Conformal Image Viewpoint Invariant

Abstract

In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space ${\mathbb{R}}^{3,1}$ leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ${\coprod }_{\alpha }{H}_{\alpha }$ of ${\mathbb{R}}^{3,1}$ by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ${\coprod }_{\alpha }{H}_{\alpha }$ for the Minkowski metric of the ambient space.

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 ${\mathbb{P}}_2\mathbb{R}$ 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 ${\mathbb{R}}^4$ 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 ${\mathbb{R}}^{3,1}$, 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 ${\mathbb{P}}_1\mathbb{C}$ 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 $d\cup \left\{w\right\}$ where d is an affine straight line of ${\mathbb{R}}^2$ and w is the point at infinity representing its direction. In this setting, two parallel lines, $d_1\cup \left\{w\right\}$ and $d_2\cup \left\{w\right\}$, in the real projective plane associated with the object are sent to two projective lines $d_1^{\prime }\cup \left\{w_1\right\}$ and $d_2^{\prime }\cup \left\{w_2\right\}$ such that their directions $w_1$ and $w_2$ are not necessarily equal. In this case, the affine lines $d_1^{\prime }$ and $d_2^{\prime }$ are not parallel in the image plane; they have a point of intersection in ${\mathbb{R}}^2$ 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 ${\mathbb{P}}_2\mathbb{R}$ is the quotient space, as follows:

$$({\mathbb{R}}^3\setminus \left\{0\right\})/\mathcal{R}$$

(1)

where the equivalence relation $\mathcal{R}$ is defined by the following:

$$x\mathcal{R}y⟺\exists \lambda \in {\mathbb{R}}^{*},x=\lambda y$$

(2)

We denote by $\pi :({\mathbb{R}}^3\setminus \left\{0\right\})⟶{\mathbb{P}}_2\mathbb{R}$ the canonical surjection and $[X:Y:Z]$ 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 $GL(3,\mathbb{R})$ of the linear mappings of ${\mathbb{R}}^3$ induces a mapping $\tilde{A}$ of the projective plane by the formula: $\tilde{A}\left(\pi \left(x\right)\right)=\pi \left(A\left(x\right)\right)$. Since the mappings A and $\lambda A$ (for $\lambda \in {\mathbb{R}}^{*}$) induce the same mappings of the real projective plane, the group of the homographies of ${\mathbb{P}}_2\mathbb{R}$ (called the projective group) is defined as the quotient $PGL(3,\mathbb{R})$ of the group $GL(3,\mathbb{R})$ by the equivalence relation $\mathcal{R}$.

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 ${\mathbb{P}}_2\mathbb{R}$, 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 ${\mathbb{P}}_2\mathbb{R}$. 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 ${\mathbb{R}}^3$ denoted by $(O,x_1,x_2,x_3)$. To the camera, another space frame is associated and is denoted by $(C,x_1^{\prime },x_2^{\prime },x_3^{\prime })$ such that its origin C is the optical center of the camera. These two frames are related by an isometry of ${\mathbb{R}}^3$, 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 $\left(CM\right)$ (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:

$$I:D\subset {\mathbb{P}}_2\mathbb{R}⟶\mathbb{R}$$

(3)

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 $PGL(3,\mathbb{R})$ of the homographies of ${\mathbb{P}}_2\mathbb{R}$. 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 ${\mathbb{P}}_2\mathbb{R}$ represented by the following matrix:

$$\left(\begin{array}{ccc}2cos\pi /4& -2sin\pi /4& 1\\ 2sin\pi /4& 2cos\pi /4& 1\\ 1/2& 0& 1\end{array}\right).$$

(4)

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 ${\mathbb{R}}^2$ of the action of the group of homographies of the projective plane ${\mathbb{P}}_2\mathbb{R}$. Since the sub-group of homographies that keeps the hyperplane at infinity of ${\mathbb{P}}_2\mathbb{R}$ globally invariant is isomorphic (by restriction) to the affine group $GA\left({\mathbb{R}}^2\right)$ of ${\mathbb{R}}^2$, then a homography of ${\mathbb{P}}_2\mathbb{R}$ preserves the affine plane if and only if it is reduced to an affine transformation of ${\mathbb{R}}^2$.

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 ${\mathbb{R}}^2$ to itself is the restriction to ${\mathbb{R}}^2$ of a homography of ${\mathbb{P}}_2\mathbb{R}$ defined by a $3\times 3$ matrix given by the following:

$$\left(\begin{array}{ccc}a& b& t_1\\ c& d& t_2\\ 0& 0& 1\end{array}\right)$$

(5)

It is actually the composition of a linear transformation of ${\mathbb{R}}^2$ with a translation. In particular, the group $GA\left({\mathbb{R}}^2\right)$ is of dimension 6; however, the group $PGL(3,\mathbb{R})$ is of dimension 8. It is known that every affine transformation of ${\mathbb{R}}^2$ 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 $I_0$ of the object, through the following formula:

$$I\circ A=I_0,$$

(6)

where A is an affine transformation. This is justified by the fact that every homography of ${\mathbb{P}}_2\mathbb{R}$ is locally approximated by an affine transformation of ${\mathbb{R}}^2$. In fact, if k is a homography, we can apply the first-order Taylor–Young formula and we obtain the following:

$$\begin{array}{ccc}\hfill k(x_1,x_2)& =& \underset{A(x_1,x_2)}{\underbrace{k(a,b)+D_{(a,b)}k.(x_1-a,x_2-b)}}+o(\parallel (x_1-a,x_2-b)\parallel )\hfill \\ & =& A(x_1,x_2)+o(\parallel (x_1-a,x_2-b)\parallel ),\hfill \end{array}$$

(7)

where $D_{(a,b)}k$ is the differential function of k at $(a,b)$. The affine transformation A is then a good approximation of k around the point $(a,b)$.

