Pose estimation of a single circle using default intrinsic calibration

Abstract

Circular markers are planar markers which offer great performances for detection and pose estimation. For an uncalibrated camera with an unknown focal length, at least the images of at least two coplanar circles are generally required to recover their poses. Unfortunately, detecting more than one ellipse in the image must be tricky and time-consuming, especially regarding concentric circles. On the other hand, when the camera is calibrated, one circle suffices but the solution is twofold and can hardly be disambiguated. Our contribution is to put beyond this limit by dealing with the uncalibrated case of a camera seeing one circle and discussing how to remove the ambiguity. We propose a new problem formulation that enables to show how to detect geometric configurations in which the ambiguity can be removed. Furthermore, we introduce the notion of default camera intrinsics and show, using intensive empirical works, the surprising observation that very approximate calibration can lead to accurate circle pose estimation.

Full text

Pose estimation of a single circle using default
intrinsic calibration
Damien MA RI YANAYAGAM, Pierre GURDJOS,
Sylvie CHAMBON and Vincent CHARVILLAT
IRIT (INP-ENSEEIHT), Toulouse, France
Email: dmariyan@enseeiht.fr
Florent BRU NE T
Ubleam, Toulouse, France
Email: florent.brunet@ubleam.com
Abstract—Circular markers are planar markers which offer
great performances for detection and pose estimation. For an
uncalibrated camera with an unknown focal length, at least the
images of at least two coplanar circles are generally required
to recover their poses. Unfortunately, detecting more than one
ellipse in the image must be tricky and time-consuming, especially
regarding concentric circles. On the other hand, when the camera
is calibrated, one circle suffices but the solution is twofold and
can hardly be disambiguated. Our contribution is to put beyond
this limit by dealing with the uncalibrated case of a camera
seeing one circle and discussing how to remove the ambiguity.
We propose a new problem formulation that enables to show
how to detect geometric configurations in which the ambiguity
can be removed. Furthermore, we introduce the notion of default
camera intrinsics and show, using intensive empirical works, the
surprising observation that very approximate calibration can lead
to accurate circle pose estimation.
I. INTRODUCTION
The problem of estimating the pose of a camera (or dually
of a 3D object) from a set of 2D projections in a single
view has been widely studied in the computer vision literature
for a long time [11]. The “minimal” problem i.e., which
requires the minimal amount of information necessary, is
known as the perspective-3-point-problem (P3P) and consists
in recovering the pose of a calibrated camera from three 3D-
2D point correspondences. Many solutions are available for
the general case when more information is available. When the
environment in the scene can be controlled, artificial features
with known positions are very often deployed in the scene.
They are used in a wide range of applications, especially when
a reliable reference is needed to, e.g., in cluttered or textureless
environments. The most popular artificial features are probably
coplanar features [4] whose layout in 2D space defines a so-
called planar marker. The mapping between a planar marker
and its image is a 2D projective transformation known as
(world-to-image) homography and can be estimated from at
least four world-to-image correspondences (the most simple
planar marker is a square). Once the camera is calibrated, the
decomposition of the homography matrix allows to recover
the pose of the camera (or dually that of the plane).
Other well-known artificial markers that have been recently
investigated again are those consisting of coplanar circles [1],
[5], [8], [13]. The knowledge of at least two circle images
(without any information on their parameters on the support
plane) allows to compute a world-to-image homography with-
out ambiguity for all the spatial configurations of two circles
except one [5].
Given a single circle image, it is well-known that a twofold
solution exists for the normal to the support plane (and so for
the pose) only if the camera is calibrated [13]. In this work,
our contribution is to put beyond this limit by dealing with the
case of an uncalibrated camera seeing one circle. Actually, our
starting point came from the surprising observation, learned
from empirical works, that very approximate calibration can
lead to accurate circle pose estimation. Our idea is to use de-
fault intrinsics by designing a generic camera model delivering
a default focal length based on off-line calibration of several
smartphone cameras.
Our first contribution is to run extensive experiments that
assess how the inaccuracy of the calibration impacts the quality
of the pose estimation. We found out that exact calibration may
not be required as small variations on the focal length does
not affect the reprojection error of other reference coplanar
points especially when the marker is far from the camera. Our
second contribution is to provide a new geometrical framework
to state the pose problem in which the issue of how to remove
the twofold ambiguity can be thoroughly investigated.
We review the related works in section II. Then in sec-
tion III, we remind the problem of recovering the pose from the
projection of a circle before introducing the solution proposed
in section IV. The idea is to introduce a new way of computing
the vanishing line (dual to the plane normal) from one circle
image. Thanks to it, as the general method leads to two
possible solutions, we show how under some assumptions
about the geometric configuration we can recover the correct
one. Then, as we suppose that we work with uncalibrated
images, we explain how we select parameter values to obtain
what we called default camera intrinsic parameters. Finally in
section V, we evaluate our method in the context of augmented
reality.
II. RELATED WORK
A lot of existing works suggest to use a set of features
encoded in a planar pattern to simplify the pose estimation.
Fiala et al. introduced a fiducial system [4] and proposed the
case of a special planar square marker. Recent efficient algo-
rithms allow to detect ellipses precisely and, in consequence,
arXiv:1804.04922v1 [cs.CV] 13 Apr 2018

