3

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

Open Access Database www.i-techonline.com

Source: Computer Vision, Book edited by: Xiong Zhihui,

ISBN 978-953-7619-21-3, pp. 538, November 2008, I-Tech, Vienna, Austria

Computer Vision

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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* = A∧B ∧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)