By restricting the viewpoint changes to the affine transformations, by considering the group $GA\left({\mathbb{R}}^2\right)$ instead of the entire projective group $PGL(3,\mathbb{R})$, 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 ${\mathbb{P}}_2\mathbb{R}$ and the use of the group $PGL(3,\mathbb{R})$ for the representation of the viewpoint changes. Here, we consider the following compactification: ${\mathbb{R}}^2\simeq \mathbb{C}\hookrightarrow {\mathbb{P}}_1\mathbb{C}$ and the corresponding group $PGL(2,\mathbb{C})$ (see [26,36]). The complex projective line ${\mathbb{P}}_1\mathbb{C}$ is homeomorphic to the Alexandroff compactification $\widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$ of the plane ${\mathbb{R}}^2\simeq \mathbb{C}$. Recall that this compactification can be described by the projection, P, called stereographic projection (see Figure 4); the image of a point $X\ne N$ of the sphere $S^2$ (N is the north pôle ) is the unique point of intersection x of the line $\left(NX\right)$ with the plane. In order to explicitly calculate $P^{-1}$, we consider a point $x=(x_1,x_2,0)$ and we search for a point on the line $\{\lambda x+(1-\lambda \left)N\right\}$ that belongs to the sphere. We easily obtain the following:

$$P^{-1}:x_1{e}_1+x_2{e}_2⟼{\displaystyle \frac{2}{x_1^2+x_2^2+1}}(x_1{e}_1+x_2{e}_2)+{\displaystyle \frac{x_1^2+x_2^2-1}{x_1^2+x_2^2+1}}{e}_3.$$

(8)

When the magnitude of x tends to infinity, we obtain $P^{-1}(\infty )=N$. Thus, the north pôle encodes the infinity of the plane.

The complex projective line ${\mathbb{P}}_1\mathbb{C}=({\mathbb{C}}^2\setminus \left\{0\right\})/\mathcal{R}$ is a complex manifold of dimension 1 whose charts are given by the following:

$$\begin{array}{c}\hfill {\phi }_X:\{[X:Y]\in {\mathbb{P}}_1\mathbb{C},X\ne 0\}⟶\mathbb{C}\\ \hfill {\phi }_X\left([X:Y]\right)=(Y/X)\\ \hfill {\phi }_Y:\{[X:Y]\in {\mathbb{P}}_1\mathbb{C},Y\ne 0\}⟶\mathbb{C}\\ \hfill {\phi }_Y\left([X:Y]\right)=(X/Y).\end{array}$$

(9)

A homography of $PGL(2,\mathbb{C})$ is represented, up to the multiplication by a non-zero complex number, by a matrix of $GL(2,\mathbb{C})$

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

(10)

or equivalently, by enabling $X/Y=z$ (considering the chart ${\phi }_Y$), via the following mapping:

$$h:z⟼{\displaystyle \frac{az+b}{cz+d}}$$

(11)

where $h(-d/c)=\infty$ and $h(\infty )=a/c$.

In [26], the author described a complex conformal model of the camera that enabled describing the viewpoint changes via the Möbius transformations of ${\mathbb{P}}_1\mathbb{C}$, i.e., the homographies of $PGL(2,\mathbb{C})$. In this model, an image is a mapping, i.e.,

$$\begin{array}{ccc}\hfill I_h:h\left(D\right)\subset \widehat{\mathbb{C}}& ⟶& \mathbb{R}\hfill \\ \hfill h\left(z\right)& ⟼& I_0\left(z\right)\hfill \end{array}$$

(12)

where $I_0:D\subset \mathbb{C}⟶\mathbb{R}$ is a reference image describing the planar object and $h\in PGL(2,\mathbb{C})$.

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 ${\mathbb{P}}_1\mathbb{C}$ send a generalized circle into a generalized circle; hence, they do not preserve the affine planar lines as the homographies of ${\mathbb{P}}_2\mathbb{R}$ 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 ${\mathbb{P}}_2\mathbb{R}$ 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 $Cl(E,Q)$ associated with a real vector space E, of finite dimension n, equipped with a quadratic form Q, is an associative algebra that contains $\mathbb{R}$ as a sub-algebra and E as a subspace, where we have the following:

$$v^2=Q\left(v\right)$$

(13)

for all v in E.

The algebra $Cl(E,Q)$ is the sum of the sub-spaces (of dimension ${\mathrm{C}}_n^k$) $Cl{(E,Q)}^k$ of the k-vectors. Then for every a in $Cl(E,Q)$, we have the following:

$$a=\sum _{k=0}^n{\langle a\rangle }_k$$

(14)

where ${\langle a\rangle }_k$ is the k-vector part of a.

Next, we give some possible operations. If $\lambda$ is a real number and a and b belong to $Cl(E,Q)$, then we have the following:

$$\begin{array}{cc}& \lambda \wedge a=a\wedge \lambda =\lambda a\\ & \lambda ·a=a·\lambda =0\end{array}$$

(15)

Also,

$$\begin{array}{cc}& a·b={\langle ab\rangle }_{|k-l|}\\ & a\wedge b={\langle ab\rangle }_{k+l}\end{array}$$

(16)

The left and right contractions are two new products denoted by ⌋ and ⌊, respectively, defined by the following:

$$\begin{array}{cc}& a\rfloor b={\sum }_{k,l}{\langle {\langle a\rangle }_k{\langle b\rangle }_l\rangle }_{l-k}\\ & a\lfloor b={\sum }_{k,l}{\langle {\langle a\rangle }_k{\langle b\rangle }_l\rangle }_{k-l}\end{array}$$

(17)

The Clifford algebra of the space ${\mathbb{R}}^{p,q}$, which is the space ${\mathbb{R}}^n$ equipped with the quadratic form of signature $(p,q)$ with $p+q=n$, is denoted by ${\mathbb{R}}_{p,q}$. In particular, the Euclidean space is denoted by ${\mathbb{R}}^n$, and its geometric algebra is denoted by ${\mathbb{R}}_n$. One can easily check that ${\mathbb{R}}_{0,1}$ is isomorphic to the commutative field of complex numbers and ${\mathbb{R}}_{0,2}$ is isomorphic to the non-commutative field of quaternions. The algebras ${\mathbb{R}}_{1,1}$ and ${\mathbb{R}}_{2,0}$ are both isomorphic to the algebra $\mathbb{R}\left(2\right)$ of the 2 × 2 real matrices.

The outer product of two vectors u and v is the element of ${\mathbb{R}}_{p,q}$, called bivector, defined by the following:

$$u\wedge v:=uv-u·v$$

(18)

where $uv$ denotes the geometric product of u and v and $u·v$ denotes their dot product. An isotropic blade A is a blade that squares to zero: $A^2=0$.

