Multiview Chirality

# Multiview Chirality

## Abstract.

Given an arrangement of cameras , the chiral domain of is the subset of that lies in front it. It is a generalization of the classical definition of chirality. We give an algebraic description of this set and use it to generalize Hartley’s theory of chiral reconstruction [4] to views and derive a chiral version of Triggs’ Joint Image [13, 12].

## 1. Introduction

In computer vision, chirality refers to the constraint that for a scene point to be visible in a camera, it must lie in front of it [4]. There is now a mature theory of multiview geometry that ignores this constraint [3]; it is not an exaggeration to say that the theory of chiral multiview geometry is still in its infancy and most of the basic questions remain unanswered. We will discuss three:

1. When can a nonchiral reconstruction be made chiral?

2. Given a set of cameras, what is the set of images of world points that lie in front of them?

3. Given image matches, when does there exist a chiral reconstruction corresponding to them?

In his seminal paper, Hartley not only introduced the term chirality, but also gave a complete answer to the first question for two views in the projective case [4]. His results are constructive and efficient, i.e. they require solving up to two linear programs that are linear in the size of the reconstruction. They do not generalize to more than two views. Concurrently, Werner et al. also discovered some of the same results [14].

If we ignore chirality, the answer to question 2 is known as the joint image [13]. Starting with the seminal work of Longuet-Higgins [6], there is now a complete algebraic and set theoretic characterization of the joint image [1, 8, 2, 3, 7, 5, 11]. In the chiral case, Werner et al. provide a number of necessary conditions, but a complete characterization is not available [16, 17].

Hartley raises the third question and answers it for two views in the form of a sign condition on a projective reconstruction. Werner et. al. also consider the third question and answer it for two views in image space, considering both minimal and nonminimal configurations [18, 15]. Nistér & Schafflitzky consider the minimial problem in the Euclidean case [9]. There is no existence theory of chiral reconstruction for views.

Our paper makes three contributions.

1. We introduce the chiral domain of an arrangement of cameras – a multiview generalization of the classical definition of chirality – that covers all of (not just finite points), and give an algebraic description of this set (Section 3).

2. We give a complete answer to question 1 for projective ((Section 4.1) and Eucildean (Section 4.2) reconstructions in an arbitrary number of views. Like Hartley’s solution for two views, our solution is also constructive and efficient. Indeed we recover Hartley’s results as special cases of ours (Section 4.3).

3. We give a complete answer to question 2, i.e. we algebraically describe the Euclidean closure of the chiral joint image for an arbitrary number of views.

Our results are complete in the sense that, except for the assumption of distinct camera centers, we do not make any other genericity assumptions. We will not address question 3 in this paper. We begin by describing the notation used in this paper and some necessary background.

## 2. Background and Notation

The sets of nonnegative integers, nonnegative real numbers, and positive real numbers are , and , respectively. denotes n-dimensional projective space over the reals, which is modulo the equivalence relation where if is a scalar multiple of . If , then we say that and are equal in , or is identified with . We use to denote coordinate wise equality in . The projectivization of a set is the set .

In multiview geometry, we focus on and , where is a compactificaction of with respect to the embedding , . So points whose last coordinate is nonzero are said to be finite, whereas points whose last coordinate is form the hyperplane at infinity. We write the plane at infinity as where we fix the normal .

We denote points in and by allowing the context to decide where lies. Similarly we denote points in and by . The dehomogenization of a finite point is denoted as .

A projective camera is a matrix of rank . The camera A is finite if . The center of the camera is the unique point such that . The camera is finite if and only if its center is finite. All cameras considered in this paper are finite. For consistency, we will choose the representative for the center of camera .

The world , which is to be imaged by , is modeled as the affine patch in with . This allows the identification of a finite point with the world point , and a world point with the finite point . The image of , in the camera is . The rational map , , is defined for all except the center of 1.

The principal plane of a finite camera is the hyperplane where is the third row of , i.e. it is the set of points in that image to infinite points in . Note that the camera center lies on . We regard as an oriented hyperplane in with normal vector , which we call the principal ray of . The factor makes sure that if we pass from to for some nonzero scalar , the normal vector of the principal plane does not change sign.

The depth of a finite point in a finite camera is essentially the projection of along the principal ray, see [3]. Formally, it is defined as

 (1) depth(q;A):=(1|det(G)|∥G3,∙∥)(n⊤Aq)(n⊤∞q).

Notice that is unaffected by scaling of and of .

