1. Introduction
Clifford algebra is a universal geometric language that introduces powerful mathematical tools enabling computational efficiency in designing and manipulating geometric objects and developing new insights within a unified algebraic framework. It has been widely used in different fields of mathematics [
1,
2,
3,
4,
5,
6], in theoretical physics [
7,
8,
9,
10], and in computer science [
11,
12,
13,
14,
15,
16,
17,
18,
19]. It was also used in many applications in image processing particularly to extend the usual definitions of Fourier transform to color images and more generally to multi-channel images; see [
20].
In computer vision, many previous works dealt with developing geometric models of an image [
21,
22,
23,
24,
25,
26,
27,
28,
29]. In the work shown in [
28,
30], the authors described invariant detectors and descriptors under viewpoint changes of the same planar object. Nevertheless, the algorithms that they developed were useful only when the viewpoint changes are the affine transformations and the plane displacements, and not efficient when the object has undergone a perspective distortion. In fact, the transformations that fully describe every viewpoint change are the homographies of the real projective plane
and cannot be reduced to the affine transformations of the Euclidean plane. In particular, the perspective distortions are the main viewpoint changes that should be considered; they do not preserve the parallelism, therefore they are not affine. These transformations are parameterized by the latitude angle of the camera, which is the parameter that produces the perspective deformations of the object. Moreover, powerful modeling needs necessarily to take into consideration the points at infinity.
In this paper, we consider the conformal Clifford algebras ([
31,
32]) as the main framework for our image and viewpoint modelings. These algebras correspond to the vector space
equipped with the Minkowski metric. The crucial point is that this formalism enables us to deal with the points at infinity in a vectorial way. We use the conformal model since it is a linearization of the classical homogenization of the projective model. The Euclidean plane is embedded into a vector space of dimension 4, where two vectors encode the origin and the infinity of the plane. These two vectors are isotropic for the Minkowski metric and the image of the embedding is a horosphere. Our main goal here is to introduce an invariant by the change of the image viewpoints. For this, we use a slightly modified version of our conformal image representation setting that is fully described in our previous work [
33]. We apply a result of [
34] where for every surface of the Minkowski space
, we identify an invariant functional that remains constant under the conformal transformations of the ambient Minkowski metric. We explain the conformal complex model of the image described in [
26] that makes use of the homographies (also called Möbius transformations) of the complex projective line
in order to encode the viewpoint changes. We illustrate how these transformations preserve the generalized circles of the plane not the affine straight lines, and therefore they are not adapted for the modeling of the viewpoint changes.
In this paper, we also deal with a scene observed by a pinhole camera, thus the viewpoint changes are the transformations that preserve the straight lines, namely the homographies of the real projective plane. In fact, these transformations preserve the real projective lines
where
d is an affine straight line of
and
w is the point at infinity representing its direction. In this setting, two parallel lines,
and
, in the real projective plane associated with the object are sent to two projective lines
and
such that their directions
and
are not necessarily equal. In this case, the affine lines
and
are not parallel in the image plane; they have a point of intersection in
called a vanishing point. This is due to a perspective distortion of the object and is caused by a non-zero latitude parameter of the camera, which is a slanted view of the object. The set of vanishing points is a line called the vanishing line. Two lines in the image plane that intersect at the vanishing line correspond to two parallel lines of the object plane and of any frontal view of the object. The perspective distortions are fully described in our previous work [
33].
This paper is organized as follows. In
Section 2, we give a quick overview of some image and viewpoint change representations: the basic homographies of the real projective plane, their restrictions to the affine geometry, and finally, the homographies of the complex projective line. In
Section 3, this is followed by a review of some basic definitions of the Clifford algebras and the conformal model of the Euclidean plane. We also describe the spinor representations of some basic Möbius transformations. In
Section 4, we describe our image and viewpoint change representations in the projective geometry and the conformal Clifford algebra and introduce our viewpoint change modeling as a conformal transformation of the disjoint union of horospheres. Finally, in
Section 5, we apply a theorem of [
34] in order to obtain a conformal viewpoint invariant.
2. Quick Overview of Some Image Representations
The projective geometry is the main framework where the perspective can be described. The real projective plane
is the quotient space, as follows:
where the equivalence relation
is defined by the following:
We denote by
the canonical surjection and
the homogeneous coordinates defined up to a non-zero multiplicative scalar, such that at least one of the
X,
Y, or
Z is nonzero.
Every linear mapping A of the group of the linear mappings of induces a mapping of the projective plane by the formula: . Since the mappings A and (for ) induce the same mappings of the real projective plane, the group of the homographies of (called the projective group) is defined as the quotient of the group by the equivalence relation .
