1.3.1 Europe:

In the EU, legislation in the form of

Directive 2003

[18]

was introduced in 2003.

Although the initial requirements of this directive aim to
reduce collisions caused by the blind-zones of LGVs and
improve road safety for new vehicles circulating from 2006

2007 onwards, the legislation does not cover the existing
ﬂeet of lorries in the EU. However, as it has been
estimated that existing ﬂeets will not be fully replaced until
2023, Directive 2007

[19]

was introduced in 2007.

This legislation requires the retroﬁtting of the required
indirect vision systems (IVSs) to all existing ﬂeets within
24 months of enactment of the bill (i.e. by July 2009).

The shaded areas in

Fig. 3

show the areas of a left-hand

drive LGV’s environment that must be visible to the driver
via the use of IVS, as required by these directives.
Although the ﬁgure shows a left-hand drive vehicle, for
right-hand drive vehicles within the jurisdiction of the
legislation, the areas of required coverage are reversed.
Examples of IVSs include additional mirrors to the
standard rear-view mirrors (internal and external), as well
as camera– monitor devices. However, practical problems
arise with the use of additional mirrors as the extra
mirrors can themselves introduce additional blind-zones, by
obstructing direct forward vision, as well as having
additional cost and styling implications.

There is a clear provision for the use of camera– monitor

devices for the coverage of vehicle blind-zones in this
directive. In fact, the use of camera– monitor devices over
mirrors is often not only desirable, but also necessary in
certain situations. For example, it is practically impossible
to cover the area at the rear of an LGV with mirrors alone,
and so camera– monitor systems are the only practical
solution.

1.3.2 Japan:

In Japan, legislation has also been proposed,

which would require medium and large vehicles to be
equipped with devices that allow the driver to detect
objects in the vehicle’s blind-zones, either directly or
indirectly using mirrors or camera– monitor devices (

Fig. 4

)

[20]

. For the purpose of the proposed legislation, a cylinder

1 m high with a diameter of 0.3 m placed anywhere within
the coverage areas must be at least partially visible to the
driver of the LGV directly, by mirror or by camera.
However, in this legislation, it is proposed that objects
within blind-zones caused by A-pillars and external mirrors
need not be visible to the driver of the vehicle

[7]

1.3.3 US:

In the US, proposed legislation, in the form of

S.694 (The Cameron Gulbransen Kids and Cars Safety
Act of 2007)

[21]

, is designed to protect against children

being injured or killed in non-trafﬁc incidents, such as
when the vehicle is reversing. Relating to the potential use
of cameras, the S.694 bill requires a ‘ﬁeld of view (FOV) to

Figure 3

Area required by Directive 2003

EC to be

visible to drivers of (left-hand drive) LGVs via the use of
IVSs (not including areas covered by the standard and
wide-angle wing-mirrors)

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enable the driver of a motor vehicle to view areas behind the
motor vehicle to reduce death and injury resulting from
backing incidents, particularly incidents involving small
children and disabled persons’. Unlike the EU and Japanese
legislation, however, the US legislation fails to describe in
any technical detail the methods by which the objectives of
the bill are to be implemented.

1.4 Wide-angle cameras to cover
vehicle blind-zones

As we shown in previous sections, the blind-zones on some
vehicles can be quite large, particularly for vehicles such as
SUVs and LGVs. The aim of this paper is to demonstrate
that a good quality, undistorted image of a vehicle’s blind-
zones can be displayed to the driver of the vehicle using a
wide-angle camera system. As shown in

Fig. 5

, standard

lens camera systems (e.g. 45

FOV lenses) are unable to

fully cover the blind-zone of some SUVs. Considering that
camera– monitor systems generally display a ‘reference’
point (i.e. part of the body of the vehicle) on screen, a
standard lens camera with an FOV of 45

can only cover,

perhaps, 1 m of the SUV blind-zone.

Fig. 5

illustrates

how the use of a wide-angle lens camera system (e.g.

100

FOV lenses) enables the entire SUV rearward

blind-zone to be covered.