circles become features worth of interest. The four projective
and affine parameters of the world-to-image homography (the
remaining four define a similarity on the world plane [6, p42])
can be recovered by detecting the images of two special point
of the support plane, known as circular points (e.g., see [6,
p52-53]) which are common to all circles. Gurdjos et al. [5]
relied on the notion of pencil of circle images to formulate
the problem of detecting the images of the circular points as a
problem of intersection of lines, obtained from the degenerate
members of the pencil. Kim et al. [8] proposed algebraic and
geometric solutions in the case of concentric circles. Calvet
et al. [2] described a whole fiducial system using concentric
circles which allows to accurately detect the position of the
image of the circles common center under highly challenging
conditions. In a same vein, Huang et al. [7] proposed to use
the common self-polar triangle of concentric circles.
When using circular markers it is also possible to simplify
the model of the camera to only depend on a sole focal length
parameter. Chen et al. [3] autocalibrate the focal length using
two or more coplanar circles. The problem to solve contains
two parameters: one ellipse and the focal length. Then, two
correspondences between a circle and an ellipse are necessary
to estimate the focal length. Based on the same method,
Bergamasco et al. [1] designed a marker composed of small
circles spread on the edge of two or more concentric rings.
The image of each circle is used with a vote system to estimate
the focal length and the image of the external rings.
Two circles on a planar marker (except if one encloses
the other) is the minimum to fully estimate the homography
without any other assumptions. However in some applications
e.g., dealing with concentric circles, detecting the images of
two or more circles can be tricky. First because the lack of
points induces an inaccurate estimation and, secondly because
it is time consuming. When the camera has already been
calibrated, it is possible to compute the homography from one
circle image with two ambiguities. Pagani et al. [9] introduced
a method quite similar to the solution proposed by Chen et
al. [3], where the ambiguity is solved by minimizing a distance
between the image of the marker rectified and the expected
pattern on all possible poses.
III. POSE ESTIMATION FROM THE IMAGE OF ONE CIRCLE
We remind here some geometrical background on the prob-
lem of pose estimation from the image of a single circle.
We consider an Euclidean projective camera, represented by
a3×4-matrix P∼KR I|T, where the rotation matrix
R∈SO(3)1and the translation vector T∈R3describe
the pose of the camera, i.e., respectively its orientation and
position in the object 3D frame. The upper triangular order-3
matrix Kis the calibration matrix as defined in [6, page 157].
Assume that Pis a plane with equation Z= 0 in the
world frame. The pose of Pin the camera frame is given
by the vector [N=r3,−d]>, where r3, the third column of
R, defines the unit norm Nof P, and dis the orthogonal
1SO(3) refers to the 3D rotation group.
distance to P. The restriction to Pof the projection mapping
is an homography whose matrix writes H∼KR e1|e2|T,
where e1and e2are the first two columns of I. In the projective
plane, any conic can be represented in 2D homogeneous coor-
dinates by a real symmetric order-3matrix. Under perspective
projection, any circle of P, assuming its quasi-affine invariance
[6, p515] i.e., that all its points lie in front of the camera,
is mapped under the homography Hto an ellipse by the
projection equation C=H−>QH>, where Q∈Sym3
2is the
circle matrix and C∈Sym3is the ellipse matrix.
For reasons that will become clearer later, we want to
parameterize the homography H,from only the knowledge of
the circle image Cand the vanishing line v∞of P. Let SP
be a similarity on the world plane that puts the circle Qinto
a unit circle centered at the origin and SIbe a similarity on
the image plane Pthat puts Cinto a canonical diagonal form
C0= diag(C0
11, C 0
22, C 0
33). Using an approach similar to [2]
with the notation [u, v, 1]>∼C−1v∞, it can be shown that,
under the assumption of a camera with square pixels, we have
H∼S−1
IMSPwhere
M=