Let denote an arrangement of cameras. We use the shorthand for the principal ray of camera . Given a pair of cameras with centers and , let denote the image of in . The points are called epipoles. The line through is called the base line of the pair of cameras . All points on the base line (except for the centers themselves) will image in the two cameras at their respective epipoles .

The (polyhedral) cone spanned by a set of vectors is the set of all nonnegative linear combinations of the vectors in , so

 (2) KU:=\textupcone(u1,…,uℓ)={u1x1+⋯+uℓxℓ | x1,…,xℓ≥0}.

The interior of is denoted by . The dual cone to is the set of all vectors that make nonnegative inner product with every vector of , so

 (3) K∗U:={y | ∀ x∈KU:y⊤x≥0}.

The interior of is the set . Checking membership in the dual cone is equivalent to checking the feasibility of the inequality system which in turn amounts to solving a linear program.

## 3. The Chiral Domain of an Arrangement of Cameras

In this section we define chirality for all points in with respect to one or more finite cameras. Our definition encompasses Hartley’s definition of chirality [4], [3, Chapter 21], which is restricted to finite points in ; it is a compactification of this classical definition.

The depth of a finite point in a finite camera defined in Equation 1 is if and only if . This happens if and only if lies on the principal plane . Otherwise, and the sign of is the same as the sign of the product which is either positive or negative. It is then natural to say that a finite point is in front of the camera if , see [4]. Since only the sign of matters, it is usual to refer to this sign as the chirality of in , denoted as , which is either or .

To extend the definition of chirality to all points in , not just finite points, we rely on the natural topology in induced by the quotient map in which if and only if . In this quotient topology, a set is open if and only if its preimage is open in in the Euclidean topology. Thus is a limit point of a sequence if and only if all open sets containing the line contain some line . The closure of a set is the set of all limit points of sequences in .

###### Definition 1 (Chiral Domain of A and Chirality).

Let be an arrangement of finite cameras. Then the chiral domain of , denoted as is the closure of the set

 {q∈P3|q finite, ∀A∈A, depth(q,A)>0}.

Moreover, a point is said to have chirality 1 with respect to , denoted as , if and only if .

###### Remark 1.

Note that is nonempty if and only if it has nonempty interior. Indeed, if it is nonempty, there is a finite point that has positive depth in all cameras in the arrangement. Since the depth depends continuously on the finite point, there is a neighborhood of finite points with positive depth in all cameras. This neighborhood is in the interior of .

###### Theorem 1.

Let be an arrangement of finite cameras. Then if and only if the row space of the matrix with columns intersects the positive orthant .

In particular, for all arrangements of cameras such that and the principal rays are linearly independent, .

###### Proof.

The set if and only if there is a finite point with positive depth in all cameras, or equivalently a such that lies in the positive or negative orthant. Thus if and only if the row space of has an intersection with .

If and the columns of are linearly independent then has row rank equal to , and the rows of span . So the rowspace of intersects .

This result is tight in the sense that there are examples of arrangements of four cameras such that is empty. Furthermore, Theorem 1 provides an efficient method for checking if is nonempty by checking the feasibility of a linear program whose size scales linearly with the number of cameras. We now give an algebraic description of .

###### Theorem 2.

Let be an arrangement of finite cameras and assume that there is a point with positive depth in all cameras. Then the chiral domain of , namely the set of all points with chirality with respect to , is

 (4) P3A={q∈P3 | ∀i,j:(n⊤∞q)(n⊤iq)≥0, (n⊤iq)(n⊤jq)≥0},

where is the principal ray of .

###### Proof.

Let be the set of finite points in that have positive depth in all cameras in . Since is defined by strict linear inequalities, it is the interior of the polyhedral cone defined by the inequalities and . We can define the semialgebraic set by the quadratic inequalities , where ranges over all -element subsets of . The projectivization of is the Euclidean closure of , given that is nonempty.

###### Remark 2.

The equality in (4) is only valid when is nonempty. The right hand side can be nonempty even if is empty. This is because the nonstrict inequalities admit all points that are on the principal planes of some cameras in and have nonnegative depth in the others.

###### Remark 3.

Theorem 2 describes using quadratic inequalities. However, is a polyhedral set in the sense that it is equal to the projectivization of a polyhedral cone .

###### Remark 4.

In fact, is not equivalent to for all , at least when is an infinite point. Indeed, specializing Theorem 2 to one camera, we get , which implies that if for one camera . So an infinite point always has chirality one with respect to a single camera. However, there are arrangements where not every infinite point has chirality with respect to , e.g. two co-incident cameras facing in opposite directions.