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 $Pin(p,q)$ of versors is a double cover of the orthogonal group $O(p,q,\mathbb{R})$. Also, the group $Spin(p,q)$ of spinors (products of even numbers of unit vectors) is a double cover of the group $SO(p,q,\mathbb{R})$. The identity-connected component of this latter group, denoted by $S{O}^{+}(p,q,\mathbb{R})$, is the group of the rotations of ${\mathbb{R}}^{p,q}$. It is encoded by its double cover i.e., the group $Spi{n}^{+}(p,q)$ of the spinors (or “rotors”) defined by the following:

$$\begin{array}{ccc}\hfill Spi{n}^{+}(p,q)& =& \{\sigma \in Spin(p,q),\sigma {\sigma }^{†}=1\}\hfill \\ \hfill & =& \{\sigma ={\mathsf{\Pi }}_{i=1}^{2k}{u}_i,|u_i^2|=1\mathrm{and}\sigma {\sigma }^{†}=1\}\hfill \end{array}$$

(19)

More generally, let $E=a_1\wedge \cdots \wedge a_k$ be a non-zero and non-isotropic k-blade ($E^2\ne 0$). We associate with E a unique sub-space $S_E$ that is the set of the vectors v given by the following equation:

$$v\wedge E=0$$

(20)

This is the sub-space spanned by the vectors $a_1\cdots a_k$. Let E be a k-blade and u be a vector of ${\mathbb{R}}^{p,q}$. The projection ${\pi }_E\left(u\right)$ of u on the sub-space associated with E is given by the following formula:

$${\pi }_E\left(u\right)=(u·E)E^{-1}$$

(21)

and the rejection ${\pi }_E^{\perp }\left(u\right)$ of u with respect to this sub-space, also called the projection of u on the sub-space of ${\mathbb{R}}^{p,q}$ orthogonal to $S_E$, is given by the following:

$${\pi }_E^{\perp }\left(u\right)=(u\wedge E)E^{-1}.$$

(22)

Let the isotropic basis $\{e_{\infty ,\alpha },e_{0,\alpha }\}$ of ${\mathbb{R}}^{1,1}$

$$\begin{array}{ccc}\hfill e_{\infty ,\alpha }& =& (\alpha ,\alpha )\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill e_{0,\alpha }& =& {\displaystyle \frac{1}{2}}(-{\displaystyle \frac{1}{\alpha }},{\displaystyle \frac{1}{\alpha }}).\hfill \end{array}$$

(24)

These two vectors are isotropic: $e_{\infty ,\alpha }^2=0$ and $e_{0,\alpha }^2=0$ for the Minkowski metric. Consider the bivector $E=e_{\infty }\wedge e_0$ encoding the space ${\mathbb{R}}^{1,1}$.

The space ${\mathbb{R}}^{3,1}$ can be decomposed into the following direct sum:

$${\mathbb{R}}^{3,1}={\mathbb{R}}^2\oplus {\mathbb{R}}^{1,1}$$

(25)

called a conformal split, where $\{e_1,e_2\}$ denotes an orthonormal basis of ${\mathbb{R}}^2$. In this case, $\{e_1,e_2,e_{\infty ,\alpha },e_{0,\alpha }\}$ is a basis of ${\mathbb{R}}^{3,1}$. This enables to encode the projection on the Euclidean plane ${\mathbb{R}}^2$ by ${\pi }_E^{\perp }$.

Definition 1.

The horosphere denoted by $H_{\alpha }$, also called the conformal model of ${\mathbb{R}}^2$ associated with the basis $\{e_{\infty ,\alpha },e_{0,\alpha }\}$, is the intersection set of the null cone, as follows:

$$N^3=\{X\in {\mathbb{R}}^{3,1};X^2=0\}$$

(26)

and the hyperplane $P^3(e_{\infty ,\alpha },e_{0,\alpha })$ with normal $e_{\infty ,\alpha }$ and containing $e_{0,\alpha }$, is defined as follows:

$$P^3(e_{\infty ,\alpha },e_{0,\alpha })=\{X\in {\mathbb{R}}^{3,1};e_{\infty ,\alpha }·(X-e_{0,\alpha })=0\},$$

(27)

where · denotes the scalar product of ${\mathbb{R}}^{3,1}$. In other words, the horosphere is the following set of normalized isotropic vectors of ${\mathbb{R}}^{3,1}$:

$$H_{\alpha }=\{X\in {\mathbb{R}}^{3,1};X^2=0andX·e_{\infty ,\alpha }=-1\}.$$

(28)

It is given by $H_{\alpha }={\phi }_{\alpha }\left({\mathbb{R}}^2\right)$ where ${\phi }_{\alpha }$ is the bijection ${\phi }_{\alpha }:{\mathbb{R}}^2⟶H_{\alpha }$ that sends $x_1{e}_1+x_2{e}_2$ to the following:

$$X_{\alpha }={\phi }_{\alpha }(x_1{e}_1+x_2{e}_2)=x_1{e}_1+x_2{e}_2+{\displaystyle \frac{1}{2}}({x_1}^2+{x_2}^2)e_{\infty ,\alpha }+e_{0,\alpha }.$$

(29)

This bijection is a global parametrization of the horosphere.

Remark 1.

Note that the definition of a horosphere of ${\mathbb{R}}^{3,1}$ is similar to that of the sphere $S^2$ of ${\mathbb{R}}^3$ canonically embedded in the Minkowski space ${\mathbb{R}}^{3,1}$. They are both defined as the intersection set of the null cone and a hyperplane of ${\mathbb{R}}^{3,1}$. Indeed, $S^2$ is defined by the following:

$$S^2=N^3\bigcap P^3(e_{-},e_{-}),$$

(30)

where $e_{-}=(0,1)\in {\mathbb{R}}^{1,1}$ and $P^3(e_{-},e_{-})$ is the hyperplane with normal $e_{-}$ and containing the same vector $e_{-}$.

Proposition 1.

Two distinct horospheres $H_{\alpha }$ and $H_{\beta }$ (for $\alpha \ne \beta$) do not intersect.

Proof. 

