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Paracatadioptric Geometry using Conformal Geometric Algebra

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Paracatadioptric Geometry using Conformal Geometric Algebra

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In this work a comprehensive geometric model for paracatadioptric sensors has been presented. The model is based on the equivalence between paracatadioptric projection and the inversion. The main reason for using the inversion is that it can be represented by a versor (i.e. a special group of multivectors) in the CGA. The advantage of this representation is that it can be applied not only to points but also to point-pairs, lines, circles, spheres and planes.
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Paracatadioptric Geometry using
Conformal Geometric Algebra
Carlos López-Franco
Computer Science Department, University of Guadalajara,
Mexico
1. Introduction
Computer vision provides non-contact measurements of the world, extending the robot
ability to operate in circumstances and environments which can not be accurately
controlled. The use of visual observations to control the motions of robots has been
extensively studied, this approach is referred in literature as visual servoing.
Conventional cameras suffer from a limited field of view. One effective way to increase the
field of view is to use mirrors in combination with conventional cameras. The approach of
combining mirrors with conventional cameras to enhance sensor field of view is referred as
catadioptric image formation.
In order to be able to model the catadioptric sensor geometrically, it must satisfy the
restriction that all the measurements of light intensity pass through only one point in the
space (effective viewpoint). The complete class of mirrors that satisfy such restriction where
analyzed by Baker and Nayar [1]. In [2] the authors deal with the epipolar geometry of two
catadioptric sensors. Later, in [3] a general model for central catadioptric image formation
was given. Also, a representation of this general model using the conformal geometric
algebra was showed in [4].
Visual servoing applications can be benefit from sensors providing large fields of view. This
work will show how a paracatadioptric sensor (parabolic mirror and a camera) can be used
in a visual servoing task for driving a nonholonomic mobile robot.
The work is mainly concerned with the use of projected lines extracted from central
catadioptric images as input of a visual servoing control loop. The paracatadioptric image of
a line is in general a circle but sometimes it could be a line. This is something that should be
taken into account to avoid a singularity in the visual servoing task.
In this work we will give a framework for the representation of image features in parabolic
catadioptric images and their transformations. In particular line images in parabolic
catadioptric images are circles. While of course conics and therefore circles can be
represented in the projective plane we will provide a much more natural representation
utilizing the conformal geometric algebra (CGA).
In CGA the conformal transformations are linearized using the fact that the conformal
group on Rn is isomorphic to the Lorentz group on Rn+1. Hence nonlinear conformal
transformations on Rn can be linearized by representing them as Lorentz transformations
and thereby further simplified as versor representation. These versors can be applied not
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only to points but also to all the CGA entities (spheres, planes, circles, lines and point-pairs).
In this model a b c represents the circle thought the three points. If one of these points is a
null vector e representing the point at infinity, then a b e represents the straight line
trough a and b as a circle through infinity. This representation could not be available
without the concept of null vector.
In contrast with other catadioptric sensors the paracatadioptric sensors have certain
properties that make them very interesting. One important property is that paracatadioptric
projection is conformal, this motivate us to use the conformal geometric algebra to represent
it. Thus the paracatadioptric projection can be expressed linearly using versors of CGA. The
advantage of our framework is that the projection can be applied to circles and lines in the
same way as it does for points. That advantage has a tremendous consequence since the
nonlinearity of the paracatadioptric image has been removed. As a result the input features
for the visual servoing control loop can be handled effectively in order to design an efficient
vision-based control scheme.
The rest of this paper is organized as follows: The next section will give a brief introduction
to the conformal geometric algebra. In section 3 we show the equivalence between
inversions on the sphere and the parabolic projections. Then, in section 4 we show the
paracatadioptric image formation using the proposed framework. Later, in section 5 an
application of a paracatadioptric stereo sensor is given. Finally, the conclusions are in
section 6.
2. Conformal geometric algebra
In general, a geometric algebra [5] Gn is a n-dimensional vector space Vn over the reals R.
The geometric algebra is generated defining the geometric product as an associative and
multilinear product satisfying the contraction rule
(1)
where ε is 1, 0 or 1 and is called the signature of ei. When ei 0 but its magnitude |ei| is
equal to zero, ei is called null vector.