One of the reasons behind the use of these homographies in computer vision is that they preserve straight lines. This means that the image of a projective line of the real projective plane by a homography is a projective line. In
, these are the unique bijections that preserve the straight lines. More details about the projective geometry can be found in [
22,
35].
2.1. The Pinhole Camera Model
This is the basic camera model that is the most used in computer vision. The geometry where it is described is precisely the one of the real projective plane
. It mainly describes the perspective projection that sends a point of the object plane to the image plane (see
Figure 1).
The points of the object plane are expressed in the frame of the space denoted by . To the camera, another space frame is associated and is denoted by such that its origin C is the optical center of the camera. These two frames are related by an isometry of , which is the composition of a rotation and a translation. It is parameterized by the 6 extrinsic camera parameters.
Every point M of the object plane corresponds to point m of the image plane, which is the intersection of the image plane with the line (C is the optical center of the camera). The perspective projection of M is described by the coordinates of m in the camera frame.
2.2. Image and Real Projective Geometry
A grayscale image can be defined in real projective geometry as a mapping, as follows:
such that
D is a domain of the real projective plane where the image plane is embedded. The group representing the viewpoint changes is the group
of the homographies of
. This latter group encodes well the perspective distortions of the object caused by the different viewpoints of the camera.
Figure 2 illustrates the action of the homography of
represented by the following matrix:
The use of this group to find invariants (detectors and descriptors) by viewpoint changes is not easy in general. One way to overcome this difficulty consists of the restriction to the affine Euclidean plane of the action of the group of homographies of the projective plane . Since the sub-group of homographies that keeps the hyperplane at infinity of globally invariant is isomorphic (by restriction) to the affine group of , then a homography of preserves the affine plane if and only if it is reduced to an affine transformation of .
2.3. The Affine Model of the Camera
It is a simplified version of the projective model that has various applications, for example in [
22,
28].
As previously explained, an affine transformation from
to itself is the restriction to
of a homography of
defined by a
matrix given by the following:
It is actually the composition of a linear transformation of
with a translation. In particular, the group
is of dimension 6; however, the group
is of dimension 8. It is known that every affine transformation of
changes two parallel lines into parallel lines (see
Figure 3).
In [
28], the authors described an image
I of a planar object by means of a frontal image
of the object, through the following formula:
where
A is an affine transformation. This is justified by the fact that every homography of
is locally approximated by an affine transformation of
. In fact, if
k is a homography, we can apply the first-order Taylor–Young formula and we obtain the following:
where
is the differential function of
k at
. The affine transformation
A is then a good approximation of
k around the point
.
By restricting the viewpoint changes to the affine transformations, by considering the group
instead of the entire projective group
, the authors of [
28] developed an algorithm called Affine-SIFT to find invariants by the affine transformations of the plane. But it appears that this algorithm is very sensitive to the perspective distortions of the objects.
2.4. The Conformal Complex Model
In this section, we describe an alternative approach to the previous one, which involves the compactification of the Euclidean plane by
and the use of the group
for the representation of the viewpoint changes. Here, we consider the following compactification:
and the corresponding group
(see [
26,
36]). The complex projective line
is homeomorphic to the Alexandroff compactification
of the plane
. Recall that this compactification can be described by the projection,
P, called stereographic projection (see
Figure 4); the image of a point
of the sphere
(
N is the north pôle ) is the unique point of intersection
x of the line
with the plane. In order to explicitly calculate
, we consider a point
and we search for a point on the line
that belongs to the sphere. We easily obtain the following:
When the magnitude of
x tends to infinity, we obtain
. Thus, the north pôle encodes the infinity of the plane.
The complex projective line
is a complex manifold of dimension 1 whose charts are given by the following:
A homography of
is represented, up to the multiplication by a non-zero complex number, by a matrix of
or equivalently, by enabling
(considering the chart
), via the following mapping:
where
and
.
In [
26], the author described a complex conformal model of the camera that enabled describing the viewpoint changes via the Möbius transformations of
, i.e., the homographies of
. In this model, an image is a mapping, i.e.,
where
is a reference image describing the planar object and
.
These Möbius transformations are conformal transformations, meaning that they preserve the angles. The distortions are then conformal in this model. Also, it is known that the Möbius transformations of
send a generalized circle into a generalized circle; hence, they do not preserve the affine planar lines as the homographies of
do (see
Figure 5). It seems then that this choice of transformations does not adequately describe the viewpoint changes, especially when they produce significant conformal distortions.