Fig. 6

shows a sample placement of two wide-angle

cameras mounted on an LGV. Camera 1 is a 135

wide-

angle camera, located approximately half-way down the
length of the LGV, and 3 m off the ground plane. The
optical axis of camera 1 is tilted at 15

from the side of the

LGV trailer. Camera 2 is a 135

wide-angle camera,

located in the middle of the front cabin at about 2 m off
the ground plane. The optical axis of camera 2 is tilted at
20

from the front face of the cabin. With both cameras

corrected for distortion,

Fig. 7

shows the areas in the

vicinity of the vehicle that can be displayed to the driver.
Such a camera system would cover all the blind zones of
the LGV shown in

Fig. 2

, and would meet the requirements

of both the EU Directive 2003

EC (

Fig. 3

) and the

proposed Japanese legislation (

Fig. 4

Certain areas around LGVs need very wide wide-angle

lens cameras (e.g. ﬁsh-eye lens cameras) to display the
entire blind-zone to the driver. However, problems arise
because of the deviation of wide-angle lens cameras from
the rectilinear pin-hole camera model, because of
geometric distortion effects caused by lens elements. Fish-
eye cameras deviate substantially from the pin-hole model,
introducing high levels of geometric nonlinear distortion.
Because of this distorted representation of the real-world
scene onscreen, there is potential for obstacles and
VRUs to not be recognised by the driver. Additionally,
the distortion may cause the driver to mis-judge the
distance to objects, because of the nonlinearity of the view
presented.

Figure 4

Proposed Japanese legislation: area in which a

cylinder (1 m high, 0.3 m diameter) must be at least
partially visible to the driver of the (right-hand drive) LGV
[20]

LGV

7.5 t

LGV

7.5 t

Figure 5

Coverage area for standard and wide-angle lens

cameras mounted on the rear of a typical SUV

Example coverage area for standard lens camera (e.g. 45

FOV)

Example coverage area for wide-angle lens camera (e.g. 100

FOV)

Figure 6

Example of a potential wide-angle camera

placement on an LGV that would meet the requirements
of the EU Directive 2003

EC and the proposed

Japanese legislation

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Thus, camera calibration and distortion correction are

important tasks for automotive camera applications. Not
only do they make images captured by the camera more
visually intuitive to the human observer, they are often also
necessary for computer vision tasks that require the
extraction of geometric information from a given scene.
The following sections describe some of the effects of
wide-angle and ﬁsh-eye lens distortion.

Radial lens distortion and

its correction

Radial lens distortion causes points on the image plane in the
wide-angle

ﬁsh-eye camera to be displaced in a nonlinear

fashion from their ideal position in the rectilinear pin-hole
camera model, along a radial axis from the centre of
distortion in the image plane (

Fig. 8

). The visual effect of

this displacement in ﬁsh-eye optics is that the image will a
higher resolution in the foveal areas, with the resolution
decreasing nonlinearly towards the peripheral areas of the
image.

For normal and narrow FOV cameras, the effects of radial

distortion can be considered negligible for most applications.
However, in wide-angle and ﬁsh-eye lenses, radial distortion
can cause severe problems, not only visually but also
for further processing in applications such as object
detection, recognition and classiﬁcation. Additionally, the
radial distortion introduced by these lenses does not preserve
the rectilinearity of an object in its transformation from real-
world coordinates to the image plane. Straight lines in the
real world can usually be approximated as circular sections in
the distorted image plane

[22 – 24]

, as evidenced in

Fig. 9

The models described in this section are relationships between

the distorted radial distance,

, and the corresponding

undistorted radial distance,

. Both are measured from the

COD (described in Section 3.1). Models of radial distortion
can be grouped into two main categories: polynomial and non-
polynomial models.

2.1 Polynomial models of radial
distortion

The use of polynomials to model radial distortion in lenses is
well established

[25– 33]

. From an embedded point of view,

polynomials are desirable as they do not require the
implementation of computationally demanding numerical
algorithms, in contrast to log and tan-based functions that
are required for non-polynomial models (however, through
the use of look-up tables, this advantage is lessened).