The geometric product of a geometric algebra Gp,q,r for two basis ei and ej is defined as
(2)
Thus, given a n-dimensional vector space Vn with an orthonormal basis {e1, e2, ...en} its
corresponding geometric algebra is generated using the geometric product. We can see that
for a n-dimensional vector space, there are 2n ways to combine its basis using the geometric
product. Each of this product is called a basis blade. Together they span all the space of the
geometric algebra Gn.
We also denote with Gp,q a geometric algebra over V p,q where p and q denote the signature of
the algebra. If p 0 and q = 0 the metric is euclidean Gn, if non of them are zero the metric is
Paracatadioptric Geometry using Conformal Geometric Algebra
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pseudoeuclidean. A generic element in Gp,q is called a multivector. Every multivector M can
be written in the expanded form
(3)
where represents the blade of grade i. The geometric product (denoted juxtaposition)
of vectors is defined with a scalar part called the inner product and an outer product, which
is a non scalar quantity, for example the outer product of a and b is
(4)
The conformal geometric algebra [5] is the geometric algebra over an homogeneous
conformal space. This framework extends the functionality of projective geometry to include
circles and spheres. Furthermore, the dilations, inversions, rotations and translations in 3D
becomes rotations in the 5D conformal geometric algebra. These transformations can not
only be applied to points or lines, but to all the conformal entities (points, lines, planes,
point pairs, circles and spheres).
In the conformal geometric algebra (CGA) we firstly add two extra vector basis e+ and e to
our R3 Euclidean space {e1, e2, e3, e, e+}, where = 1 and = 1. We denote this algebra
with G4,1 to show that four basis vectors square to +1 and one basis vector square to 1. In
addition we define
(5)
we note that they are null vectors since . The vector e0 can be interpreted as
the origin of the coordinate system, and the vector e as the point at infinity. A few useful
properties (easily proved) about this basis are
(6)
Where E = e+ e is called the Minkowski plane.
To specify a 3-dimensional Euclidean point in an unique form in this 5-dimensional space,
we require the definition of two constraints. The first constraint is that the representation
must be homogeneous, that is X and X represent the same point in the Euclidean space.
The second constraint requires that the vector X be a null vector (X2 = 0). The equation that
satisfies these constraints is
(7)
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where x Rn and X Nn+1. Note that this is a bijective mapping. From now and in the rest
of the paper the points X are named conformal points and the points x are named Euclidean
points. A conformal point (7) lies on the intersection of the null cone Nn+1 and the
hyperplane Pn+1(e, e0), that is
(8)
A sphere on the CGA can be defined as
(9)
To test for incidence of a point X with an entity S expressed in the inner product null space
(IPNS) we use
(10)
A point in the CGA is represented using (9), but setting the radius r = 0 we get (7). Also note
that if x = 0 in (7) we get the point e0 of (5) corresponding to the origin of Rn. One interesting
thing about the conformal points is that their inner product
(11)
is a directly representation of the Euclidean distance between the two points. Thus, the inner
product has now a geometric meaning due to the concept of null vectors. Therefore, the
square of a conformal point X2 = XX = X ·X +X X = X ·X = 0 represents the Euclidean
distance with itself.
In the CGA two multivectors represent the same object if they differ by just a non zero scalar
factor, that is
(12)
Thus, multiplying (9) with a scalar λ we get
(13)
if we calculate the inner product of the above equation with respect to the point at infinity e
(14)
Paracatadioptric Geometry using Conformal Geometric Algebra
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we have recover the scalar factor λ. Therefore, if we want to normalize a sphere S (i.e. λ = 1)
we apply
(15)
The same equation can be applied to points, remember that they are nothing but spheres of
zero radius. For other objects the common normalization is by its magnitude that is
(16)
where represents the i-vector part of the multivector M.