Let two horospheres, i.e.,

$$\begin{array}{ccc}\hfill H_{\alpha }& =& N^3\cap P^3(e_{\infty ,\alpha },e_{0,\alpha })\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill H_{\beta }& =& N^3\cap P^3(e_{\infty ,\beta },e_{0,\beta }).\hfill \end{array}$$

(32)

Since $e_{\infty ,\alpha }=(\alpha ,\alpha )$ and $e_{\infty ,\beta }=(\beta ,\beta )$ are collinear vectors, then the hyperplanes $P^3(e_{\infty ,\alpha },e_{0,\alpha })$ and $P^3(e_{\infty ,\beta },e_{0,\beta })$ are parallel and not equal ($\alpha \ne \beta$). Thus,

$$H_{\alpha }\cap H_{\beta }=\varnothing .$$

(33)

 ☐

The Möbius Transformations

One of the main results of the conformal model is that it enables the linearization of the Möbius transformations of ${\mathbb{R}}^n$. Recall first that the Möbius group of the space ${\mathbb{R}}^n$ is the group of diffeomorphisms generated by the orthogonal transformations, the dilations, the translations, and inversions. Every Möbius transformation g is represented by a Vahlen matrix (see [38]), i.e.,

$$\left[\sigma \right]=\left(\begin{array}{cc}{\sigma }_{11}& {\sigma }_{12}\\ {\sigma }_{21}& {\sigma }_{22}\end{array}\right)$$

(34)

by the formula

$$g\left(x\right)=({\sigma }_{11}x+{\sigma }_{12}){({\sigma }_{21}x+{\sigma }_{22})}^{-1},$$

(35)

for all $x\in {\mathbb{R}}^n$, where ${\sigma }_{11},{\sigma }_{12},{\sigma }_{21}$, and ${\sigma }_{22}$ are versors of the Clifford algebras of ${\mathbb{R}}_{n,0}$ such that ${\sigma }_{11}{\sigma }_{12}^{†}$, ${\sigma }_{12}{\sigma }_{22}^{†}$, ${\sigma }_{21}{\sigma }_{22}^{†}$ and ${\sigma }_{11}{\sigma }_{21}^{†}$ belong to ${\mathbb{R}}^n$ and ${\sigma }_{11}{\sigma }_{22}^{†}-{\sigma }_{12}{\sigma }_{21}^{†}$ is a non-zero real number. We have the following results (see [31]):

Theorem 1.

Let g be a Möbius transformation with the Vahlen matrix $\left[\sigma \right]$. The versor, i.e.,

$$\sigma =e_{\infty }(-e_0{\sigma }_{11}+{\sigma }_{12})-e_0({\sigma }_{21}+e_{\infty }{\sigma }_{22})$$

(36)

satisfies

$$\sigma [x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty }+e_0]{\left({\sigma }^{*}\right)}^{-1}={\mu }_g\left(x\right)[g\left(x\right)+{\displaystyle \frac{1}{2}}g{\left(x\right)}^2{e}_{\infty }+e_0]$$

(37)

where

$${\mu }_g\left(x\right)=({\sigma }_{21}x+{\sigma }_{22}){({\sigma }_{21}^{*}x+{\sigma }_{22}^{*})}^{†}$$

(38)

is the scalar component of the product $-e_{\infty }{\sigma }^{*}x{\sigma }^{-1}$ (see [37] for precise definitions of the main involution ∗ and the reversion † of the Clifford algebra).

Now, we explicitly express the versor representations of some particular planar Möbius transformations that we will use later in our viewpoint modeling:

• Translations: the translation $T_t:x\mapsto x+t$ is encoded by the versor:

  $${\sigma }_t=1-{\displaystyle \frac{1}{2}}t{e}_{\infty }.$$

  (39)
  It is easy to check that Equation (37) is reduced in this case to the following:

  $${\sigma }_t(x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty }+e_0){\sigma }_{-t}=T_t\left(x\right)+{\displaystyle \frac{1}{2}}{\left(T_t\left(x\right)\right)}^2{e}_{\infty }+e_0.$$

  (40)
• Rotations: let $R_{\gamma }$ be the rotation about the origin of angle $\gamma$. Its versor representation is as follows:

  $${\sigma }_{\gamma }=e^{-{\displaystyle \frac{\gamma }{2}}{e}_1\wedge e_2}.$$

  (41)
  By expanding this exponential form and since $e_1\wedge e_2$ squares to $-1$, the versor ${\sigma }_{\gamma }$ can be written as follows:

  $${\sigma }_{\gamma }=cos{\displaystyle \frac{\gamma }{2}}-e_1\wedge e_{2}sin{\displaystyle \frac{\gamma }{2}}.$$

  (42)
  Hence, we easily check through some trigonometric identities the following:

  $${\sigma }_{\gamma }(x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty }+e_0){\sigma }_{-\gamma }=R_{\gamma }x+{\displaystyle \frac{1}{2}}{\left(R_{\gamma }x\right)}^2{e}_{\infty }+e_0.$$

  (43)
• Dilations: let the dilation $x\mapsto e^{-\eta }x$ and let the versor be as follows:

  $${\sigma }_{\eta }=e^{{\displaystyle \frac{\eta }{2}}E},$$

  (44)

  where $E=e_{\infty }\wedge e_0$ as previously denoted. This versor can be written as follows:

  $${\sigma }_{\eta }=cosh{\displaystyle \frac{\eta }{2}}+Esinh{\displaystyle \frac{\eta }{2}}$$

  (45)

  since $E^2=1$. Using some formulas of the hyperbolic functions as well as the following relations:

  $$E{e}_{\infty }=-e_{\infty }E=-e_{\infty }\mathrm{and}E{e}_0=-e_{0}E=e_0,$$

  (46)

  we obtain the following:

  $$\begin{array}{ccc}\hfill {\sigma }_{\eta }{e}_{\infty }{\sigma }_{-\eta }& =& e^{-\eta }{e}_{\infty }\hfill \end{array}$$

  (47)

  $$\begin{array}{ccc}\hfill {\sigma }_{\eta }{e}_0{\sigma }_{-\eta }& =& e^{\eta }{e}_0.\hfill \end{array}$$

  (48)
  Equation (37) is given in this case by the following:

  $${\sigma }_{\eta }(x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty }+e_0){\sigma }_{-\eta }=e^{\eta }(e^{-\eta }x+{\displaystyle \frac{1}{2}}{\left(e^{-\eta }x\right)}^2{e}_{\infty }+e_0).$$