With the exception of the division model, the models

described in this section are the functions of the undistorted
radial distance, that is

is a function of

. It is usually

necessary to convert a distorted image to an undistorted
image, and thus obtaining

as a function of

desirable. Problems arise with polynomial models because
of the fact that there is no general analytical method to
invert them, that is, there is no general method to invert a
forward model to an inverse model for use in the radial
distortion correction. However, back-mapping (described in

Figure 8

Displacement of points in radial distortion, where

P is the, r

is the distorted radius and r

is the undistorted

radius

Figure 7

Areas of coverage if the video from the cameras

mounted in

Fig. 6

were corrected for geometric distortion

Figure 9

Photographic example of ﬁsh-eye distortion

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Section 2.3.2) provides a means by which the forward model
can be used to convert a distorted image to an undistorted
rectilinear image.

2.1.1 Standard polynomial model:

The standard

model for radial distortion is an odd-ordered polynomial,
as described by Slama in

[25]

and, subsequently, used in

[26, 29, 32, 34]

þ þ

(1)

where

is the distorted radius,

the undistorted radius and

the coefﬁcients of the polynomial. It is generally considered

that (1) cannot accurately describe the level of distortion
introduced by wide-angle lenses

[35, 36]

. However, for

standard lenses, it is usually considered that a ﬁfth-order
polynomial is sufﬁcient for the radial distortion correction

[32]

(2)

Although there is no exact analytical method of inverting (1),
an inverse to the ﬁfth-order version (2) can be approximated
as described by Mallon and Whelan

[32]

2
1

2
2

(3)

2.1.2 Division model for radial distortion:

Fitzgibbon introduces the so-called ‘division model’ in

[31]

þ þ

(4)

It should be noted that this is inherently an inverted model,
that is, it models the undistorted radial distance of a point as a
function of the distorted radial distance of that point.
Although (4) is similar in form to (1), it is important to
note that this is not an approximation to an inversion of
the

standard

polynomial

model.

Rather,

both

are

approximations to the camera’s true distortion curve. The
division model suffers from problems similar to those in
(1), that is, that there is no general method for inversion,
and it cannot accurately describe the distortion introduced
by wide-angle

ﬁsh-eye lenses.

The division model is often used when circle-ﬁtting is

employed to calibrate a speciﬁc lens, as this allows
distortion estimation to be reformulated as circle-ﬁtting for
which many algorithms are available

[23, 24]

2.1.3 Polynomial ﬁsh-eye transform (PFET):

It has

been noted already that (1) and (4) can be used to describe
distortion in standard, non-ﬁsh-eye lenses, but that these
models are insufﬁcient to accurately describe the level of
distortion introduced by wide-angle

ﬁsh-eye lenses.

A polynomial that uses both odd and even coefﬁcients can

be used to accurately model the radial distortion introduced
by a ﬁsh-eye lens

[30, 33, 37, 38]

. Basu and Licardie (30,

37) refer to this as the polynomial ﬁsh-eye transform (PFET)

þ þ

(5)

The beneﬁt of using the coefﬁcients beyond the ﬁfth order is
generally considered negligible with the PFET

[30, 37]

2.2 Non-polynomial models of
radial distortion

In this section, we introduce several ﬁsh-eye distortion
models that are not based on a polynomial approximation
of the ﬁsh-eye lens distortion curve. One of the more
notable advantages of using non-polynomial models over
the polynomial models is that they are, in general, readily
inverted using analytical methods. This is particularly useful
where distortion models are used in the correction of wide-
angle camera systems.

2.2.1 Perspective model:

The principle of central

projection states that for a rectilinear pin-hole camera, the
following is true

tan (

)

(6)

where

is the apparent focal length of the ﬁsh-eye camera

and

the angle of a light ray from the optical axis, in

radians. The principle of equidistant projection states that
for a lensed camera, the following is true

(7)

[39]

, the ‘perspective model’ of radial distortion is derived

from these two principles

tan

(8)

This is the conversion between the rectilinear space described
by (6) and the distorted space described by (7). The apparent
focal length

is not necessarily the same as the actual focal

length of the ﬁsh-eye camerasystem, since the ﬁsh-eye
optics often include several different groups of lenses that
affect the actual physical focal length of the ﬁsh-eye camera.