## 4. Chiral Reconstructions

A reconstruction of a collection of image correspondences

 P:={(p1k,…,pmk)∈(R2)m,k=1,…,n}

is a set of world points and cameras such that . Reconstructions can be transformed by homographies of to other reconstructions of . Since chirality is not a projective invariant, the new reconstruction may become chiral or lose chirality. This naturally leads to the question: When can a reconstruction of a given collection of image correspondences be turned into a chiral reconstruction of ?. We will treat the projective and Euclidean cases separately.

### 4.1. Projective Reconstructions

A projective reconstruction of is a pair consisting of an arrangement of finite cameras and a set of points such that for some scalars . Recall that for all . Since all of the image correspondences have last coordinate , , and since all cameras are finite, for any . This implies that no point can lie on the principal plane of any camera .

###### Definition 2.

A chiral reconstruction of is a projective reconstruction of such that each is a finite camera and for all .

In the context of two cameras, [4] and [15] call a projective reconstruction of a weak realization, and a chiral reconstruction a strong realization. In fact, while our definition of chiral reconstruction requires finite cameras, by allowing world points to be infinite, we extend the notion of a strong realization.

We first state a lemma (proof in the Appendix) that describes the effect of a homography on a reconstruction. Recall that, for the center of a finite camera , we choose the representative in

 cA=[−G−1t1].
###### Lemma 1.

Let be a finite camera with center . Let with last row and . Then

1. After the homography, the plane at infinity is .

2. The camera is finite if and only if . Its center then is .

3. The principal ray of is .

4. For all , we have .

We now address the question of when a given projective reconstruction of can be transformed to a projectively equivalent reconstruction

that is chiral, by a homography of .

###### Theorem 3.

Given a projective reconstruction of , set . Then there is a with last row such that is a chiral reconstruction if and only if one of the following sets or is nonempty:

 (5) S1 ={h | ∀i,j,k, (h⊤qk)(h⊤ci)σik≥0, (h⊤ci)(h⊤cj)σikσjk>0} (6) S2 ={h | ∀i,j,k, (h⊤qk)(h⊤ci)σik≤0, (h⊤ci)(h⊤cj)σikσjk>0}
###### Proof.

The reconstruction of is chiral if and only if for each , lies in the chiral domain of the camera arrangement . Therefore, from Theorem 2, Lemma 1, and the requirement that cameras in the chiral reconstruction need to be finite, i.e., for all , is chiral if and only if there exist such that for all ,

 (7) (n⊤∞Hqk)(n⊤AiH−1Hqk) =δ(h⊤qk)(h⊤ci)(n⊤iqk)≥0, (8) (n⊤AiH−1Hqk)(n⊤AjH−1Hqk) =(h⊤ci)(h⊤cj)(n⊤iqk)(n⊤jqk)>0.

Recall that we write as shorthand for . Substituting for , we get the two sets and to account for the sign of . This shows that feasibility of (7) and (8) is equivalent to one of or being nonempty. Any tuple can be completed to a where is the last row of and .

We now introduce the notion of a signed reconstruction.

###### Definition 3.

A signed reconstruction of is a projective reconstruction of in which for each camera , there exist constants such that for all . We say that a projective reconstruction can be signed if there exist such that in and is a signed reconstruction.

###### Lemma 2.

Suppose that a projective reconstruction of is projectively equivalent to a chiral reconstruction of . Then for each pair , the product is constant for all , and can be signed.

###### Proof.

Let be a projective reconstruction of . For either or to be nonempty it is necessary that for each pair , the product is constant for every . In this case, we show that can be signed. For each , define if or if . By construction, for all . After this change, we still have is constant for all . Then it follows that for each , is constant for all , and is a signed reconstruction of .

Note that signing a projective reconstruction only amounts to changing the sign of some world points. It does not affect the cameras or chirality of the world points in these cameras. We are now ready to state our main result.

###### Theorem 4.

Given a signed reconstruction of , there exists a chiral reconstruction if and only if

 (9) K∗Q∩(intK∗σC∪intK∗−σC)≠{0}

where , and is the interior of its dual cone, and , and is its dual cone.

###### Proof.

Since is a signed reconstruction, we may substitute the constants for , and rewrite and as

 (10) S1 ={h | ∀i,j,k, (h⊤qk)(h⊤σici)≥0, (h⊤σici)(h⊤σjcj)>0} (11) S2 ={h | ∀i,j,k, (h⊤qk)(h⊤σici)≤0, (h⊤σici)(h⊤σjcj)>0}

The set is the union of the cones and . Similarly, is the union of and . Since if and only if , and if and only if , finding a chiral reconstruction reduces to checking whether intersects one of the cones or .