An important operation that is used in the geometric algebra is called reversion, denoted by
”~” and defined by
(17)
2.1 Geometric objects representation
The geometric objects can also be defined with the outer product of points that lie on the
object. For example, with four points we define the sphere
(18)
The incidence of a point X with the sphere S* expressed in the outer product null space
(OPNS) is
(19)
Both representation of the sphere (S and S*) are dual to each other, i.e. orthogonal to each
other in the representative conformal space. Therefore, the representations are equivalent if
we multiply them by the pseudoscalar Ic = e1 e2 e3 e+ e, thus
(20)
If one of the points of the sphere (18) is the point at infinity, then the sphere becomes a plane
(a flat sphere with infinite radius)
(21)
Similarly, the outer product of three points defines a circle C* = A B C, and if one of the
points of the circle is at infinity (C* = AB e) then the circle becomes a line (a flat circle
with infinite radius), see (Table 1). The line can also be defined as the outer product of two
spheres
(22)
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which is the line passing through the centers of the spheres S1 and S2. If instead of two
spheres we take one sphere and a point
(23)
then the line passes through the center of the sphere S and point X.
A complete list of the entities and their representations are given in (Table 1). The table
shows a geometric object called point-pair, which can be seen as the outer product of two
points
(24)
The point-pair represents a 1-dimensional sphere, and it can be the result of the intersection
between: a line and a sphere, two circles or three spheres. In addition, if one of the points is
the point at infinity
(25)
then we get a special point-pair which is called flat-point. If the intersection between two
lines exists, then they intersect in one point X and also at the point at infinity. It also can be
the result of the intersection between a line and a plane.
Table 1. Entities in conformal geometric algebra
2.2 Conformal transformations
A transformation of geometric objects is said to be conformal if it preserves angles. Liouville
was the first that proved that any conformal transformation on Rn can be expressed as a
composite of inversions and reflections in hyperplanes. The CGA G4,1 allows the computation
of inversions and reflections with the geometric product and a special group of multivectors
called versors.
2.3 Objects rigid motion
In conformal geometric algebra we can perform rotations by means of an entity called rotor
which is defined by
(26)
Paracatadioptric Geometry using Conformal Geometric Algebra
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where l is the bivector representing the dual of the rotation axis. To rotate an entity, we
simply multiply it by the rotor R from the left and the reverse of the rotor from the right,
If we want to translate an entity we use a translator which is defined as
(27)
With this representation the translator can be applied multiplicatively to an entity similarly
to the rotor, by multiplying the entity from the left by the translator and from the right with
the reverse of the translator:
Finally, the rigid motion can be expressed using a motor which is the combination of a rotor
and a translator: M= TR, the rigid body motion of an entity is described with
For more details on the geometric algebra and CGA, the interested reader is referred to view
[5–9].
3. Paracatadioptric projection and inversion
In this section we will see the equivalence between the paracatadioptric projection and the
inversion. The objective of using the inversion is that it can be linearized and represented by
a versor in the conformal geometric algebra. This versor can be applied not only to points
but also to point-pairs, lines, circles, spheres and planes.
The next subsection shows the equivalence between the paracatadioptric projection and the
inversion, later the computation of the inversion using will be shown.
3.1 Paracatadioptric projection
The parabolic projection of a point is defined as the intersection
of the line xf (where f is the parabola’s focus) and the parabolic mirror, followed by an
orthographic projection. The orthographic projection is to a plane perpendicular to the axis
of the mirror.
The equation of a parabola with focal length p, whose focus is at the origin is
(28)
The projection of the point x to the mirror is
(29)
where λ is defined as
(30)
Finally, the point xp is projected onto a plane perpendicular to the axis of the parabola. The
reason for this is that any ray incident with the focus is reflected such that it is perpendicular
to the image plane.
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3.2 Relationship between inversion and parabolic projection
The inversion of a point x G3 with respect to sphere centered at the origin, and radius r, is
a point x G3 lying on the line defined by the point x and the origin, if x’ is the inverse of
the point x then
(31)
When the sphere is centered at the point c G3, the inverse of the point x is defined as
(32)
As we already mention, the parabolic projection of a point x can be found with (30). Now,
given a sphere centered at the origin with radius p, see Fig. 1. The projection of the point x
on the sphere is simply
(33)
Fig. 1. Parabolic projection.
Note that there are three similar triangles in Fig. 1, which can be seen clearly in Fig. 2. The three
triangles are : NSQ, NPS and SPQ. Therefore
which is the exactly the inversion of a point with respect to a circle centered at N and radius
Thus we have that
(34)
Thus, the parabolic projection of a point x is equivalent to the inversion of a point , where
the point lies on a sphere s centered at the focus of the parabola and radius p, with respect to
a sphere s0, centered at n and radius 2p, see Fig. 3. To prove this equivalence we can use the
definitions of the parabolic projection and inversion. With the definition of inversion (31) we
have the following equation
(35)
where n = pe3 G3.