This quick overview clearly shows that it is not efficient to use the previous viewpoint and viewpoint change modelings. This is particularly due to the space where we let the group of transformations act, which is the compactification of the Euclidean plane. Moreover, good modeling must take into consideration the representation of the point at infinity. The approach that we are adopting in this work stands out radically compared with the previous ones as we use a vector representation of the space of viewpoints. This is made by means of the powerful tools of the conformal Clifford algebra that we introduce in the next section. Our idea in this paper is to embed the domain of the image into a higher dimensional real space (of dimension 4 instead of 2) and to consider the conformal transformations of this new space in order to represent the viewpoint changes.
3. Clifford Algebra and Conformal Model
In this section, we review some definitions and backgrounds of the conformal Clifford algebra. For further details about geometric algebra, the reader is referred to [
31,
37].
The geometric algebra
associated with a real vector space
E, of finite dimension
n, equipped with a quadratic form
Q, is an associative algebra that contains
as a sub-algebra and
E as a subspace, where we have the following:
for all
v in
E.
The algebra
is the sum of the sub-spaces (of dimension
)
of the
k-vectors. Then for every
a in
, we have the following:
where
is the
k-vector part of
a.
Next, we give some possible operations. If
is a real number and
a and
b belong to
, then we have the following:
Also,
The left and right contractions are two new products denoted by ⌋ and ⌊, respectively, defined by the following:
The Clifford algebra of the space
, which is the space
equipped with the quadratic form of signature
with
, is denoted by
. In particular, the Euclidean space is denoted by
, and its geometric algebra is denoted by
. One can easily check that
is isomorphic to the commutative field of complex numbers and
is isomorphic to the non-commutative field of quaternions. The algebras
and
are both isomorphic to the algebra
of the 2 × 2 real matrices.
The outer product of two vectors
u and
v is the element of
, called bivector, defined by the following:
where
denotes the geometric product of
u and
v and
denotes their dot product. An isotropic blade
A is a blade that squares to zero:
.
A
k-versor is a multi-vector that can be factorized into the geometric product of
k-unit vectors. The set of all versors is a group under the geometric product. A spinor is an even versor, i.e., a versor with an even
k. More precisely, the group
of versors is a double cover of the orthogonal group
. Also, the group
of spinors (products of even numbers of unit vectors) is a double cover of the group
. The identity-connected component of this latter group, denoted by
, is the group of the rotations of
. It is encoded by its double cover i.e., the group
of the spinors (or “rotors”) defined by the following:
More generally, let
be a non-zero and non-isotropic
k-blade (
). We associate with
E a unique sub-space
that is the set of the vectors
v given by the following equation:
This is the sub-space spanned by the vectors
. Let
E be a
k-blade and
u be a vector of
. The projection
of
u on the sub-space associated with
E is given by the following formula:
and the rejection
of
u with respect to this sub-space, also called the projection of
u on the sub-space of
orthogonal to
, is given by the following:
Let the isotropic basis
of
These two vectors are isotropic:
and
for the Minkowski metric. Consider the bivector
encoding the space
.
The space
can be decomposed into the following direct sum:
called a conformal split, where
denotes an orthonormal basis of
. In this case,
is a basis of
. This enables to encode the projection on the Euclidean plane
by
.
Definition 1. The horosphere denoted by , also called the conformal model of associated with the basis , is the intersection set of the null cone, as follows:and the hyperplane with normal and containing , is defined as follows:where · denotes the scalar product of . In other words, the horosphere is the following set of normalized isotropic vectors of :It is given by where is the bijection that sends to the following:This bijection is a global parametrization of the horosphere. Remark 1. Note that the definition of a horosphere of is similar to that of the sphere of canonically embedded in the Minkowski space . They are both defined as the intersection set of the null cone and a hyperplane of . Indeed, is defined by the following:where and is the hyperplane with normal and containing the same vector . Proposition 1. Two distinct horospheres and (for ) do not intersect.
Proof. Let two horospheres, i.e.,
Since
and
are collinear vectors, then the hyperplanes
and
are parallel and not equal (
). Thus,
☐
4. Conformal Image Representation
We start first by describing the perspective distortions, in the real projective geometry, undergone by the object plane due to the variations of the latitude angle of the pinhole camera model. The latitude is defined as the angle between the orthogonal axis to the object plane and the optical axis of the camera (see
Figure 6). For more details, the reader can refer to [
33]).
4.1. The Projective Modeling
The perspective distortions are encoded by the homographies of
, denoted by
for
, given by the following:
where
is the
matrix defined by the following:
Figure 7 shows the perspective distortion with
. If
, the corresponding homography is the identity. In this case, the image is called frontal (the latitude parameter is null). The inverse of the matrix
is the matrix, i.e.,
Since we are working in the homogeneous model, we can disregard the factor
. The obtained matrix is a matrix representing the inverse
of the perspective distortion
. We have the following definition:
Definition 2 (Projective image representation).