  (49)
  This means that the dilation $D_{\delta }:x\mapsto \delta x$ with $\delta >0$ is represented by the versor as follows:

  $${\sigma }_{\delta }=e^{-{\displaystyle \frac{E}{2}}ln\delta }=cosh\left({\displaystyle \frac{ln\delta }{2}}\right)-Esinh\left({\displaystyle \frac{ln\delta }{2}}\right),$$

  (50)

  such that

  $$\delta {\sigma }_{\delta }(x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty }+e_0){\sigma }_{{\delta }^{-1}}=\delta x+{\displaystyle \frac{1}{2}}{\left(\delta x\right)}^2{e}_{\infty }+e_0.$$

  (51)

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 ${\mathbb{P}}_2\mathbb{R}$, denoted by $h_{\theta }$ for $\theta \in [0,\pi /2]$, given by the following:

$$\begin{array}{ccc}\hfill h_{\theta }:{\mathbb{P}}_2\mathbb{R}& ⟶& {\mathbb{P}}_2\mathbb{R}\hfill \\ \hfill \pi \left(x\right)& ⟼& \pi (M_{\theta }.x),\hfill \end{array}$$

(52)

where $M_{\theta }$ is the $3\times 3$ matrix defined by the following:

$$M_{\theta }=\left(\begin{array}{ccc}cos\theta & 0& 0\\ 0& 1& 0\\ -sin\theta & 0& 1\end{array}\right).$$

(53)

Figure 7 shows the perspective distortion with $\theta =\pi /4$. If $\theta =0$, the corresponding homography is the identity. In this case, the image is called frontal (the latitude parameter is null). The inverse of the matrix $M_{\theta }$ is the matrix, i.e.,

$$M_{\theta }^{-1}={\displaystyle \frac{1}{cos\theta }}\left(\begin{array}{ccc}1& 0& 0\\ 0& cos\theta & 0\\ sin\theta & 0& cos\theta \end{array}\right).$$

(54)

Since we are working in the homogeneous model, we can disregard the factor $1/cos\theta$. The obtained matrix is a matrix representing the inverse $h_{\theta }^{-1}$ of the perspective distortion $h_{\theta }$. We have the following definition:

Definition 2 (Projective image representation).

An image is given by a mapping, as follows:

$$I_{\psi ,\theta }:R_{-\psi }{h}_{\theta }\left({\mathbb{R}}^2\right)⟶\mathbb{R}$$

(55)

defined on the domain $R_{-\psi }{h}_{\theta }\left({\mathbb{R}}^2\right)$ of ${\mathbb{P}}_2\mathbb{R}$ where θ is the latitude of the camera, $\psi \in [0,2\pi ]$ is the angle of rotation of the camera around its optical axis, and $R_{-\psi }$ is the rotation of the angle $(-\psi )$.

The set

$$h_{\theta }\left({\mathbb{R}}^2\right)=\{[x_{1}cos\theta :x_2:-x_{1}sin\theta +1]\in {\mathbb{P}}_2\mathbb{R}\}$$

(56)

is called the perspective plane. It is strictly included in ${\mathbb{P}}_2\mathbb{R}$ and contains the points at infinity of homogeneous coordinates $[cot\theta :x_2:0]$. To every image, $I_{\psi ,\theta }$, defined on $R_{-\psi }{h}_{\theta }\left({\mathbb{R}}^2\right)$, a frontal image $I_0$ is associated with the following:

$$\begin{array}{ccc}\hfill I_0:{\mathbb{R}}^2& ⟶& \mathbb{R}\hfill \\ \hfill x& ⟼& I_{\psi ,\theta }\circ R_{-\psi }{h}_{\theta }\left(x\right).\hfill \end{array}$$

(57)

If the rotation parameter $\psi$ is null, we obtain the following:

$$\begin{array}{c}\hfill I_0=I_{\theta }\circ h_{\theta }.\end{array}$$

(58)

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 $R_{-{\psi }_1}{h}_{{\theta }_1}\left({\mathbb{R}}^2\right)$ and $R_{-{\psi }_2}{h}_{{\theta }_2}\left({\mathbb{R}}^2\right)$, and a homography of ${\mathbb{P}}_2\mathbb{R}$ whose restriction is as follows:

$$k:R_{-{\psi }_1}{h}_{{\theta }_1}\left({\mathbb{R}}^2\right)⟶R_{-{\psi }_2}{h}_{{\theta }_2}\left({\mathbb{R}}^2\right)$$

(59)

to $R_{-{\psi }_1}{h}_{{\theta }_1}\left({\mathbb{R}}^2\right)$ can be written as follows:

$$\begin{array}{ccc}\hfill k& =& R_{-{\psi }_2}{h}_{{\theta }_2}f{\left(R_{-{\psi }_1}{h}_{{\theta }_1}\right)}^{-1}\hfill \end{array}$$

(60)

where f is a composition of a rotation, a translation, and a dilation of ${\mathbb{R}}^2$.

More precisely, given two images, $I_{{\psi }_1,{\theta }_1}$ and $I_{{\psi }_2,{\theta }_2}$, 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, $I_0^1$ and $I_0^2$. This is represented in Figure 9 by a commutative diagram.

4.2. The Conformal Modeling

We suppose that the rotation parameter $\psi$ of the camera around its optical axis is null. We focus on the main parameter $\theta$ encoding the perspective distortions.

Definition 4.

An image is the data of a horosphere $H_{{\alpha }_{\theta }}$, parametrized by ${\phi }_{{\phi }_{{\alpha }_{\theta }}}$ (Equation (29)) and encoding the camera latitude angle θ, and a mapping ${\overline{I}}_{{\alpha }_{\theta }}$ defined on $H_{{\alpha }_{\theta }}$ by the following:

$$\begin{array}{ccc}\hfill {\overline{I}}_{{\alpha }_{\theta }}:H_{{\alpha }_{\theta }}& ⟶& \mathbb{R}.\hfill \\ \hfill X_{{\alpha }_{\theta }}={\phi }_{{\alpha }_{\theta }}\left(x\right)& ⟼& I_{\theta }\circ h_{\theta }\left(x\right),\hfill \end{array}$$

(61)

where $I_{\theta }$ is the projective image associated with ${\overline{I}}_{{\alpha }_{\theta }}$ and $h_{\theta }$ its perspective distortion. This enables us to write ${\overline{I}}_{{\alpha }_{\theta }}\circ {\phi }_{{\alpha }_{\theta }}=I_{\theta }\circ h_{\theta }$. By considering the points at infinity of the perspective plane, we can let ${\alpha }_{\theta }=cot\theta$ (see [33]).