−1C0
22uv −u
0−C0
11u2+ 1 −v
−C0
11u C0
22v1



r0 0
0−1 0
0 0 s


with r= (−C0
22
C0
11
(C0
11u2+C0
22v2+C0
33))1/2(1)
and s=−C0
22(1 −C0
11u2)1/2
Note that the matrices SPand SIcan be completely deter-
mined by the circle image Cand M, except for an unknown
2D rotation around the circle centre on P. Recovering this
rotation is not the goal of this paper. Some simple solution
like placing a visible mark on the edge of the marker works
generally well in many cases.
Our main task will be to recover the vanishing line v∞
of the plane, as explained in the sequel. Note that the vector
xc= [u, v, 1]>defined above is that of the image of the circle
centre which is the pole of v∞w.r.t. the dual ellipse of C.
IV. SUPPORT PLANE’S VANISHING LINE ESTIMATION
We warn the reader that parts written in italics in this section
requires a proof that is not provided due to lack of space.
However all proofs will appear in an extended paper version.
A. A twofold solution in the calibrated case
In the case of calibrated image, an equivalent problem of
computing the pose of its support plane Pis that of recovering
the vanishing line of P. Let Qbe the matrix of a circle on
a plane P, and ψ=H>ωHbe that of the back-projection
onto Pof the image of the absolute conic [6, p. 81], where
ω=K−>K−1.It is easy to show that ψrepresents also a
virtual3circle (as does ω).
2Sym3refers to the space of order-3 real symmetric matrices.
3Virtual conics have positive definite matrices, so, no real points on them.

Let {αi}i=1..3denotes the set of generalized eigenvalues
of the matrix-pair (Q,ψ), i.e., the three roots of the char-
acteristic equation det(Q−αψ)=0. The set of matrices
{Q−αψ}α∈R∪{∞} defines a conic pencil [5] which includes
three degenerate conics with matrices Di=Q−αiψ.
These rank-2 matrices represent line-pairs and have the form
Di=li
a(li
b)>+li
b(li
a)>, where li
aand li
bare vectors of
these lines. Such line-pair matrix Dican be easily decomposed
and vectors of its lines recovered albeit it is impossible to
distinguish li
afrom li
b.It can be shown that the projective
signatures4of the three degenerate members always are (1,1),
(2,0) and (0,2). Assume, without loss of generality, that the
degenerate conic D2is the one with signature (1,1).A first
key result is that D2is a pair of two distinct real lines, one
of which being the line at infinity l∞= [0,0,1]>; the other
one being denoted by lo. The other two degenerate conics D1
and D3—with signatures (2,0) and (0,2)— are pairs of two
conjugate complex lines. Consequently, the three (so-called)
base points xi, where lines in a pair meet, are real. Moreover,
their vectors are the generalized eigenvectors of (Q,ψ)and
satisfy Dixi=0.
Similarly, in the image plane, if Cdenotes the image of
the circle Q, the set of matrices {C−βω}β∈R∪{∞} defines
also a conic pencil whose members are the images of the
pencil {Q−αψ}α∈R∪{∞}. Hence, the line-pair in {C−βω}
that includes the image of l∞i.e., the vanishing line v∞, can
always be identified since it is the only degenerate member
with signature (1,1). Nevertheless, at this step, it is impossible
to distinguish v∞from the other line vo, image of lo.
Assume that all matrices Q,C,ψand ωare normalized
to have a unit determinant. It is known that, in this case,
parameters in pencils satisfy α=β, so, the generalized
eigenvalues of the matrix-pair (Q,ψ)are exactly the same
as those of (C,ω).It can be shown that these eigenvalues
can always be sorted such that λ1≥λ2≥λ3, where
D2=C−λ2ωis the (sole) degenerate conic with signature
(1,1). Remind that D2is the conic which contains v∞plus
vo, which are two a priori indistinguishable lines denoted by
v1,2. Because the matrix D2is real, symmetric, rank-2and
order-3,its generalized eigen-decomposition using the base
point vectors x1,x3∈R3writes as following:
D2=hx1
kx1k
x3
kx3kiλ1−λ20
0λ3−λ2