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The inverse of this model is

tan

(9)

2.2.2 Fish-eye transform:

Basu and Licardie proposed

the FET in

[30, 37]

. This model is based on the observation

that the ﬁsh-eye image has a higher resolution in the foveal
areas and a lower resolution towards the peripheral areas

ln 1

(10)

where

is a simple scalar and

controls the amount of distortion

across the image. The inverse of this model is given by

(11)

2.2.3 FOV model:

Devernay and Faugeras describe the

FOV model in

[40]

, based on a simple optical model of a

ﬁsh-eye lens

tan

(12)

where

is the apparent angular FOV of the ﬁsh-eye camera.

As with

in the perspective model,

may not correspond

with the actual camera-system FOV, since the complex
ﬁsh-eye optics involved may not exactly follow this model.
Additionally, they point out that this model may not always
be sufﬁcient to model the complex distortion of ﬁsh-eye
lenses. In these cases, Devernay and Faugeras state that
(1) can be used with

0 before applying (12).

The inverse of this model is

tan(

)

2 tan(

(13)

2.3 Radial distortion correction

Radial distortion correction is the process by which points in the
distorted ﬁsh-eye image are transformed to points in the
undistorted image. It is possible that radial distortion can be
optically corrected using an appropriate combination of
compensating lenses. However, according to Bogner in

[41]

the amount of distortion that can be corrected by lenses is
physically limited by the refractive, reﬂective and transmissive
characteristics of the materials from which they are made.
The best wide-angle optics produce acceptable rectilinear
images at FOVs up to 110

. Therefore for wide-angle lenses

with FOVs greater than 110

, it is necessary to perform some

form of post-processing to convert the image to the rectilinear
model. Indeed, often it is preferable to use post-processing to
correct the distortion of cameras with FOVs of up to 110

, as

the optics to convert from the radially distorted image to the
rectilinear image can be expensive.

2.3.1 Calibration:

In general, the correction of radial

distortion involves a calibration procedure to determine the
parameters of one of the ﬁsh-eye models described in the
previous sections, to ﬁt the distortion of a particular ﬁsh-
eye lens camera. The distortion can then be corrected by
inverting the model and transforming each pixel in the
image according to that model (the exception is the
division model described by (4), which is already an inverse
model). This results in an ‘undistorted’ image.

The majority of calibration procedures require the use of a

calibration diagram with known geometry in 3D space.
Features from the calibration diagram, such as corners,
dots, lines and circles or any other feature that are easily
extracted from the image, are used to calibrate the camera.
This is known as photogrammetric calibration, and there
are numerous published methods using calibration diagrams

[26, 30, 36 – 38, 42 – 44]

. Alternatively, a self-calibration

method can be employed, whereby the calibration system
has no a priori knowledge of the scene. With this
method, the necessary information is extracted from an
arbitrary scene via point correspondences in multiple view
geometry, circle-ﬁtting or other suitable techniques

[22, 29,

31, 45, 46]

2.3.2 Vacant pixels:

Because of the essential ‘stretching’

effect of distortion correction (undistortion), the resultant
image will contain many vacant pixels that will not been
mapped (i.e. will not been ﬁlled with a re-mapped pixel
value) during the undistortion procedure, as shown by the
dark pixels in

Fig. 10

. Interpolation methods can be used

to overcome this, as is implemented in

[39, 42, 47]

As an alternative to simple forward stretching and

interpolation, a technique known as ‘back-mapping’ (or
‘inverse-mapping’) may be used [33]. Instead of mapping
every pixel in the distorted space to the undistorted space,
back-mapping does the inverse. Back-mapping calculates
the location of the pixel in undistorted space and uses a
forward transform model to determine that pixel’s location
in distorted space. This overcomes the problem of vacant

Figure 10

This is a corrected version of

Fig. 9

Dark lines in this image is the pattern that is created by the vacant
pixels in an undistorted image

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pixels because every pixel in the undistorted image will be
assigned a value, as shown in

Fig. 11

A problem with back-mapping arises because each

pixel in distorted space may be spread over several pixels
in the undistorted space during the radial distortion
correction

and

back-mapping

procedure.

help

overcome this problem, bilinear interpolation can be built
into the back-mapping algorithm by allowing the location
of each pixel in distorted space to be estimated with non-
integer accuracy using a weighted average of the
surrounding pixels.