###### Remark 5.

Checking (9) amounts to checking whether one of two linear programs is feasible. This is the higher dimensional analog of checking the feasibility of Hartley’s chiral inequalities for two views [4]. The cones in Theorem 4 are polyhedral and the number of their generators scales linearly with the size of the scene. As a result, Theorem 4 provides a polynomial time method for constructively checking when a multiview projective reconstruction can be transformed into a chiral reconstruction.

###### Remark 6.

Theorem 4 also implies that if a chiral reconstruction exists, there is one in which all world points are finite and do not lie on principal planes. The argument is as follows: Since the produced in Theorem 4 lies in and , it follows that for all , i.e., transformed world points do not lie on principal planes of transformed cameras. Observe that for all , meaning , which means that for any signed reconstruction , the cone is full dimensional. Consequently, any in or can be perturbed to a vector in and remain in or .

### 4.2. Euclidean Reconstructions

In the previous section, we asked when a projective reconstruction can be transformed to a chiral reconstruction. We now ask the same question for a Euclidean reconstruction of , by which we mean a reconstruction in which each camera has the form where .

Proposition 10 in [4] shows that we can assume by applying an appropriate similarity, without affecting chirality. Under this assumption, the following two theorems (whose proofs appear in the Appendix) answer the above question for and views respectively.

###### Theorem 5.

Let be a signed Euclidean reconstruction of with distinct centers. There exists a chiral Euclidean reconstruction of if and only if or .

###### Theorem 6.

Let be a signed Euclidean reconstruction of with cameras, distinct centers, and . There exists a chiral Euclidean reconstruction of if and only if .

These theorems are specializations of Theorem 4. Their proofs are based on the observation that restricting the cameras to be Euclidean restricts the class of homographies in Theorem 4 to four () and two () discrete choices respectively. The four choices for correspond to the well known twisted pair transformations and the two choices for correspond to reflection.

### 4.3. Connections to Hartley’s Work on Two-View Chirality

Our development of multiview chirality was inspired by the seminal paper of Hartley on chirality for two views [4]. In this section we show that the main theorems on two view chirality from [3] and [4] follows from our present work.

We first show the connection between our work and Hartley’s notion of quasi-affine transformations that appears in both [4] and [3, Chapter 21]. In [3, Definition 21.3], a homography is said to be quasi-affine with respect to a set , with elements having last coordinate , if no point in the convex hull of is sent to infinity by . We observe that this is equivalent to saying that , the last row of , lies in or . To accommodate infinite points, we make a more general definition of a quasi-affine transformation.

###### Definition 4.

A linear map is quasi-affine with respect to if the last row of is in . Further, is strictly quasi-affine with respect to if .

Geometrically, is quasi-affine with respect to if lies in one of the closed halfspaces defined by the hyperplane , which is the plane sent to infinity by the homography . If lies in a open halfspace of (as in Hartley’s setup) then is strictly quasi-affine with respect to .

Recall that in a signed reconstruction we have fixed the sign of the last coordinates of all and of all , and all points in and are considered to be in . We first show that Theorem 4 can be interpreted in terms of quasi-affine transformations.

###### Theorem 7.

Suppose is a signed reconstruction of . Then there exists a chiral reconstruction of if and only if there is a homography that is quasi-affine with respect to and strictly quasi-affine with respect to .

###### Proof.

The intersection is nonempty if and only if is nonempty because the negative of a vector in will lie in . The statement now follows from Theorem 4.

In the rest of this section we focus on two view reconstructions and derive several results from Hartley’s work. Recall that if we have a two view reconstruction of such that and , then and . Recall also that . The products have the same sign for all if and only if have the same sign for all , i.e., is constant for all . We will assume that the centers of and are distinct. In this language, Theorem 17 in [4] (also [15, Theorem 1]) says the following:

###### Theorem 8.

[4, Theorem 17] A projective reconstruction of can be transformed by a homography to a chiral reconstruction if and only if have the same sign for all .

###### Proof.

The only if direction was proved in Lemma 2.

For the “if” direction, suppose have the same sign for all . We first note that is a nonzero element of either or . Indeed, and if or then . Otherwise if , then .

Also, since the centers and are distinct, is not a scalar multiple of , hence is a pointed cone, i.e., does not contain a line. This implies that is full-dimensional and hence has an interior. The same is true for .

Without loss of generality suppose . Since , we have that for all , and so . Let be a neighborhood of contained in . Since is also in , there is some that lies in the . This is in , so by Theorem 4, has a chiral reconstruction.