(36)
Paracatadioptric Geometry using Conformal Geometric Algebra
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Fig. 2. The three similar triangles of the stereographic projection
thus, the projected point is
(37)
The constant is equal to
(38)
which is exactly the same value of the scalar λ from the parabolic projection (30). Therefore,
we can conclude that the parabolic projection (29) and the inversion (37) of the point x are
equivalent.
Fig. 3. Equivalence between parabolic projection and inversion
3.3 Inversion and the conformal geometric algebra
In the conformal geometric algebra, the conformal transformations are represented as
versors [7]. In particular, the versor of the inversion is a sphere, and it is applied in the same
way as the rotor, or the translator. Given a sphere of radius r centered at c represented by
the vector
(39)
the inversion of a point X with respect to S is
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(40)
To clarify the above equation, let us analyze the special case when S is a unit sphere,
centered at origin. Then S reduces to
(41)
thus (40) becomes
(42)
The first term e+xe+ is equal to
(43)
The term is equivalent to
(44)
Finally, the last term is
(45)
Rewriting (42) we have
(46)
From the above equation we recognize the Euclidean point
(47)
which represents the inversion of the point x. The case of the inversion with respect to an
arbitrary sphere is
(48)
where f(x) is equal to (37), the inversion in Rn. The value σ represents the scalar factor of the
homogeneuos point.
The interesting thing about the inversion in the conformal geometric algebra is that it can be
applied not only to points but also to any other entity of the CGA. In the following section
we will see how the paracatadioptric image formation can be described in terms of the CGA.
Paracatadioptric Geometry using Conformal Geometric Algebra
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4. Paracatadioptric image formation and conformal geometric algebra
In the previous section we saw the equivalence between the parabolic projection and the
inversion. We also saw how to compute the inversion in the CGA using a versor, in this case
the versor is simply the sphere where the inversion will be carried out. In this section we
will define the paracatadioptric image formation using the CGA.
Given a parabolic mirror with a focal length p, the projection of a point in the space through
the mirror followed by an orthographic projection can be handled by two spheres. Where
the first sphere is centered at the focus of the mirror, and its radius is p. This sphere can be
defined as
(49)
The second sphere S0 can be defined in several ways, but we prefer to define it with respect
to a point N on the sphere S (i.e. N · S = 0). If we compare the point equation (7) with the
sphere equation (9), we can observe that the sphere has an extra term thus if we
extract it to the point N we get a sphere centered at N and with radius r. The sphere S0 is
defined as
(50)
where 2p is the radius of the sphere. With these two spheres the image formation of points,
circles and lines will be showed in the next subsections.
4.1 Point Images
Let X be a point on the space, its projection to the sphere can be found by finding the line
passing through it and the sphere center, that is
(51)
then, this line is intersected with the sphere S
(52)
Where Zs is a point-pair the paracatadioptric projection of this point can
be found with
(53)
which is also a point pair The paracatadioptric projection of the point
closest to X, can be found with
(54)
and then with
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(55)
The point Xc is the projection of the point X onto the catadioptric image plane, which is
exactly the same point obtained through a parabolic and orthographic projections (Fig. 4).
Fig. 4. Point projection onto the catadioptric image plane
4.2 Back projection of point images
Given a point Xc on the catadioptric image (Fig. 4), its projection to the sphere is simply
(56)
Since the point Xs and the sphere center lie on the line , it can be calculated as
(57)
The original point X lies also on this line, but since we have a single image the depth can not
be determined and thus the point X can no be calculated.
4.3 Circle images
The circle images can be found in the same way as for points images. To see that, let
X1,X2,X3 be three points on the sphere S, the circle defined by them is
(58)
which can be a great or a small circle. The projection of the circle onto the catadioptric image
is carried out as in (55)
(59)
Where could be a line, but there is no problem since it is represented as a circle with one
point at infinity.
The back projection of a circle (or line) that lie on the catadioptric image plane, can be
found easily with
Paracatadioptric Geometry using Conformal Geometric Algebra
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(60)
In Fig. 5 the projection of circles on the sphere to the catadioptric image plane is shown.
Fig. 5. Projection of circles on the sphere.