Definition 5.

A conformal viewpoint change is represented by a horosphere change through the mapping

$$\tilde{F}:H_{{\alpha }_{{\theta }_1}}⟶H_{{\alpha }_{{\theta }_2}}$$

(62)

that sends

$$X_{{\alpha }_{{\theta }_1}}=x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty ,{\alpha }_{{\theta }_1}}+e_{0,{\alpha }_{{\theta }_1}}$$

(63)

to

$$\tilde{F}\left(X_{{\alpha }_{{\theta }_1}}\right)=f\left(x\right)+{\displaystyle \frac{1}{2}}{\left[f\left(x\right)\right]}^2{e}_{\infty ,{\alpha }_{{\theta }_2}}+e_{0,{\alpha }_{{\theta }_2}},$$

(64)

where f is a composition of a rotation, a translation, and a dilation of the plane. In other words, the following identity on ${\mathbb{R}}^2$ holds:

$$\tilde{F}\circ {\phi }_{{\alpha }_{{\theta }_1}}={\phi }_{{\alpha }_{{\theta }_2}}\circ f.$$

(65)

Proposition 2.

The mapping $\tilde{F}$ is the restriction to the horosphere $H_{{\alpha }_{{\theta }_1}}$ of a linear conformal transformation of ${\mathbb{R}}^{3,1}$. Therefore, $\tilde{F}$ is a conformal transformation of the induced metric by the ambient Minkowski space ${\mathbb{R}}^{3,1}$ on the horosphere.

Remark 2.

Recall that a transformation $F:{\mathbb{R}}^{3,1}⟶{\mathbb{R}}^{3,1}$ of class $C^1$ is said to be conformal if and only if for all $Z\in {\mathbb{R}}^{3,1}$ there exists $\gamma \left(Z\right)\ne 0$ such that we have the following:

$$\left[dF\right(Z).X]·\left[dF\right(Z).Y]=\gamma \left(Z\right)X·Y,$$

(66)

$\forall X,Y\in {\mathbb{R}}^{3,1}$, where $dF$ is the differential function of F. If F is linear, then this definition is equivalent to the following:

$$(F.X)·(F.Y)=\gamma X·Y,$$

(67)

$\forall X,Y\in {\mathbb{R}}^{3,1},$ where $\gamma \ne 0$. In particular, an orthogonal transformation is conformal.

Proof. 

Let $F_1:H_{{\alpha }_{{\theta }_1}}⟶H_{{\alpha }_{{\theta }_1}}$ and $F_2:H_{{\alpha }_{{\theta }_1}}⟶H_{{\alpha }_{{\theta }_2}}$ be two mappings that send the following:

$$X_1=x_1{e}_1+x_2{e}_2+{\displaystyle \frac{1}{2}}({x_1}^2+{x_2}^2)e_{\infty ,{\alpha }_{{\theta }_1}}+e_{0,{\alpha }_{{\theta }_1}}$$

(68)

to

$$F_1\left(X_1\right)=f(x_1{e}_1+x_2{e}_2)+{\displaystyle \frac{1}{2}}{\left[f(x_1{e}_1+x_2{e}_2)\right]}^2{e}_{\infty ,{\alpha }_{{\theta }_1}}+e_{0,{\alpha }_{{\theta }_1}}$$

(69)

and

$$F_2\left(X_1\right)=x_1{e}_1+x_2{e}_2+{\displaystyle \frac{1}{2}}({x_1}^2+{x_2}^2)e_{\infty ,{\alpha }_{{\theta }_2}}+e_{0,{\alpha }_{{\theta }_2}},$$

(70)

respectively. We have

$$\tilde{F}=F_2\circ F_1.$$

(71)

Since f is the composition of a rotation, a translation, and a dilation, with a ratio of $\rho \ne 0$, there exists a versor $\sigma$ in ${\mathbb{R}}_{3,1}$ such that

$$F_1\left(X_1\right)=\rho \sigma X_1{{\sigma }^{*}}^{-1}\left(\mathrm{Equation}\left(37\right)\right).$$

(72)

Consequently, $F_1$ is the restriction to $H_{{\alpha }_{{\theta }_1}}$ of a conformal transformation of ${\mathbb{R}}^{3,1}$. Moreover, $F_2$ is the restriction to $H_{{\alpha }_{{\theta }_1}}$ of the conformal transformation of ${\mathbb{R}}^{3,1}$ that sends a vector:

$$x_1{e}_1+x_2{e}_2+x_3{e}_{\infty ,{\alpha }_{{\theta }_1}}+x_4{e}_{0,{\alpha }_{{\theta }_1}}$$

(73)

to the vector, as follows:

$$x_1{e}_1+x_2{e}_2+x_3{e}_{\infty ,{\alpha }_{{\theta }_2}}+x_4{e}_{0,{\alpha }_{{\theta }_2}}.$$

(74)

In fact, one can easily verify that this last transform is orthogonal since $e_{\infty ,{\alpha }_{{\theta }_1}}·e_{0,{\alpha }_{{\theta }_1}}=e_{\infty ,{\alpha }_{{\theta }_2}}·e_{{\alpha }_{{\theta }_2}}=-1$. Hence, $\tilde{F}$ is the restriction to $H_{{\alpha }_{{\theta }_1}}$ of a conformal transformation of ${\mathbb{R}}^{3,1}$.  ☐

4.3. Equivalent Definitions

In this subsection, we simplify the notations by letting ${\alpha }_{\theta }=:\alpha$ while keeping in mind that $\alpha =cot\theta$ encodes the latitude angle $\theta$. We will introduce small modifications to the previous conformal representations in order to model every viewpoint change by a conformal map of the surface ${\coprod }_{\alpha }{H}_{\alpha }$ of ${\mathbb{R}}^{3,1}$ to itself. This enables us, in the following section, to apply a conformal invariance result associated with a sub-manifold of ${\mathbb{R}}^{3,1}$, in order to encode a viewpoint invariance:

Definition 6.