x>
1
kx1k
x>
3
kx3k

(2)
from which it can be shown that
v1,2=pλ1−λ2
x1
kx1k±pλ2−λ3
x3
kx3k(3)
The two solutions to the normal to Pare given by
N1,2=K>v1,2in the camera frame, and (3) explains the
known doublefold ambiguity in the plane pose [3].
4The signature of a conic is σ(C) = (max(p, n),min(p, n)), where pand
ncount the positive and negative eigenvalues of its (real) matrix C. It is left
unchanged by projective transformations.
B. About removing the twofold ambiguity
We have seen that there are two solutions for the vanishing
line (or the plane normal in the calibrated case) which are in
general not distinguishable. In this section, we discuss whether
known configurations allows the ambiguity to be removed.
We extend the new theoretical framework proposed in §IV-A
that involves the point q(on the support plane P) where the
optical axis cuts Pplus the line Lobtained by intersecting P
and the principal plane5of the camera (Lis orthogonal to the
orthogonal projection of the optical axis onto P). Now, let L0
denote the line parallel to Lthrough the circle centre. Within
this geometrical framework, we can claim, for instance, that
a sufficient condition for the ambiguity to be solved is given
by the two following conditions:
(i) qand the orthogonal projection on Pof the camera
centre lie on the same side of L0;
(ii) the point, intersection of the orthogonal projection on P
of the optical axis and L0, lies outside the circle centered
at qwith same radius as Q.
Figure 1 illustrates this important result. We are convinced
that future investigations using this framework can help to
reveal more configurations in which the ambiguity can be
removed. We are now giving more geometrical insights in-
dciating how to determine such configurations, via three
propositions. The first is the second key result which is the
building brick of our approach:
Proposition 1 (second key result) The line loin D2sepa-
rates the two base points x1and x3. Hence, denoting by ¯x
the normalized vector ¯x =x/x3, the following inequalities
hold (l>
∞¯x1)(l>
∞¯x3)>0and (lo>¯x1)(lo>¯x3)<0.
These two inequalities hold under any affine transformation
but not under a general projective transformation.
How the conditions in proposition 1 can be helpful in
removing the plane pose ambiguity? Can we state a corollary
saying that, in the image plane, under some known geometric
configuration, we know which the line voin C−λ2ω,
image of lo,always separates points z1and z3, images of
base points x1and x3, while the other does not? That is,
if we a priori know sign(v>
o¯z1)(v>
o¯z3)can we guarantee
that (v>
o¯z1)(v>
o¯z3) = −(v>
∞¯z1)(v>
∞¯z3)? If yes, since the
vectors of these base points are the generalized eigenvectors
of (C,ω)associated to parameters λj,j∈ {1,3}and can be
straightforwardly computed, we could remove the ambiguity
by choosing as vanishing line v∞the “correct” line in C−λ2ω.
We claim the following proposition for this corollary to hold,
whose (omitted) proof directly follows from the properties of
quasi-affineness w.r.t. the base points [6].
Proposition 2 When x1and x3lie either both in front or
both behind the camera i.e., on the same half-plane bounded
by L, we have (v>
o¯z1)(v>
o¯z3)<0and (v>
∞¯z1)(v>
∞¯z3)>0.
Otherwise (v>
o¯z1)(v>
o¯z3)>0and (v>
∞¯z1)(v>
∞¯z3)<0.
5The 3D plane through the camera centre and parallel to the image plane.