An image is given by a mapping, as follows:defined on the domain of where θ is the latitude of the camera, is the angle of rotation of the camera around its optical axis, and is the rotation of the angle . The set
is called the perspective plane. It is strictly included in
and contains the points at infinity of homogeneous coordinates
. To every image,
, defined on
, a frontal image
is associated with the following:
If the rotation parameter
is null, we obtain the following:
This is illustrated in
Figure 8. Based on the previous definition, we introduce the viewpoint changes.
Definition 3. A viewpoint change is given by two perspective plane rotations and , and a homography of whose restriction is as follows:to can be written as follows:where f is a composition of a rotation, a translation, and a dilation of . More precisely, given two images,
and
, of a planar object, the homography,
k, is the mapping between their domains and the similarity,
f, describes the mapping between the domains of their associated frontal images,
and
. This is represented in
Figure 9 by a commutative diagram.
4.2. The Conformal Modeling
We suppose that the rotation parameter of the camera around its optical axis is null. We focus on the main parameter encoding the perspective distortions.
Definition 4. An image is the data of a horosphere , parametrized by (Equation (29)) and encoding the camera latitude angle θ, and a mapping defined on by the following:where is the projective image associated with and its perspective distortion. This enables us to write . By considering the points at infinity of the perspective plane, we can let (see [33]). Definition 5. A conformal viewpoint change is represented by a horosphere change through the mappingthat sendstowhere f is a composition of a rotation, a translation, and a dilation of the plane. In other words, the following identity on holds: Proposition 2. The mapping is the restriction to the horosphere of a linear conformal transformation of . Therefore, is a conformal transformation of the induced metric by the ambient Minkowski space on the horosphere.
Remark 2. Recall that a transformation of class is said to be conformal if and only if for all there exists such that we have the following:, where is the differential function of F. If F is linear, then this definition is equivalent to the following: where . In particular, an orthogonal transformation is conformal. Proof. Let
and
be two mappings that send the following:
to
and
respectively. We have
Since
f is the composition of a rotation, a translation, and a dilation, with a ratio of
, there exists a versor
in
such that
Consequently,
is the restriction to
of a conformal transformation of
. Moreover,
is the restriction to
of the conformal transformation of
that sends a vector:
to the vector, as follows:
In fact, one can easily verify that this last transform is orthogonal since
. Hence,
is the restriction to
of a conformal transformation of
. ☐
4.3. Equivalent Definitions
In this subsection, we simplify the notations by letting while keeping in mind that encodes the latitude angle . We will introduce small modifications to the previous conformal representations in order to model every viewpoint change by a conformal map of the surface of to itself. This enables us, in the following section, to apply a conformal invariance result associated with a sub-manifold of , in order to encode a viewpoint invariance:
Definition 6. An image involves the data of a horosphere and a mapping defined on the disjoint union of horospheres, as follows:such that where is a projective image and its corresponding perspective distortion and if , which is if . Definition 7. A viewpoint change is described by an involution defined by the following:where f is a composition of a rotation, a translation, and a dilation of the Euclidean plane. The involution encodes a homography , as well as its inverse .
Proposition 3. The image corresponding to the viewpoint change of is given by the horosphere and the mapping: Proposition 4. The previous viewpoint change representation is conformal to the canonical metric on the disjoint union of horospheres .
Proof. In fact its restrictions to every horosphere is conformal to the canonical metric on (see Proposition 2) then is conformal on the disjoint union of horospheres. ☐
6. Conclusions
In this paper, we introduced a conformal viewpoint invariant based on powerful representations of the viewpoints and the viewpoint changes in the conformal Clifford algebras. We made use of the conformal model of the Euclidean plane and the spinor representations of the similarities of the real projective plane in order to show that the viewpoint changes can be described by conformal transformations of the disjoint union of horospheres for the ambient Minkowski metric of . In our viewpoint modeling, we give particular interest to the perspective distortions since they are the main viewpoint changes as they produce perspective deformations of the object. These are parameterized by the latitude angle of the camera, which is the angle between the vertical axis and the optical axis of the camera.
In order to achieve the full perspective invariance, we must simulate with enough accuracy the perspective distortion of the image. This corresponds to sampling with high precision the interval , where the latitude angle belongs. Hence, the resulting samples constitute an accurate discrete representation of this interval. This ensures that the actual perspective distortion is well approximated.