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only by the feature extraction process, but by the sub-
sequent tracking stage that will not be approached in
this paper. Then, painting the pipe would be required
even if the rectification process would not exist. Pre-
viously calibrating the pair of cameras is something
that would be inconvenient because the cameras are
mounted during the maintenance task and the techni-
cians are not trained for it.
The current methods for rectification do not work
properly in a noisy environment with a reduced num-
ber of feature correspondences, especially when the
calibration is unknown. Thus, this paper presents a
novel rectification technique for stereo rigs that op-
erates even with a reduced number of feature cor-
respondences. The state of the art techniques (such
as (Fusiello and Irsara, 2008)) need at least six cor-
respondences, while the proposed technique requires
only three. However, the proposed solution requires
the following restrictions on the cameras rig: cam-
eras’ projection planes must be coplanar; and cameras
must have equal intrinsic parameters.
Tests were carried out in order to evaluate the
proposed technique. Three different sets of tests
were applied to measure the error related to the tech-
nique. The first one considered a synthetic test that
was proposed only with points numerically disturbed
by a random generated noise. In the second test, a
real structured environment was built and a carefully
mounted stereo rig was used.
The third test occurred in a real scenario. As stated
before, the technique was tested in a deep underwater
environment, where images were captured by a ROV.
2 RELATED WORK
Rectification of stereo images is a frequently in-
vestigated topic by the computer vision community.
These researches began by photogrammetrists, such
as (Slama et al., 1980), which were further developed
by computer vision researchers aiming to facilitate the
feature matching between images from a stereo rig.
Rectification techniques can be classified into two
categories: calibrated, and uncalibrated. Calibrated
techniques assume that cameras’ intrinsic and extrin-
sic parameters are known and the rectifying homogra-
phies are estimated only by taking into account these
parameters. In (Fusiello et al., 2000), a simple and
effective calibrated method is presented.
Uncalibrated techniques estimate the rectifying
homographies by using a set of corresponding 2D
points between the images and/or epipolar restrictions
(such as the fundamental matrix). These techniques
are more used than the calibrated ones because in
most of the real problems the rectification is required
in a stage before the cameras poses are known. How-
ever, it is a more complex problem with non-linear
solutions, which requires the use of approximations
and optimization methods. Such methods are required
because there are infinite pairs of rectifying homo-
graphies, although it is convenient to choose the one
which produces less image deformation. Some un-
calibrated techniques can be found in (Hartley, 1998),
(Loop and Zhang, 1999), (Isgro and Trucco, 1999),
(Fusiello and Irsara, 2008).
When the epipole is close to or inside the image,
image deformation tends to be large. In these cases
planar rectifications are not enough, therefore it is
necessary to use different techniques such as cylin-
drical rectification (Roy et al., 1997) or polar rectifi-
cation (Pollefeys et al., 1999).
In the scenario presented by this paper only part of
cameras parameters are previously known, which en-
forces the use of an uncalibrated technique. However,
uncalibrated techniques require at least six accurate
corresponding points between the images (Fusiello
and Irsara, 2008), requirement that may not always
be fulfilled by the application. The proposed tech-
nique overcomes this limitation by relying on some
restrictions imposed on the stereo rig. In practice,
these constrain the way in which cameras must be
relatively positioned and oriented. If the stereo rig
is mounted so that the cameras’ projection planes are
coplanar, epipoles will be localized close to the infin-
ity, enabling a planar rectification to solve the prob-
lem.
3 BACKGROUND
In this section, some concepts that are at the core of
the proposed rectification technique will be presented
as well as the adopted notation. These concepts are
more extensively explained in (Hartley and Zisser-
man, 2004), (Loop and Zhang, 1999).
3.1 Epipolar Geometry
Given two pinhole cameras P and P
0
with their re-
spective projection matrices defined as P = K[I|0] and
P
0
= K
0
R[I|−C]. I is a 3 × 3 identity matrix. Camera
P has its projection center at the origin of the coordi-
nate system 0 = [0, 0, 0]
>
. Camera P
0
has its projec-
tion center at C = [x
c
, y
c
, z
c
]
>
, defined in Euclidean
coordinates. Furthermore, matrices K and K
0
are the so
called calibration matrices, which encapsulate cam-
eras’ intrinsic parameters. A simplified calibration
matrix has the form diag( f , f , 1), where f is the lens
UncalibratedImageRectificationforCoplanarStereoCameras
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