4.4 Line images
To find the paracatadioptric projection of a line L* in the 3D space (Fig. 59), we first project
the line to the sphere S. The plane defined by the line L* and the sphere S is
(61)
then the projection of the line L* onto the sphere is
(62)
where Cs is a great circle. Finally the paracatadioptric projection of L* can be found with the
inversion of the circle Cs, that is
(63)
Fig. 6. Projection of a line in the space
5. Robot control using paracatadioptric lines
The problem to be solved is the line following with a mobile robot. The mobile robot is a
nonholonomic system with a paracatadioptric system. We assume that the camera optical
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axis is superposed with the rotation axis of the mobile robot. Thus, the kinematic screw is
only composed with a linear velocity v along the e1 axis and an angular velocity ω.
The problem will be solved using paracatadioptric image of lines. One of those lines is the
paracatadioptric image of the desired line Cd and the other one is the current
paracatadioptric image of the tracked line C. These lines will be projected to the sphere and
then to a perspective plane Πp, in this planes the image projection are straight lines. Finally,
with the lines on the perspective plane we will compute the angular and lateral deviations.
Consider the paracatadioptric image Cd of the desired 3D line the inverse
paracatadioptric projection of Cd can be found with
(64)
Then the plane where the circle lies is defined as
(65)
Finally, the intersection of the plane with the perspective plane is
(66)
this line is the projection of the paracatadioptric image line into the perspective plane
. The perspective plane can be defined as
(67)
where and
(68)
The expression S Se represents the line passing through the centers of both spheres (S
and S0). The value of the scalar δ can be defined arbitrarily.
The current paracatadioptric line C can be projected into the line L in the perspective plane
in similar way using the above equations, see Fig. 7.
Fig. 7. a) Paracatadioptric projection of the desired and current line. b) Projection of the
paracatadioptric lines into the perspective plane.
The lines and , on the perspective plane, define a rotor which can be computed with
(69)
Paracatadioptric Geometry using Conformal Geometric Algebra
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where represents the geometric product of the two lines. The angle between the lines
is then
(70)
which represents the angular deviation. The lateral deviation can be found with the signed
distance between the lines, the signed distance between the lines is
(71)
The angular and lateral deviations are used in a dynamic controller ,proposed in [10], to
generate the robot’s angular velocity. The dynamic controller is
(72)
where the control gains are defined as
(73)
(74)
The value of is left free to specify faster or slower systems, and where ξ is usually set to
1/2. The trajectories of the paracatadioptric images and the current paracatadioptric line
are show in Fig. 8. These trajectories confirm that task is correctly realized. In Fig. 9 the angular
an lateral deviations of the current paracatadioptric image with respect to the desired image
are shown. These figures show that both deviations are well regulated to zero.
a) b)
Fig. 8. a) Tracked line in the paracatadioptric image. b) Trajectory of the projected lines in
the paracatadioptric image.
6. Conclusions
In this work a comprehensive geometric model for paracatadioptric sensors has been
presented. The model is based on the equivalence between paracatadioptric projection and the
inversion. The main reason for using the inversion is that it can be represented by a versor (i.e.
a special group of multivectors) in the CGA. The advantage of this representation is that it can
be applied not only to points but also to point-pairs, lines, circles, spheres and planes.
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The paracatadioptric projection and back-projection of points, point-pairs, circles, and lines is
simplified using the proposed framework. This will allow an easier implementation of
paracatadioptric sensors in more complex applications.
Fig. 9. a) Angular deviation. b) Lateral Deviation.
The proposed framework has been used to control a nonholonomic robot, with a
paracatadioptric sensor. The input to the control scheme are the paracatadioptric images of
the desired and current lines. With help of the proposed model the paracatadioptric images
are back projected to sphere, and then projected to a perspective plane. Then, the lines on
the perspective plane are used to compute the angular and lateral deviations. Finally, with
these values the angular velocity of the robot can be computed using a dynamic controller.
The application showed that is not necessary to have 3D measurements of the scene to solve
the task, indeed it is possible to solve it from image data only.