Now let us investigate a formal condition saying when x1
and x3lie on the same half-plane bounded by L. Consider
an Euclidean representation of the projective world in which
the origin is the point qat which the optical axis cuts the
plane P. Let the X-axis be parallel to the line Land the Y-
axis is the orthogonal projection of the optical axis onto P.
Consequently, the Z-axis is directed by the normal to P, as
shown in figure 1. Let C= [0,−cos θ, sin θ]>,θ∈[0,π
2[,
be the 3D cartesian coordinates of the camera centre, where
π−θis the angle between the Y-axis and the optical axis
in the Y Z-plane (note that we choose the scale such that the
camera centre is at distance 1from the origin). Therefore the
direction of the optical axis is given by −C.
Fig. 1: Proposed parametrization for detecting the ambiguity.
In the 2D representation of the projective plane P(i.e., of
the XZ-plane), let the circle have centre (xc, yc)and radius
R. Let d= [0,1,cos θ]>is the vector of line L. It can be
shown, using a symbolic software like MAPLE6, that:
Proposition 3 Base points x1and x3lie, in the world plane,
on the same side of Lif and only if
cosθ(y2
c−R2)(yc+cosθ)+yccosθ(1+x2
c)+x2
c+y2
c≤0(4)
Since cosθ > 0, if yc > 0and y2
c−R2>0then x1and
x3lie on opposite sides of L. The former inequality says
that qmust lie on the same side of L0, the line parallel
to Lthrough the circle centre, as the orthogonal projection
of the camera centre onto P. The latter inequality says the
point (0, yc)must lie outside the circle centered at q(0,0)
with same radius Ras Q. As we are in the “otherwise” part
of proposition 2, the vanishing line is given by the line that
does not separate the image of the base points. Since (0, yc)
represents the intersection of the orthogonal projection on P
of the optical axis and L0, this is the result announced at the
beginning of this section.
C. Defining default intrinsics for the camera
In the previous sections we have seen that, providing that
the camera intrinsics are known, there is a twofold solution
for the vanishing line. Recovering accurate intrinsics of a
camera requires generally a calibration procedure. In many
applications, the model of the camera can be simplified to
reduce the number of parameters. A very common model is
6https://fr.maplesoft.com/
that of a camera with square pixels and principal point at
the centre of the image plane. Consequently, the focal length
is the sole unknown, e.g., for self-calibration purposes [10].
The focal length value is sometimes available through EXIF
data, stored in digital images or video files, through camera
hardware on top level API (Android, iOS) or through data
provided by manufacturer on websites. Focal length, denoted
f, in pixels (what we need) can be obtained from this data if
we find the field of view in angle or the focal length equivalent
in 35mm. However the focal length is very often given in
millimetre without the sensor size required to obtain the focal
length in pixels.
We consider here the case where it is impossible to calibrate
the camera by none of the methods mentioned above. So
how to do? We propose to design a generic camera model
delivering default intrinsics (i.e., focal length) and based on
off-line calibration of several smartphone cameras. If a camera
can generally take any focal length value, the optics and the
sensor of smartphones are constrained by the device size and
the desired field of view. Why doing that? We found out that
surprisingly enough, that it is not necessary to have very accu-
rate intrinsics to estimate the vanishing line given the image of
a single circle. In fact, as shown in the experimental section V,
this estimation is very robust to intrinsics fluctuation.
After calibrating a dozen of camera devices and obtaining
data from manufacturers of twenty more smartphones, we
estimate a gaussian model of the focal length equivalent
in 35mm, as shown in figure 2. In our case we obtained
Focal 35 mm equivalent (mm)
20 25 30 35 40 45 50
Density values
0
0.02
0.04
0.06
0.08
0.1
0.12
Calibrated camera dataset
Estimation with gaussian distribution
Fig. 2: Focal calibration of different camera parameters
experimentally an average focal length of f35 = 29.95mm
with a variance of σ2
f35 = 11.86. More precisely, we estimate
a gaussian function (in blue) based of the focal values collected
or estimated (in red) from different smartphone device brands.
V. EX PE RI MENTAL R ES ULTS
A. Test Description
The goal of the test presented in this section is to evaluate
the proposed method to estimate the pose of a camera. We
performed those tests on synthetic and real images in the
conditions illustrated in figure 3. In order to limit the poses
used in experiments, we made some hypotheses. First, we
suppose that the camera focus on the centre of the marker,