Since the constructed at the end of the proof of Theorem 8 lies in we can strengthen Theorem 7 in the case of two views as follows.

###### Corollary 1.

Suppose is a signed reconstruction of . Then there exists a chiral reconstruction of if and only if there is a homography that is strictly quasi-affine with respect to both and .

Part (ii) of Theorem 21.7 in [3] says the following. Suppose is a projective reconstruction of for which there is a projectively equivalent chiral reconstruction, and have the same sign for all . Then the homography that yields a chiral reconstruction can be chosen to be strictly quasi-affine with respect to . Indeed, by Theorem 8, since can be transformed to a chiral reconstruction by a homography, we must have that have the same sign for all . If now we also have that have the same sign for all , then it follows that have the same sign for all . Therefore, is already signed. The result now follows from Corollary 1.

Hartley’s work was done with the aim of upgrading a two view projective reconstruction to a metric reconstruction. In follow up work, Nistér addresses this question for multiple views [10]. He does this by transforming the projective reconstruction into one which is quasi-affine with respect to the camera centers. As can be seen from Theorem 7 above, quasi-affineness with respect to the camera centers is a necessary condition for chirality. He does not enforce quasi-affineness with respect to the scene points, because they are often noisy and their chirality may change as part of the metric upgrade. Nistér argues shows that enforcing the quasi-affineness on camera centers makes the iterative algorithm used to perform the subsequent metric upgrade easier and more reliable.

## 5. The Chiral Joint Image

In this final section, we address question 2 from the Introduction, and algebraically describe the set of images of world points that have chirality one with respect to an arrangement of cameras. The algebraic study of this set in the nonchiral case leads to the multiview constraints. Our study will lead to the chiral multiview constraints, the semi-algebraic analog of multiview constraints.

World points are imaged in an arrangement of finite cameras via the rational map2

 (12)

Triggs calls the joint image [12, 13] and Heyden-Åström call it the natural descriptor [5]. The Zariski closure of in by . Trager et al. refer to it as the joint image variety of  [11] and characterize it as follows:

###### Theorem 9 ([11, Proposition 1]).

Given an arrangement of cameras , with distinct camera centers, let , and . Then .

Recall that the epipolar and trifocal constraints cut out the joint image variety [1].

While the Zariski topology is natural for algebraic sets, it is too coarse for semi-algebraic sets. So we will be working with the Euclidean topology 3 on , and write for the Euclidean closure of the joint image. Luckily, the Euclidean and Zariski closure of the joint image are the same. The proof of the following theorem can be found in the Appendix.

###### Theorem 10.

As a result, we only consider closure in the Euclidean topology going forward. Our interest in this section is in the following set.

###### Definition 5 (Chiral Joint Image).

The chiral joint image of a camera arrangement is , the image of the chiral domain of under .

The rest of this section is devoted to the algebraic description of , the Euclidean closure of the chiral joint image. We begin by defining two sets.

###### Definition 6.

Given an arrangement of finite cameras , define

 (13) CA:={p∈(P2)m ∣∣∣[]ldet(Gi)pi3(ai×aj)⊤(bij×aj)≥0,det(Gi)det(Gj)pi3pj3((bij×ai)⊤(bij×aj))≥0},

where , is a direction of the baseline connecting the centers of cameras and , and .

Equation 13 is well-defined on because every inequality defining has even degree in the coordinates on the -factors. In fact, the three inequalities are all biquadratic, i.e. of degree . Moreover, the sign does not depend on the choice of the order of the cameras in the arrangement because this choice is implicit in and explicit in the terms in the inequalities. So a relabeling of the cameras will not change the signs involved.

We will see that is obtained from the description of by eliminating the world points. A first guess might be that the . Indeed this is roughly true, but as we will see in Theorem 11 the precise statement requires a bit more care. The problem comes from the image of points that lie on the baselines connecting pairs of cameras. This leads us to the following definition.

###### Definition 7.

Given a camera arrangement define and .

The set consists of images of points in that lie on the baselines of pairs of cameras in . We cannot hope to determine the chirality of points in (with respect to ) from only their image coordinates. This is because, given a pair of cameras , every point on the baseline gets imaged to the pair of epipoles , but different world points on the baseline can have different chiralities with respect to the camera pair depending on their individual orientations.

Our main result is the following theorem, which relates the intersection to the chiral joint image minus the set .

###### Theorem 11.

Let be an arrangement of finite cameras with distinct centers and assume that the chiral domain is nonempty. Let