Other geometric considerations

3.1 Centre of distortion

Radial lens distortion is the displacement of image points
along a radial axis from a single point on the image plane.
This point is known as the COD, which does not
necessarily align with the centre of the image sensor.
Therefore in order to be able to fully model and correct
radial lens distortion, it is necessary to accurately determine
the position of the COD on the image plane. In physical
terms, the COD can often be assumed to be the point at
which the optical axis of the camera lens system intersects
the image plane. Inaccurate estimation of the COD will
introduce additional radial distortion (after radial distortion
correction has been applied), as well as a degree of tangential
distortion, described in Section 3.2 (

Fig. 12

). The

estimation of the COD is only relevant in wide-angle
camera

systems

that

display

radial

lens

distortion.

According to Ruiz

et al.

[48]

, the location of the COD in

cameras with small to moderate FOV, in which no
distortion is readily apparent, is of minor importance. A
range of methods to estimate the COD are described in

[30, 33, 34, 37, 38]

3.2 Tangential distortion

According to Mallon and Whelan

[32]

, tangential distortion

causes a geometric shift of the image along, and tangential to,

the radial direction through the COD, as demonstrated in

Fig. 13

There are two primary causes of tangential distortion:

inaccurate COD estimation and thin prism distortion.
According to Weng

et al.

[27]

, thin prism distortion arises

from imperfection in lens design and manufacturing as well
as camera assembly, and causes a degree of both radial
and tangential distortion. Slama

[25]

and Shah and

Aggarwal

[36]

, both give mathematical models to deal

with tangential distortion. Stein

[49]

demonstrates that

low levels of tangential distortion can be compensated
for by using COD estimation. Most other published
research makes the assumption that other causes of
tangential distortion can be considered negligible

[24, 26,

32, 40, 50]

3.3 Uneven illumination

In cameras with wide FOV, such as those with ﬁsh-eye and
wide-angle lenses, there is often a noticeable nonlinear loss

Figure 12

Example of the shift of the centre of distortion

away from the centre of the image

The dotted lines show the distortion with the COD at the image
centre

Figure 13

Effect of tangential distortion

Figure 11

Example of undistorted image using back-

mapping with bilinear interpolation to overcome the
problem of vacant pixels

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of illuminance towards the periphery of an image because of
the structure of the camera lens system. This is noticeable in

Fig. 9

, where the extremities of the image are considerably

darker than the central area. One common cause of this
uneven illumination is the fact that the camera pupil
appears elliptical when viewed from an angle off the optical
axis. A commonly used model for this cause of uneven
illumination is the ‘cosine fourth’ (or cos

) model. Under

this model, the illumination of a given scene point is
reduced by cos

, where

is the off-axis angle of the scene

point

[51]

cos

(14)

where

the illuminance for a scene point on the optical axis

and

the illuminance for a scene point off the optical axis.

However, Welford

[51]

states that this simple model does

not fully describe the uneven illumination in a camera system.

Aggarwal

et al.

[52]

and Welford [51] describe another

position-dependent cause of the loss of illuminance, which
is attributed to vignetting. It is based on the fact that,
although an on-axis beam emanating from a distant point
fully illuminates the camera aperture, an off-axis beam
emanating from a distant point is obstructed by the lens
elements and unable to fully ﬁll the aperture.

However, Aggarwal et al.

[52]

observed that there are still

disparities between observed uneven illumination effects for
given cameras and the uneven illumination that would be
predicted by the standard cos

and vignetting models. They

state that the these models do not take into consideration
an effect they refer to as pupil aberration, which refers to
the nonlinear refraction of rays that results in a signiﬁcant
non-uniform light distribution across the aperture. Other
possible causes include light being obstructed by the ﬁeld-
stop, lens-rim or other components of the camera.

There have been several other model-based approaches to

correct for uneven illumination. For example, Asada

et al.

[53]

developed a variable cone model to correct for uneven

illumination and Yu

et al.

[54]

proposed to use a

hyperbolic function to describe the uneven illumination for
each scanline in the image.

A non-model-based approach was proposed by Leong

et al.