Fig. 3: Setup: reference chessboard and pose annotation
i.e. the principal axis of the camera passes through the centre
of the marker, see figure 3. Then, the angle γhas been set to
zero. In deed, we can simulate any angle by rotating the image
using the assumption that the principal point is centred on the
image and that the principal axis is orthogonal to the image
plane. Finally, the angle βhas been fixed to zero as estimating
the 2D rotation around the plane normal is out of the scope
of this article. The remaining variables whose variations are
studied in our test are the angle αand the distance r.
We know that introducing generic camera parameter, as
proposed in section IV-C, should have a negative impact on
the accuracy of the pose estimation. Consequently, one of the
objectives of this experiment is to evaluate the sensitivity of
the proposed method to inaccurate camera focal parameter.
The observation of the distribution of focal length of various
smartphone camera, see figure 2, reveals that all 35mm focal
equivalent are included in [−30%,+30%] of the average value.
So, five different values that span this range are used in
the experiment: {0.7,0.85,1.0,1.15,1.3}. In order to generate
synthetic images, we have simulated a synthetic camera of
focal αx= 1280 and resolution of 1280 ×720 pixels. To
obtain real images, we have used the camera of a smartphone
which have been calibrated with openCV library7. In both
cases, we suppose that ellipses have been firstly detected in the
images, i.e. contour points are first detected and then ellipses
are estimated [12]. We try to evaluate the impact of errors of
this estimation to the quality of the results. In consequences,
in our synthetic tests, we have also simulated noises on the
detections of the ellipses, i.e. errors on the pixels that belong to
the ellipse. More precisely, edge points of the ellipse have been
translated with a zero mean gaussian variance of σx= 1.0.
Finally, we evaluate the quality of the results obtained by
using three different measurements relative to the pose and the
reprojection accuracy:
a) Error on the normal of the plane relative to the camera;
b) Error on the position of the marker;
c) Error of reprojection of 3D points close to the marker.
Each curve illustrates the results obtained by applying a mod-
ifier on focal length used for pose estimation. The resulting
errors are displayed as function of the distance rin the interval
[15 ×D, 50 ×D]where Dis the diameter of the marker.
This interval is related to the distances used for being able
to detect and to recognize a marker for an augmented reality
7https://opencv.org/
application, i.e. the distance where the marker occupies, at
least 80 pixels. We also show results for three different angle
values, α∈ {15,30,45}, displayed in three sub-figures.
B. Analysis of the results
Results on synthetic images are presented in figure 4. In 4a,
we show the error on the estimation of the orientation for
the pose. We can notice that as the distance of the marker
to the camera increases, the error on pose orientation also
increases. This relation is even more remarkable when the
angle is the lowest between the marker plane and the camera,
i.e. the graph on the left. In 4b, we can see that in the calibrated
case the accuracy in position stays low and does not depend
on the distance to the camera and the angles between the
marker plane and the camera. In the uncalibrated cases, as
expected the detection of the ellipses becomes less accurate
when the distance increases and, consequently, the quality
of the estimation of the marker position is also affected. In
fact, the error in position increases linearly when the distance
increases. This observation is quite intuitive. In deed observing
a marker with a zoom or taking its image closer leads to very
similar shape of the marker. The error on the reprojection of
3D points, presented in 4c, illustrates that, with a focal length
well estimated, the higher the distance, the higher the errors.
Whereas, when the focal length is not well estimated, the
higher the distance, the lower the error and, more important,
this error is quite near the error obtained when the focal length
is correctly estimated. It means that using generic parameter is
not affecting the quality of the reprojection in a context where
the marker is far from the camera.
The figure 5 allows us to present similar conclusions on
real images. The 3D point reprojection error is presented. The
error in calibrated case slightly increases with the distance
as observed in figure 4a. When the marker is close to the
camera, the error of reprojection when the camera is not
correctly calibrated is high but it drastically decreases when
the distance to the camera increases, and, finally, this error is
of the same order as that obtained with the calibrated case.
This observation is not really a surprise as the projection of a
distant object loses its perspective with distance. Again, this
result illustrates the interest of using generic camera parameter
in augmented reality.
VI. CONCLUSION
In this paper, we introduced a method to estimate the
pose of a camera from the image of a circular marker in a
calibrated case. If, in general case, two solutions are found,
some assumptions on geometric configuration can help to
distinguish the correct pose. Moreover, we demonstrated the
interest of using default camera parameters, in the context of
augmented reality. In particular, the results presented showed
that, in a case of a distant marker, the 3D reprojection errors
is low enough. Future work would be to use more information
in the marker environment to increase the stability of the
detection of the marker and the pose estimation and to allow
decoding from longer distance.

15 20 25 30 35 40 45 50
normal error (degree)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 alpha = 15°
distance d (circle diameter unit)
15 20 25 30 35 40 45 50
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 alpha = 30°
focal -30%
focal -15%
focal calibrated
focal +15%
focal +30%
15 20 25 30 35 40 45 50
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 alpha = 45°
(a) Orientation of the normal
15 20 25 30 35 40 45 50
position error (circle diameter unit)
0
5
10
15
20
25
30
distance d (circle diameter unit)
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0
5
10
15
20
25
30
15 20 25 30 35 40 45 50
0
5
10
15
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25
30
(b) Position of the marker
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3D points reprojection error (pixel)
1
1.5
2
2.5
3
3.5
4
4.5
5
distance d (circle diameter unit)
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1
1.5
2
2.5
3
3.5
4
4.5
5
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1
1.5
2
2.5
3
3.5
4
4.5
5
(c) Reprojection of 3D points
Fig. 4: Error with synthetic images.
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3D points reprojection error (pixel)
0
1
2
3
4
5
6
7
8alpha [11,15] °
distance d (circle diameter unit)
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0
1
2
3
4
5
6
7
8alpha [32,36] °
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0
1
2
3
4
5
6
7
8alpha [45,47] °
Fig. 5: Error with ellipses detected on real image: 3D points reprojection.
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