An image involves the data of a horosphere $H_{\widehat{\alpha }}$ and a mapping ${\overline{I}}_{\widehat{\alpha }}$ defined on the disjoint union ${\coprod }_{\alpha }{H}_{\alpha }$ of horospheres, as follows:

$$\begin{array}{ccc}\hfill {\overline{I}}_{\widehat{\alpha }}:\coprod _{\alpha }{H}_{\alpha }& ⟶& \mathbb{R},\hfill \end{array}$$

(75)

such that ${\overline{I}}_{\widehat{\alpha }}\circ {\phi }_{\widehat{\alpha }}=I_{\widehat{\theta }}\circ h_{\widehat{\theta }}$ where $I_{\widehat{\theta }}$ is a projective image and $h_{\widehat{\theta }}$ its corresponding perspective distortion and ${\overline{I}}_{\widehat{\alpha }}\left(X_{\alpha }\right)=0$ if $\alpha \ne \widehat{\alpha }$, which is ${\overline{I}}_{\widehat{\alpha }}\left(X_{\alpha }\right)=0$ if $X_{\alpha }\notin H_{\widehat{\alpha }}$.

Definition 7.

A viewpoint change is described by an involution $\tilde{F}$ defined by the following:

$$\begin{array}{ccc}\hfill \tilde{F}:\coprod _{\alpha }{H}_{\alpha }& \to & \coprod _{\alpha }{H}_{\alpha }\hfill \\ \hfill x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty ,{\alpha }_1}+e_{0,{\alpha }_1}& \mapsto & f\left(x\right)+{\displaystyle \frac{1}{2}}{\left[f\left(x\right)\right]}^2{e}_{\infty ,{\alpha }_2}+e_{0,{\alpha }_2}\hfill \\ \hfill x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty ,{\alpha }_2}+e_{0,{\alpha }_2}& \mapsto & f^{-1}\left(x\right)+{\displaystyle \frac{1}{2}}{\left[f^{-1}\left(x\right)\right]}^2{e}_{\infty ,{\alpha }_1}+e_{0,{\alpha }_1}\hfill \\ \hfill x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty ,\alpha }+e_{0,\alpha }& \mapsto & x+{\displaystyle \frac{1}{2}}{x}^2{e}_{\infty ,\alpha }+e_{0,\alpha }for\alpha \ne {\alpha }_1,{\alpha }_2\hfill \end{array}$$

(76)

where f is a composition of a rotation, a translation, and a dilation of the Euclidean plane.

The involution $\tilde{F}$ encodes a homography $k=h_{{\theta }_2}\circ f\circ h_{{\theta }_1}^{-1}$, as well as its inverse $k^{-1}=h_{{\theta }_1}\circ f^{-1}\circ h_{{\theta }_2}^{-1}$.

Proposition 3.

The image corresponding to the viewpoint change $\tilde{F}$ of ${\overline{I}}_{\widehat{\alpha }}$ is given by the horosphere $\tilde{F}\left(H_{\widehat{\alpha }}\right)=H_{\tilde{\alpha }}$ and the mapping:

$$\begin{array}{ccc}\hfill {\overline{I}}_{\tilde{\alpha }}:\coprod _{\alpha }{H}_{\alpha }& ⟶& \mathbb{R}\hfill \\ \hfill X_{\alpha }& ⟼& \left\{\begin{array}{ccc}{\overline{I}}_{\widehat{\alpha }}\circ \tilde{F}\left(X_{\alpha }\right)& if& X_{\alpha }=X_{\tilde{\alpha }}\in H_{\tilde{\alpha }}\\ 0& if& X_{\alpha }\notin H_{\tilde{\alpha }}.\end{array}\right.\hfill \end{array}$$

(77)

Proposition 4.

The previous viewpoint change representation $\tilde{F}$ is conformal to the canonical metric on the disjoint union of horospheres ${\coprod }_{\alpha }{H}_{\alpha }$.

Proof. 

In fact its restrictions ${\tilde{F}}_{|H_{\alpha }}$ to every horosphere $H_{\alpha }$ is conformal to the canonical metric on $H_{\alpha }$ (see Proposition 2) then $\tilde{F}$ is conformal on the disjoint union of horospheres.  ☐

5. Conformal Viewpoint Invariant

This section is devoted to applying the following result of [34] to our conformal representations context.

Proposition 5.

Let $h:(S,g)\to \mathbb{R}$ be a mapping defined on a surface $S\subset {\mathbb{R}}^{3,1}$ equipped with a metric g induced by ${\mathbb{R}}^{3,1}$. Then, the functional, i.e.,

$$J\left(h\right)={\int }_S{\left|{\nabla }^{g}h\left(s\right)\right|}_g^{2}d{\mu }^g$$

(78)

is a conformal invariant. This invariance means that for every conformal transformation

$$\tilde{F}:(S,g)\to (S,g),$$

we have

$$J\left(h\right)=J(h\circ \tilde{F}).$$

(79)

Note that $|{\nabla }^g{h\left(s\right)|}_g^2$ denotes the square of the norm of the tangent vector ${\nabla }^{g}h\left(s\right)$ to S at s for the metric g. That is:

$$|{\nabla }^g{h\left(s\right)|}_g^2=g_s({\nabla }^{g}h\left(s\right),{\nabla }^{g}h\left(s\right)).$$

(80)

Moreover, ${\nabla }^{g}h$ ($d{\mu }^g$ resp.) denotes the gradient of h (the canonical measure resp.) on the manifold $(S,g)$. For details, see Appendix A.

5.1. The Induced Metric G on S

Consider the surface $S={\coprod }_{\alpha }{H}_{\alpha }$ of ${\mathbb{R}}^{3,1}$. It is defined through the local parameterizations ${\phi }_{\alpha }:{\mathbb{R}}^2⟶H_{\alpha }$ for all $\alpha \in \mathbb{R}$. Let $X_{\alpha }\in H_{\alpha }$, then the tangent space to S at $X_{\alpha }$ is $T_{X_{\alpha }}S=T_{X_{\alpha }}{H}_{\alpha }$. The parameterization ${\phi }_{\alpha }$ of $H_{\alpha }$ can be explicitly written as follows:

$${\phi }_{\alpha }(x_1,x_2)=(x_1,x_2,{\displaystyle \frac{\alpha (x_1^2+x_2^2)}{2}}-{\displaystyle \frac{1}{2\alpha }},{\displaystyle \frac{\alpha (x_1^2+x_2^2)}{2}}+{\displaystyle \frac{1}{2\alpha }}).$$