7. References
Baker, S., Nayar, S.: A theory of catadioptric image formation. In Proc. Int. Conf. on
Computer Vision (1998) 35–42
Svoboda, T., P.T., Hlavac, V.: Epipolar geometry for panoramic cameras. In Proc. 5th
European Conference on Computer Vision (1998) 218–231
Geyer, C., Daniilidis, K.: A unifying theory for central panoramic systems and practical
implications. Proc. Eur. Conf. on Computer Vision (2000) 445–461
Bayro-Corrochano, E., L´opez-Franco, C.: Omnidirectional vision: Unified model using
conformal geometry. Proc. Eur. Conf. on Computer Vision (2004) 318–343
Hestenes, D., Li, H., Rockwood, A.: New algebraic tools for classical geometry. In Sommer,
G., ed.: Geometric Computing with Clifford Algebra. Volume 24., Berlin
Heidelberg, Springer-Verlag (2001) 3–26
Bayro-Corrochano, E., ed.: Robot perception and action using conformal geometric algebra.
Springer- Verlag, Heidelberg (2005)
Li, H., Hestenes, D.: Generalized homogeneous coordinates for computational geometry. In
Sommer, G., ed.: Geometric Computing with Clifford Algebra. Volume 24., Berlin
Heidelberg, Springer-Verlag (2001) 27–60
Rosenhahn, B., Gerald, S.: Pose estimation in conformal geometric algebra. Technical Report
Number 0206 (2002)
Perwass, C., Hildenbrand, D.: Aspects of geometric algebra in euclidean, projective and
conformal space. Technical Report Number 0310 (2003)
Canudas de Wit, E., S.B., Bastian, G., eds.: Theory of Robot Control. Springer-Verlag (1997)
Geyer, C., Daniilidis, K.: Paracatadioptric camera calibration. IEEE Transactions on Pattern
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Catadioptric sensors refer to the combination of lens-based devices and reflective surfaces. These systems are useful because they may have a field of view which is greater than hemispherical, providing the ability to simultaneously view in any direction. Configurations which have a unique effective viewpoint are of primary interest, among these is the case where the reflective surface is a parabolic mirror and the camera is such that it induces an orthographic projection and which we call paracatadioptric. We present an algorithm for the calibration of such a device using only the images of lines in space. In fact, we show that we may obtain all of the intrinsic parameters from the images of only three lines and that this is possible without any metric information. We propose a closed-form solution for focal length, image center, and aspect ratio for skewless cameras and a polynomial root solution in the presence of skew. We also give a method for determining the orientation of a plane containing two sets of parallel lines from one uncalibrated view. Such an orientation recovery enables a rectification which is impossible to achieve in the case of a single uncalibrated view taken by a conventional camera. We study the performance of the algorithm in simulated setups and compare results on real images with an approach based on the image of the mirror's bounding circle
Conference Paper
Omnidirectional vision systems can provide panoramic alertness in surveillance, improve navigational capabilities, and produce panoramic images for multimedia. Catadioptric realizations of omnidirectional vision combine reflective surfaces and lenses. A particular class of them, the central panoramic systems, preserve the uniqueness of the projection viewpoint. In fact, every central projection system including the well known perspective projection on a plane falls into this category. In this paper, we provide a unifying theory for all central catadioptric systems. We show that all of them are isomorphic to projective mappings from the sphere to a plane with a projection center on the perpendicular to the plane. Subcases are the stereographic projection equivalent to parabolic projection and the central planar projection equivalent to every conventional camera. We define a duality among projections of points and lines as well as among different mappings. This unification is novel and has a a significant impact on the 3D interpretation of images. We present new invariances inherent in parabolic projections and a unifying calibration scheme from one view. We describe the implied advantages of catadioptric systems and explain why images arising in central catadioptric systems contain more information than images from conventional cameras. One example is that intrinsic calibration from a single view is possible for parabolic catadioptric systems given only three lines. Another example is metric rectification using only affine information about the scene.
Aspects of geometric algebra in euclidean, projective and conformal space Paracatadioptric camera calibration
  • C Perwass
  • D Hildenbrand
  • C Geyer
  • K Daniilidis
Perwass, C., Hildenbrand, D.: Aspects of geometric algebra in euclidean, projective and conformal space. Technical Report Number 0310 (2003) Canudas de Wit, E., S.B., Bastian, G., eds.: Theory of Robot Control. Springer-Verlag (1997) Geyer, C., Daniilidis, K.: Paracatadioptric camera calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence 24 (2002) 687–695