[55]

. They observed that uneven illumination is an additive

low-frequency signal in the image. They convolve a given
image with a Gaussian kernel ﬁlter, the aim being to smooth
the image until it is devoid of all features but retains the
underlying uneven illumination pattern. This pattern is then
used to correct the uneven illumination in a given image. The
problem with this method is that it is potentially difﬁcult to
calculate the cut-off frequency of the Gaussian ﬁlter because
of the complex nature of the uneven illumination. Equally
important is the fact that if this method is employed, any
external cause of uneven illumination will be removed, which
is undesirable (it is only desirable that uneven illumination
because of the properties of the camera and lens system is
removed).

A relatively simple method to correct for uneven

illumination introduced by the camera is described in

[56]

If a Lambertian surface (i.e. a surface that reﬂects light
equally in all directions) is imaged, an intensity proﬁle can
be obtained that describes the uneven illumination for a
given camera. A uniformly illuminated white surface can be
used as an approximation to a Lambertian surface. The
maximum intensity response (or pixel value) is found, and a
correction factor is determined for each pixel location using
the following

lut

Þ ¼

ref , max

ref

(15)

where

lut

is the correction factor for a given pixel

location to be stored in a look-up table,

ref

is the

Figure 14

Intensity proﬁle of the camera used to capture

Fig. 9

Image of evenly illuminated white surface

The corresponding correction factor LUT

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intensity response for the same pixel location and

ref , max

the maximum value of

ref

. Any image taken with the

same camera can be corrected by simply multiplying each
pixel value by the corresponding LUT value

corr

Þ ¼

orig

lut

(16)

Fig. 14

shows an image of an evenly illuminated white

surface, and the corresponding correction factor LUT.

Fig. 15

shows an example of a corrected image using this

LUT.

Summary

In this paper, we discussed the rationale within the
automotive industry for the on-vehicle use of camera
systems. Speciﬁcally, with increasing numbers of vehicles
on the world’s roads, statistics show that a signiﬁcant
percentage of trafﬁc fatalities are caused by drivers who are
not aware of VRUs in their vehicle’s blind-zones.

Although customer demand for products that give

information about a vehicle’s blind-zones to the driver is
already high, pending and existing legislation is, de facto,
making the installation of such cameras on vehicles a
necessity, particularly for large SUVs and LGVs. However,
although small vehicles are exempt from current European
and Japanese legislation, the US legislation speciﬁcally
targets the smaller private vehicles.

Ultimately, camera systems, in conjunction with other

available technology, will be necessary equipment for
improved visibility around vehicles, particularly SUVs and
LGVs. Wide-angle

ﬁsh-eye camera systems are currently

the best candidates, because of their ability to display even
the largest blind-zones of a vehicle.

Considering this, we described several types of visual

distortion resulting from the use of wide-angle cameras
(particularly

ﬁsh-eye

cameras),

and

reviewed several

methods to correct for these effects. Radial distortion is
by far the most evident geometric effect of using
ﬁsh-eye cameras, although other distortions should be
considered, such as inaccurate COD estimation and uneven
illumination.

If all the necessary distortions are removed from a ﬁsh-eye

image, the result is an image that, for many applications,
accurately approximates the desired rectilinear model.

Fig. 16

shows an example of the correction of all distortions

described in this paper. Comparison of this ﬁgure with the
original distorted image in

Fig. 9

illustrates the combined

effectiveness of the methods described.

Additionally, there is the potential for advanced processing

if camera systems are used in place of mirrors, such as VRU
detection, which can actively warn the driver of the vehicle
of potential collisions. Future work could include the
examination of existing products, and potential products,
that employ this type of detection. Additionally, although
this paper did not describe it, fusion of camera systems
with other sensory devices (e.g. passive

active infra-red,

sonar and radar) is an area that deserves a full review.

Figure 16

Fig. 9

with all distortions removed

Figure 15

Fig. 9

with the uneven illumination removed

Original image

Uneven illumination removed

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Acknowledgments

This research is funded by Enterprise Ireland and Valeo Vision
Systems (formerly Connaught Electronics Ltd.) under the
Enterprise Ireland Innovation Partnerships Scheme.

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