(81)

Then,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial x_1}}& =& (1,0,x_1\alpha ,x_1\alpha )\hfill \end{array}$$

(82)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial x_2}}& =& (0,1,x_2\alpha ,x_2\alpha ).\hfill \end{array}$$

(83)

Thus, the metric at the point $X_{\alpha }$ of $H_{\alpha }$ is represented on the basis $\{{\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial x_1}},{\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial x_2}}\}$ of the tangent space $T_{X_{\alpha }}S$ by the matrix $g\left(X_{\alpha }\right)=\left(g_{ij}\left(X_{\alpha }\right)\right)$, where we have the following:

$$\begin{array}{ccc}\hfill g_{11}\left(X_{\alpha }\right)& =& g_{22}\left(X_{\alpha }\right)={\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial x_1}}·{\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial x_1}}=1\hfill \end{array}$$

(84)

$$\begin{array}{ccc}\hfill g_{12}\left(X_{\alpha }\right)& =& g_{21}\left(X_{\alpha }\right)={\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial x_1}}·{\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial x_2}}=0.\hfill \end{array}$$

(85)

This can be written as follows:

$$g\left(X_{\alpha }\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right).$$

(86)

Hence, S is equipped with the Riemannian metric g induced by the ambient metric of ${\mathbb{R}}^{3,1}$.

5.2. Viewpoint Invariant

Now, let the conformal image defined in Definition 6:

$$\begin{array}{ccc}\hfill {\overline{I}}_{\widehat{\alpha }}:\coprod _{\alpha }{H}_{\alpha }& ⟶& \mathbb{R}.\hfill \end{array}$$

(87)

Then, its image by the functional J of Proposition 5 is

$$\begin{array}{ccc}\hfill J\left({\overline{I}}_{\widehat{\alpha }}\right)& =& {\int }_S{\left|{\nabla }^g{\overline{I}}_{\widehat{\alpha }}\left(X\right)\right|}_g^{2}d{\mu }^g\hfill \\ \hfill & =& {\int }_{H_{\widehat{\alpha }}}{\left|{\nabla }^g{\overline{I}}_{\widehat{\alpha }}\left(X_{\widehat{\alpha }}\right)\right|}_g^{2}d{\mu }^g\hfill \\ \hfill & =& {\int }_{{\mathbb{R}}^2}{|\nabla ({\overline{I}}_{\widehat{\alpha }}\circ {\phi }_{\widehat{\alpha }})\left(x\right)|}^{2}dx\hfill \end{array}$$

(88)

because ${\phi }_{\widehat{\alpha }}$ is a parametrization of $H_{\widehat{\alpha }}$ (see the Appendix A). This enables us to write J as follows:

$$\begin{array}{ccc}\hfill J\left({\overline{I}}_{\widehat{\alpha }}\right)& =& {\int }_{{\mathbb{R}}^2}{|\nabla (I_{\widehat{\theta }}\circ h_{\widehat{\theta }})\left(x\right)|}^{2}dx,\hfill \end{array}$$

(89)

where $I_{\widehat{\theta }}$ is the projective image associated with the conformal image ${\overline{I}}_{\widehat{\alpha }}$ and $h_{\widehat{\theta }}$ is the perspective distortion encoded by ${\phi }_{\widehat{\alpha }}$ (Definition 6). Then, we have the following:

$$\begin{array}{ccc}\hfill J\left({\overline{I}}_{\widehat{\alpha }}\right)& =& {\int }_{{\mathbb{R}}^2}{|\nabla I_{\widehat{\theta }}^0\left(x\right)|}^{2}dx\hfill \end{array}$$

(90)

where $I_{\widehat{\theta }}^0$ is the frontal image associated with $I_{\widehat{\theta }}$ (Equation (58)). Therefore, through proposition 5, we have the following conformal invariance: Let $\tilde{F}:S\to S$ denote a viewpoint change of ${\overline{I}}_{\widehat{\alpha }}$ with $\tilde{F}\left(H_{\widehat{\alpha }}\right)=H_{\tilde{\alpha }}$ (Definition 7). Thus, by Equation (79), we obtain the following:

$${\int }_{H_{\widehat{\alpha }}}|{\nabla }^g{\overline{I}}_{\widehat{\alpha }}\left(X_{\widehat{\alpha }}\right){|}_g^{2}d{\mu }^g={\int }_{H_{\tilde{\alpha }}}{\left|{\nabla }^g({\overline{I}}_{\widehat{\alpha }}\circ \tilde{F})\left(X_{\tilde{\alpha }}\right)\right|}_g^{2}d{\mu }^g.$$

(91)

That is,

$${\int }_{H_{\widehat{\alpha }}}|{\nabla }^g{\overline{I}}_{\widehat{\alpha }}\left(X_{\widehat{\alpha }}\right){|}_g^{2}d{\mu }^g={\int }_{H_{\tilde{\alpha }}}{\left|{\nabla }^g{\overline{I}}_{\tilde{\alpha }}\left(X_{\tilde{\alpha }}\right)\right|}_g^{2}d{\mu }^g.$$

(92)

where ${\overline{I}}_{\tilde{\alpha }}$ is the image corresponding to the viewpoint change $\tilde{F}$ of ${\overline{I}}_{\widehat{\alpha }}$ (Equation (77)). This is equivalent to the following:

$${\int }_{{\mathbb{R}}^2}|\nabla (I_{\widehat{\theta }}\circ h_{\widehat{\theta }})\left(x\right){|}^{2}dx={\int }_{{\mathbb{R}}^2}{|\nabla (I_{\widehat{\theta }}\circ h_{\widehat{\theta }}\circ f^{-1})\left(x\right)|}^{2}dx,$$

(93)

where f is the planar similarity encoding the change between their associated frontal images. This allows us to write the previous invariance as follows:

$${\int }_{{\mathbb{R}}^2}|\nabla (I_{\widehat{\theta }}\circ h_{\widehat{\theta }})\left(x\right){|}^{2}dx={\int }_{{\mathbb{R}}^2}|\nabla (I_{\tilde{\theta }}\circ h_{\tilde{\theta }}\left(x\right){|}^{2}dx.$$

(94)

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 ${\mathbb{R}}^{3,1}$. 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 $[0,{\displaystyle \frac{\pi }{2}}]$, 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.